Users' Mathboxes Mathbox for David A. Wheeler < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aacllem Structured version   Visualization version   GIF version

Theorem aacllem 42318
Description: Lemma for other theorems about 𝔸. (Contributed by Brendan Leahy, 3-Jan-2020.) (Revised by Alexander van der Vekens and David A. Wheeler, 25-Apr-2020.)
Hypotheses
Ref Expression
aacllem.0 (𝜑𝐴 ∈ ℂ)
aacllem.1 (𝜑𝑁 ∈ ℕ0)
aacllem.2 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
aacllem.3 ((𝜑𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
aacllem.4 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐴𝑘) = Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋))
Assertion
Ref Expression
aacllem (𝜑𝐴 ∈ 𝔸)
Distinct variable groups:   𝐴,𝑘,𝑛   𝑘,𝑁,𝑛   𝑘,𝑋   𝜑,𝑘,𝑛
Allowed substitution hints:   𝐶(𝑘,𝑛)   𝑋(𝑛)

Proof of Theorem aacllem
Dummy variables 𝑥 𝑤 𝑦 𝐵 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aacllem.0 . 2 (𝜑𝐴 ∈ ℂ)
2 aacllem.1 . . . . . . 7 (𝜑𝑁 ∈ ℕ0)
32nn0red 11349 . . . . . 6 (𝜑𝑁 ∈ ℝ)
43ltp1d 10951 . . . . 5 (𝜑𝑁 < (𝑁 + 1))
5 peano2nn0 11330 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
62, 5syl 17 . . . . . . 7 (𝜑 → (𝑁 + 1) ∈ ℕ0)
76nn0red 11349 . . . . . 6 (𝜑 → (𝑁 + 1) ∈ ℝ)
83, 7ltnled 10181 . . . . 5 (𝜑 → (𝑁 < (𝑁 + 1) ↔ ¬ (𝑁 + 1) ≤ 𝑁))
94, 8mpbid 222 . . . 4 (𝜑 → ¬ (𝑁 + 1) ≤ 𝑁)
10 aacllem.3 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
11103expa 1264 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
12 eqid 2621 . . . . . . . . . . 11 (𝑛 ∈ (1...𝑁) ↦ 𝐶) = (𝑛 ∈ (1...𝑁) ↦ 𝐶)
1311, 12fmptd 6383 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶):(1...𝑁)⟶ℚ)
14 qex 11797 . . . . . . . . . . 11 ℚ ∈ V
15 ovex 6675 . . . . . . . . . . 11 (1...𝑁) ∈ V
1614, 15elmap 7883 . . . . . . . . . 10 ((𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ (ℚ ↑𝑚 (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ 𝐶):(1...𝑁)⟶ℚ)
1713, 16sylibr 224 . . . . . . . . 9 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ (ℚ ↑𝑚 (1...𝑁)))
18 eqid 2621 . . . . . . . . 9 (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
1917, 18fmptd 6383 . . . . . . . 8 (𝜑 → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑𝑚 (1...𝑁)))
20 eqid 2621 . . . . . . . . . . . 12 (ℂflds ℚ) = (ℂflds ℚ)
2120qdrng 25303 . . . . . . . . . . 11 (ℂflds ℚ) ∈ DivRing
22 drngring 18748 . . . . . . . . . . 11 ((ℂflds ℚ) ∈ DivRing → (ℂflds ℚ) ∈ Ring)
2321, 22ax-mp 5 . . . . . . . . . 10 (ℂflds ℚ) ∈ Ring
24 fzfi 12766 . . . . . . . . . 10 (1...𝑁) ∈ Fin
25 eqid 2621 . . . . . . . . . . 11 ((ℂflds ℚ) freeLMod (1...𝑁)) = ((ℂflds ℚ) freeLMod (1...𝑁))
2625frlmlmod 20087 . . . . . . . . . 10 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod)
2723, 24, 26mp2an 708 . . . . . . . . 9 ((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod
28 fzfi 12766 . . . . . . . . 9 (0...𝑁) ∈ Fin
2920qrngbas 25302 . . . . . . . . . . . 12 ℚ = (Base‘(ℂflds ℚ))
3025, 29frlmfibas 20099 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ DivRing ∧ (1...𝑁) ∈ Fin) → (ℚ ↑𝑚 (1...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (1...𝑁))))
3121, 24, 30mp2an 708 . . . . . . . . . 10 (ℚ ↑𝑚 (1...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (1...𝑁)))
3225frlmsca 20091 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ DivRing ∧ (1...𝑁) ∈ Fin) → (ℂflds ℚ) = (Scalar‘((ℂflds ℚ) freeLMod (1...𝑁))))
3321, 24, 32mp2an 708 . . . . . . . . . 10 (ℂflds ℚ) = (Scalar‘((ℂflds ℚ) freeLMod (1...𝑁)))
34 eqid 2621 . . . . . . . . . 10 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁))) = ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))
3520qrng0 25304 . . . . . . . . . . . 12 0 = (0g‘(ℂflds ℚ))
3625, 35frlm0 20092 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) × {0}) = (0g‘((ℂflds ℚ) freeLMod (1...𝑁))))
3723, 24, 36mp2an 708 . . . . . . . . . 10 ((1...𝑁) × {0}) = (0g‘((ℂflds ℚ) freeLMod (1...𝑁)))
38 eqid 2621 . . . . . . . . . . . 12 ((ℂflds ℚ) freeLMod (0...𝑁)) = ((ℂflds ℚ) freeLMod (0...𝑁))
3938, 29frlmfibas 20099 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ DivRing ∧ (0...𝑁) ∈ Fin) → (ℚ ↑𝑚 (0...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (0...𝑁))))
4021, 28, 39mp2an 708 . . . . . . . . . 10 (ℚ ↑𝑚 (0...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (0...𝑁)))
4131, 33, 34, 37, 35, 40islindf4 20171 . . . . . . . . 9 ((((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (0...𝑁) ∈ Fin ∧ (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑𝑚 (1...𝑁))) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0}))))
4227, 28, 41mp3an12 1413 . . . . . . . 8 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑𝑚 (1...𝑁)) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0}))))
4319, 42syl 17 . . . . . . 7 (𝜑 → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0}))))
44 elmapi 7876 . . . . . . . . 9 (𝑤 ∈ (ℚ ↑𝑚 (0...𝑁)) → 𝑤:(0...𝑁)⟶ℚ)
45 fzfid 12767 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (0...𝑁) ∈ Fin)
46 fvexd 6201 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ V)
4715mptex 6483 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ V
4847a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ V)
49 simpr 477 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤:(0...𝑁)⟶ℚ) → 𝑤:(0...𝑁)⟶ℚ)
5049feqmptd 6247 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → 𝑤 = (𝑘 ∈ (0...𝑁) ↦ (𝑤𝑘)))
51 eqidd 2622 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
5245, 46, 48, 50, 51offval2 6911 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘)( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑛 ∈ (1...𝑁) ↦ 𝐶))))
53 fzfid 12767 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
54 ffvelrn 6355 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℚ)
5554adantll 750 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℚ)
5617adantlr 751 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ (ℚ ↑𝑚 (1...𝑁)))
57 cnfldmul 19746 . . . . . . . . . . . . . . . . . . . . . 22 · = (.r‘ℂfld)
5820, 57ressmulr 16000 . . . . . . . . . . . . . . . . . . . . 21 (ℚ ∈ V → · = (.r‘(ℂflds ℚ)))
5914, 58ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 · = (.r‘(ℂflds ℚ))
6025, 31, 29, 53, 55, 56, 34, 59frlmvscafval 20103 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘)( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (((1...𝑁) × {(𝑤𝑘)}) ∘𝑓 · (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
61 fvexd 6201 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑤𝑘) ∈ V)
6211adantllr 755 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
63 fconstmpt 5161 . . . . . . . . . . . . . . . . . . . . 21 ((1...𝑁) × {(𝑤𝑘)}) = (𝑛 ∈ (1...𝑁) ↦ (𝑤𝑘))
6463a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((1...𝑁) × {(𝑤𝑘)}) = (𝑛 ∈ (1...𝑁) ↦ (𝑤𝑘)))
65 eqidd 2622 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) = (𝑛 ∈ (1...𝑁) ↦ 𝐶))
6653, 61, 62, 64, 65offval2 6911 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (((1...𝑁) × {(𝑤𝑘)}) ∘𝑓 · (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))
6760, 66eqtrd 2655 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘)( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))
6867mpteq2dva 4742 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘)( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑛 ∈ (1...𝑁) ↦ 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))))
6952, 68eqtrd 2655 . . . . . . . . . . . . . . . 16 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))))
7069oveq2d 6663 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))))
71 fzfid 12767 . . . . . . . . . . . . . . . 16 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (1...𝑁) ∈ Fin)
7223a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (ℂflds ℚ) ∈ Ring)
7355adantlr 751 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℚ)
7411an32s 846 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℚ)
7574adantllr 755 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℚ)
76 qmulcl 11803 . . . . . . . . . . . . . . . . . . . 20 (((𝑤𝑘) ∈ ℚ ∧ 𝐶 ∈ ℚ) → ((𝑤𝑘) · 𝐶) ∈ ℚ)
7773, 75, 76syl2anc 693 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · 𝐶) ∈ ℚ)
7877an32s 846 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑤𝑘) · 𝐶) ∈ ℚ)
79 eqid 2621 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))
8078, 79fmptd 6383 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
8114, 15elmap 7883 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) ∈ (ℚ ↑𝑚 (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
8280, 81sylibr 224 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) ∈ (ℚ ↑𝑚 (1...𝑁)))
83 eqid 2621 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))
8415mptex 6483 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) ∈ V
8584a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) ∈ V)
86 snex 4906 . . . . . . . . . . . . . . . . . . 19 {0} ∈ V
8715, 86xpex 6959 . . . . . . . . . . . . . . . . . 18 ((1...𝑁) × {0}) ∈ V
8887a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((1...𝑁) × {0}) ∈ V)
8983, 45, 85, 88fsuppmptdm 8283 . . . . . . . . . . . . . . . 16 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))) finSupp ((1...𝑁) × {0}))
9025, 31, 37, 71, 45, 72, 82, 89frlmgsum 20105 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))) = (𝑛 ∈ (1...𝑁) ↦ ((ℂflds ℚ) Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶)))))
91 cnfldbas 19744 . . . . . . . . . . . . . . . . . 18 ℂ = (Base‘ℂfld)
92 cnfldadd 19745 . . . . . . . . . . . . . . . . . 18 + = (+g‘ℂfld)
93 cnfldex 19743 . . . . . . . . . . . . . . . . . . 19 fld ∈ V
9493a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ℂfld ∈ V)
95 fzfid 12767 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (0...𝑁) ∈ Fin)
96 qsscn 11796 . . . . . . . . . . . . . . . . . . 19 ℚ ⊆ ℂ
9796a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ℚ ⊆ ℂ)
98 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))
9977, 98fmptd 6383 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶)):(0...𝑁)⟶ℚ)
100 0z 11385 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℤ
101 zq 11791 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ ℤ → 0 ∈ ℚ)
102100, 101ax-mp 5 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℚ
103102a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 0 ∈ ℚ)
104 addid2 10216 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥)
105 addid1 10213 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℂ → (𝑥 + 0) = 𝑥)
106104, 105jca 554 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℂ → ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
107106adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑥 ∈ ℂ) → ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
10891, 92, 20, 94, 95, 97, 99, 103, 107gsumress 17270 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (ℂfld Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = ((ℂflds ℚ) Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))))
109 simplr 792 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 𝑤:(0...𝑁)⟶ℚ)
110 qcn 11799 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤𝑘) ∈ ℚ → (𝑤𝑘) ∈ ℂ)
11154, 110syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℂ)
112109, 111sylan 488 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℂ)
113 qcn 11799 . . . . . . . . . . . . . . . . . . . . . 22 (𝐶 ∈ ℚ → 𝐶 ∈ ℂ)
11411, 113syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℂ)
115114an32s 846 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℂ)
116115adantllr 755 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℂ)
117112, 116mulcld 10057 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · 𝐶) ∈ ℂ)
11895, 117gsumfsum 19807 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (ℂfld Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
119108, 118eqtr3d 2657 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ((ℂflds ℚ) Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
120119mpteq2dva 4742 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ ((ℂflds ℚ) Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶)))) = (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)))
12170, 90, 1203eqtrd 2659 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)))
122 qaddcl 11801 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ) → (𝑥 + 𝑦) ∈ ℚ)
123122adantl 482 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 + 𝑦) ∈ ℚ)
12497, 123, 95, 77, 103fsumcllem 14457 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) ∈ ℚ)
125 eqid 2621 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
126124, 125fmptd 6383 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
12714, 15elmap 7883 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)) ∈ (ℚ ↑𝑚 (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
128126, 127sylibr 224 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)) ∈ (ℚ ↑𝑚 (1...𝑁)))
129121, 128eqeltrd 2700 . . . . . . . . . . . . 13 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) ∈ (ℚ ↑𝑚 (1...𝑁)))
130 elmapi 7876 . . . . . . . . . . . . 13 ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) ∈ (ℚ ↑𝑚 (1...𝑁)) → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))):(1...𝑁)⟶ℚ)
131 ffn 6043 . . . . . . . . . . . . 13 ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))):(1...𝑁)⟶ℚ → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) Fn (1...𝑁))
132129, 130, 1313syl 18 . . . . . . . . . . . 12 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) Fn (1...𝑁))
133 c0ex 10031 . . . . . . . . . . . . 13 0 ∈ V
134 fnconstg 6091 . . . . . . . . . . . . 13 (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
135133, 134ax-mp 5 . . . . . . . . . . . 12 ((1...𝑁) × {0}) Fn (1...𝑁)
136 nfcv 2763 . . . . . . . . . . . . . 14 𝑛((ℂflds ℚ) freeLMod (1...𝑁))
137 nfcv 2763 . . . . . . . . . . . . . 14 𝑛 Σg
138 nfcv 2763 . . . . . . . . . . . . . . 15 𝑛𝑤
139 nfcv 2763 . . . . . . . . . . . . . . 15 𝑛𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))
140 nfcv 2763 . . . . . . . . . . . . . . . 16 𝑛(0...𝑁)
141 nfmpt1 4745 . . . . . . . . . . . . . . . 16 𝑛(𝑛 ∈ (1...𝑁) ↦ 𝐶)
142140, 141nfmpt 4744 . . . . . . . . . . . . . . 15 𝑛(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
143138, 139, 142nfov 6673 . . . . . . . . . . . . . 14 𝑛(𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
144136, 137, 143nfov 6673 . . . . . . . . . . . . 13 𝑛(((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))
145 nfcv 2763 . . . . . . . . . . . . 13 𝑛((1...𝑁) × {0})
146144, 145eqfnfv2f 6313 . . . . . . . . . . . 12 (((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) Fn (1...𝑁) ∧ ((1...𝑁) × {0}) Fn (1...𝑁)) → ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) ↔ ∀𝑛 ∈ (1...𝑁)((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = (((1...𝑁) × {0})‘𝑛)))
147132, 135, 146sylancl 694 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) ↔ ∀𝑛 ∈ (1...𝑁)((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = (((1...𝑁) × {0})‘𝑛)))
148121fveq1d 6191 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))‘𝑛))
149 sumex 14412 . . . . . . . . . . . . . . 15 Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) ∈ V
150125fvmpt2 6289 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...𝑁) ∧ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) ∈ V) → ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))‘𝑛) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
151149, 150mpan2 707 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))‘𝑛) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
152148, 151sylan9eq 2675 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
153133fvconst2 6466 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
154153adantl 482 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
155152, 154eqeq12d 2636 . . . . . . . . . . . 12 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = (((1...𝑁) × {0})‘𝑛) ↔ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
156155ralbidva 2984 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (∀𝑛 ∈ (1...𝑁)((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))‘𝑛) = (((1...𝑁) × {0})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
157147, 156bitrd 268 . . . . . . . . . 10 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) ↔ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
158157imbi1d 331 . . . . . . . . 9 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0})) ↔ (∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0}))))
15944, 158sylan2 491 . . . . . . . 8 ((𝜑𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))) → (((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0})) ↔ (∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0}))))
160159ralbidva 2984 . . . . . . 7 (𝜑 → (∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))((((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))) = ((1...𝑁) × {0}) → 𝑤 = ((0...𝑁) × {0})) ↔ ∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))(∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0}))))
16143, 160bitrd 268 . . . . . 6 (𝜑 → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))(∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0}))))
162 drngnzr 19256 . . . . . . . . 9 ((ℂflds ℚ) ∈ DivRing → (ℂflds ℚ) ∈ NzRing)
16321, 162ax-mp 5 . . . . . . . 8 (ℂflds ℚ) ∈ NzRing
16433islindf3 20159 . . . . . . . 8 ((((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂflds ℚ) ∈ NzRing) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))))))
16527, 163, 164mp2an 708 . . . . . . 7 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))))
16647, 18dmmpti 6021 . . . . . . . . 9 dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (0...𝑁)
167 f1eq2 6095 . . . . . . . . 9 (dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (0...𝑁) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))–1-1→V ↔ (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V))
168166, 167ax-mp 5 . . . . . . . 8 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))–1-1→V ↔ (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V)
169168anbi1i 731 . . . . . . 7 (((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) ↔ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))))
170165, 169bitri 264 . . . . . 6 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) LIndF ((ℂflds ℚ) freeLMod (1...𝑁)) ↔ ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))))
171 con34b 306 . . . . . . . . 9 ((∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0})) ↔ (¬ 𝑤 = ((0...𝑁) × {0}) → ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
172 df-nel 2897 . . . . . . . . . . 11 (𝑤 ∉ {((0...𝑁) × {0})} ↔ ¬ 𝑤 ∈ {((0...𝑁) × {0})})
173 velsn 4191 . . . . . . . . . . 11 (𝑤 ∈ {((0...𝑁) × {0})} ↔ 𝑤 = ((0...𝑁) × {0}))
174172, 173xchbinx 324 . . . . . . . . . 10 (𝑤 ∉ {((0...𝑁) × {0})} ↔ ¬ 𝑤 = ((0...𝑁) × {0}))
175174imbi1i 339 . . . . . . . . 9 ((𝑤 ∉ {((0...𝑁) × {0})} → ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0) ↔ (¬ 𝑤 = ((0...𝑁) × {0}) → ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
176171, 175bitr4i 267 . . . . . . . 8 ((∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0})) ↔ (𝑤 ∉ {((0...𝑁) × {0})} → ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
177176ralbii 2979 . . . . . . 7 (∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))(∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0})) ↔ ∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))(𝑤 ∉ {((0...𝑁) × {0})} → ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
178 raldifb 3748 . . . . . . 7 (∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))(𝑤 ∉ {((0...𝑁) × {0})} → ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0) ↔ ∀𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
179 ralnex 2991 . . . . . . 7 (∀𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) ¬ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 ↔ ¬ ∃𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
180177, 178, 1793bitri 286 . . . . . 6 (∀𝑤 ∈ (ℚ ↑𝑚 (0...𝑁))(∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → 𝑤 = ((0...𝑁) × {0})) ↔ ¬ ∃𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
181161, 170, 1803bitr3g 302 . . . . 5 (𝜑 → (((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) ↔ ¬ ∃𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
182 eqid 2621 . . . . . . . . . . . . 13 (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))
18331, 182lssmre 18960 . . . . . . . . . . . 12 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod → (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) ∈ (Moore‘(ℚ ↑𝑚 (1...𝑁))))
18427, 183ax-mp 5 . . . . . . . . . . 11 (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) ∈ (Moore‘(ℚ ↑𝑚 (1...𝑁)))
185184a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) ∈ (Moore‘(ℚ ↑𝑚 (1...𝑁))))
186 eqid 2621 . . . . . . . . . . . 12 (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))
187 eqid 2621 . . . . . . . . . . . 12 (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) = (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))
188182, 186, 187mrclsp 18983 . . . . . . . . . . 11 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod → (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁))) = (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))))
18927, 188ax-mp 5 . . . . . . . . . 10 (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁))) = (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))
190 eqid 2621 . . . . . . . . . 10 (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) = (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))
19133islvec 19098 . . . . . . . . . . . . 13 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LVec ↔ (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂflds ℚ) ∈ DivRing))
19227, 21, 191mpbir2an 955 . . . . . . . . . . . 12 ((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LVec
193182, 189, 31lssacsex 19138 . . . . . . . . . . . . 13 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LVec → ((LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) ∈ (ACS‘(ℚ ↑𝑚 (1...𝑁))) ∧ ∀𝑧 ∈ 𝒫 (ℚ ↑𝑚 (1...𝑁))∀𝑥 ∈ (ℚ ↑𝑚 (1...𝑁))∀𝑦 ∈ (((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑥})) ∖ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘𝑧))𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑦}))))
194193simprd 479 . . . . . . . . . . . 12 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LVec → ∀𝑧 ∈ 𝒫 (ℚ ↑𝑚 (1...𝑁))∀𝑥 ∈ (ℚ ↑𝑚 (1...𝑁))∀𝑦 ∈ (((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑥})) ∖ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘𝑧))𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑦})))
195192, 194ax-mp 5 . . . . . . . . . . 11 𝑧 ∈ 𝒫 (ℚ ↑𝑚 (1...𝑁))∀𝑥 ∈ (ℚ ↑𝑚 (1...𝑁))∀𝑦 ∈ (((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑥})) ∖ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘𝑧))𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑦}))
196195a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → ∀𝑧 ∈ 𝒫 (ℚ ↑𝑚 (1...𝑁))∀𝑥 ∈ (ℚ ↑𝑚 (1...𝑁))∀𝑦 ∈ (((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑥})) ∖ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘𝑧))𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(𝑧 ∪ {𝑦})))
197 frn 6051 . . . . . . . . . . . . 13 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑𝑚 (1...𝑁)) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ (ℚ ↑𝑚 (1...𝑁)))
19819, 197syl 17 . . . . . . . . . . . 12 (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ (ℚ ↑𝑚 (1...𝑁)))
199 dif0 3948 . . . . . . . . . . . 12 ((ℚ ↑𝑚 (1...𝑁)) ∖ ∅) = (ℚ ↑𝑚 (1...𝑁))
200198, 199syl6sseqr 3650 . . . . . . . . . . 11 (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((ℚ ↑𝑚 (1...𝑁)) ∖ ∅))
201200adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((ℚ ↑𝑚 (1...𝑁)) ∖ ∅))
202 eqid 2621 . . . . . . . . . . . . . . 15 ((ℂflds ℚ) unitVec (1...𝑁)) = ((ℂflds ℚ) unitVec (1...𝑁))
203202, 25, 31uvcff 20124 . . . . . . . . . . . . . 14 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((ℂflds ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑𝑚 (1...𝑁)))
20423, 24, 203mp2an 708 . . . . . . . . . . . . 13 ((ℂflds ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑𝑚 (1...𝑁))
205 frn 6051 . . . . . . . . . . . . 13 (((ℂflds ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑𝑚 (1...𝑁)) → ran ((ℂflds ℚ) unitVec (1...𝑁)) ⊆ (ℚ ↑𝑚 (1...𝑁)))
206204, 205ax-mp 5 . . . . . . . . . . . 12 ran ((ℂflds ℚ) unitVec (1...𝑁)) ⊆ (ℚ ↑𝑚 (1...𝑁))
207206, 199sseqtr4i 3636 . . . . . . . . . . 11 ran ((ℂflds ℚ) unitVec (1...𝑁)) ⊆ ((ℚ ↑𝑚 (1...𝑁)) ∖ ∅)
208207a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → ran ((ℂflds ℚ) unitVec (1...𝑁)) ⊆ ((ℚ ↑𝑚 (1...𝑁)) ∖ ∅))
209 un0 3965 . . . . . . . . . . . . . 14 (ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅) = ran ((ℂflds ℚ) unitVec (1...𝑁))
210209fveq2i 6192 . . . . . . . . . . . . 13 ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅)) = ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘ran ((ℂflds ℚ) unitVec (1...𝑁)))
211 eqid 2621 . . . . . . . . . . . . . . . 16 (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁)))
21225, 202, 211frlmlbs 20130 . . . . . . . . . . . . . . 15 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁))))
21323, 24, 212mp2an 708 . . . . . . . . . . . . . 14 ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁)))
21431, 211, 186lbssp 19073 . . . . . . . . . . . . . 14 (ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁))) → ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘ran ((ℂflds ℚ) unitVec (1...𝑁))) = (ℚ ↑𝑚 (1...𝑁)))
215213, 214ax-mp 5 . . . . . . . . . . . . 13 ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘ran ((ℂflds ℚ) unitVec (1...𝑁))) = (ℚ ↑𝑚 (1...𝑁))
216210, 215eqtri 2643 . . . . . . . . . . . 12 ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅)) = (ℚ ↑𝑚 (1...𝑁))
217198, 216syl6sseqr 3650 . . . . . . . . . . 11 (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅)))
218217adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅)))
219 un0 3965 . . . . . . . . . . 11 (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∪ ∅) = ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
22027, 163pm3.2i 471 . . . . . . . . . . . . . 14 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂflds ℚ) ∈ NzRing)
221186, 33lindsind2 20152 . . . . . . . . . . . . . 14 (((((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂflds ℚ) ∈ NzRing) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))) ∧ 𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))) → ¬ 𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥})))
222220, 221mp3an1 1410 . . . . . . . . . . . . 13 ((ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))) ∧ 𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))) → ¬ 𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥})))
223222ralrimiva 2965 . . . . . . . . . . . 12 (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))) → ∀𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ¬ 𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥})))
224189, 190ismri2 16286 . . . . . . . . . . . . . 14 (((LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) ∈ (Moore‘(ℚ ↑𝑚 (1...𝑁))) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ (ℚ ↑𝑚 (1...𝑁))) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) ↔ ∀𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ¬ 𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥}))))
225184, 198, 224sylancr 695 . . . . . . . . . . . . 13 (𝜑 → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) ↔ ∀𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ¬ 𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥}))))
226225biimpar 502 . . . . . . . . . . . 12 ((𝜑 ∧ ∀𝑥 ∈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ¬ 𝑥 ∈ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∖ {𝑥}))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))))
227223, 226sylan2 491 . . . . . . . . . . 11 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))))
228219, 227syl5eqel 2704 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))))
229 mptfi 8262 . . . . . . . . . . . . 13 ((0...𝑁) ∈ Fin → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin)
230 rnfi 8246 . . . . . . . . . . . . 13 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin)
23128, 229, 230mp2b 10 . . . . . . . . . . . 12 ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin
232231orci 405 . . . . . . . . . . 11 (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin ∨ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ Fin)
233232a1i 11 . . . . . . . . . 10 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin ∨ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ Fin))
234185, 189, 190, 196, 201, 208, 218, 228, 233mreexexd 16302 . . . . . . . . 9 ((𝜑 ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → ∃𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁))(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 ∧ (𝑣 ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))))
235234ex 450 . . . . . . . 8 (𝜑 → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))) → ∃𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁))(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 ∧ (𝑣 ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))))))
236 ovex 6675 . . . . . . . . . . . . 13 ((ℂflds ℚ) unitVec (1...𝑁)) ∈ V
237236rnex 7097 . . . . . . . . . . . 12 ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ V
238 elpwi 4166 . . . . . . . . . . . 12 (𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁)) → 𝑣 ⊆ ran ((ℂflds ℚ) unitVec (1...𝑁)))
239 ssdomg 7998 . . . . . . . . . . . 12 (ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ V → (𝑣 ⊆ ran ((ℂflds ℚ) unitVec (1...𝑁)) → 𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁))))
240237, 238, 239mpsyl 68 . . . . . . . . . . 11 (𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁)) → 𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)))
241 endomtr 8011 . . . . . . . . . . . . . 14 ((ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁))) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)))
242241ancoms 469 . . . . . . . . . . . . 13 ((𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣) → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)))
243 f1f1orn 6146 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1-onto→ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
244 ovex 6675 . . . . . . . . . . . . . . . . 17 (0...𝑁) ∈ V
245244f1oen 7973 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1-onto→ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) → (0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
246243, 245syl 17 . . . . . . . . . . . . . . 15 ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
247 endomtr 8011 . . . . . . . . . . . . . . . . 17 (((0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁))) → (0...𝑁) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)))
248202uvcendim 20180 . . . . . . . . . . . . . . . . . . . 20 (((ℂflds ℚ) ∈ NzRing ∧ (1...𝑁) ∈ Fin) → (1...𝑁) ≈ ran ((ℂflds ℚ) unitVec (1...𝑁)))
249163, 24, 248mp2an 708 . . . . . . . . . . . . . . . . . . 19 (1...𝑁) ≈ ran ((ℂflds ℚ) unitVec (1...𝑁))
250249ensymi 8003 . . . . . . . . . . . . . . . . . 18 ran ((ℂflds ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)
251 domentr 8012 . . . . . . . . . . . . . . . . . . 19 (((0...𝑁) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∧ ran ((ℂflds ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)) → (0...𝑁) ≼ (1...𝑁))
252 hashdom 13163 . . . . . . . . . . . . . . . . . . . . 21 (((0...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((#‘(0...𝑁)) ≤ (#‘(1...𝑁)) ↔ (0...𝑁) ≼ (1...𝑁)))
25328, 24, 252mp2an 708 . . . . . . . . . . . . . . . . . . . 20 ((#‘(0...𝑁)) ≤ (#‘(1...𝑁)) ↔ (0...𝑁) ≼ (1...𝑁))
254 hashfz0 13214 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (#‘(0...𝑁)) = (𝑁 + 1))
2552, 254syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (#‘(0...𝑁)) = (𝑁 + 1))
256 hashfz1 13129 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (#‘(1...𝑁)) = 𝑁)
2572, 256syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (#‘(1...𝑁)) = 𝑁)
258255, 257breq12d 4664 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((#‘(0...𝑁)) ≤ (#‘(1...𝑁)) ↔ (𝑁 + 1) ≤ 𝑁))
259253, 258syl5bbr 274 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((0...𝑁) ≼ (1...𝑁) ↔ (𝑁 + 1) ≤ 𝑁))
260251, 259syl5ib 234 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((0...𝑁) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∧ ran ((ℂflds ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)) → (𝑁 + 1) ≤ 𝑁))
261250, 260mpan2i 713 . . . . . . . . . . . . . . . . 17 (𝜑 → ((0...𝑁) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) → (𝑁 + 1) ≤ 𝑁))
262247, 261syl5 34 . . . . . . . . . . . . . . . 16 (𝜑 → (((0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁))) → (𝑁 + 1) ≤ 𝑁))
263262expd 452 . . . . . . . . . . . . . . 15 (𝜑 → ((0...𝑁) ≈ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) → (𝑁 + 1) ≤ 𝑁)))
264246, 263syl5 34 . . . . . . . . . . . . . 14 (𝜑 → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) → (𝑁 + 1) ≤ 𝑁)))
265264com23 86 . . . . . . . . . . . . 13 (𝜑 → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
266242, 265syl5 34 . . . . . . . . . . . 12 (𝜑 → ((𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
267266expdimp 453 . . . . . . . . . . 11 ((𝜑𝑣 ≼ ran ((ℂflds ℚ) unitVec (1...𝑁))) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
268240, 267sylan2 491 . . . . . . . . . 10 ((𝜑𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁))) → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
269268adantrd 484 . . . . . . . . 9 ((𝜑𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁))) → ((ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 ∧ (𝑣 ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
270269rexlimdva 3029 . . . . . . . 8 (𝜑 → (∃𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁))(ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ≈ 𝑣 ∧ (𝑣 ∪ ∅) ∈ (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
271235, 270syld 47 . . . . . . 7 (𝜑 → (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))) → ((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V → (𝑁 + 1) ≤ 𝑁)))
272271impd 447 . . . . . 6 (𝜑 → ((ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁))) ∧ (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V) → (𝑁 + 1) ≤ 𝑁))
273272ancomsd 470 . . . . 5 (𝜑 → (((𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)–1-1→V ∧ ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ (LIndS‘((ℂflds ℚ) freeLMod (1...𝑁)))) → (𝑁 + 1) ≤ 𝑁))
274181, 273sylbird 250 . . . 4 (𝜑 → (¬ ∃𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 → (𝑁 + 1) ≤ 𝑁))
2759, 274mt3d 140 . . 3 (𝜑 → ∃𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
276 eldifsn 4315 . . . . 5 (𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) ↔ (𝑤 ∈ (ℚ ↑𝑚 (0...𝑁)) ∧ 𝑤 ≠ ((0...𝑁) × {0})))
27744anim1i 592 . . . . 5 ((𝑤 ∈ (ℚ ↑𝑚 (0...𝑁)) ∧ 𝑤 ≠ ((0...𝑁) × {0})) → (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})))
278276, 277sylbi 207 . . . 4 (𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) → (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})))
27996a1i 11 . . . . . . . . 9 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ℚ ⊆ ℂ)
2802adantr 481 . . . . . . . . 9 ((𝜑𝑤:(0...𝑁)⟶ℚ) → 𝑁 ∈ ℕ0)
281279, 280, 55elplyd 23952 . . . . . . . 8 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ (Poly‘ℚ))
282281adantrr 753 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ (Poly‘ℚ))
283 uzdisj 12409 . . . . . . . . . . . . . . . . . 18 ((0...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ∅
2842nn0cnd 11350 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ ℂ)
285 pncan1 10451 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁)
286284, 285syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
287286oveq2d 6663 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
288287ineq1d 3811 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((0...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))
289283, 288syl5eqr 2669 . . . . . . . . . . . . . . . . 17 (𝜑 → ∅ = ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))
290289eqcomd 2627 . . . . . . . . . . . . . . . 16 (𝜑 → ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅)
291133fconst 6089 . . . . . . . . . . . . . . . . . 18 ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶{0}
292 snssi 4337 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ ℚ → {0} ⊆ ℚ)
293100, 101, 292mp2b 10 . . . . . . . . . . . . . . . . . . 19 {0} ⊆ ℚ
294293, 96sstri 3610 . . . . . . . . . . . . . . . . . 18 {0} ⊆ ℂ
295 fss 6054 . . . . . . . . . . . . . . . . . 18 ((((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶{0} ∧ {0} ⊆ ℂ) → ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℂ)
296291, 294, 295mp2an 708 . . . . . . . . . . . . . . . . 17 ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℂ
297 fun 6064 . . . . . . . . . . . . . . . . 17 (((𝑤:(0...𝑁)⟶ℚ ∧ ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℂ) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
298296, 297mpanl2 717 . . . . . . . . . . . . . . . 16 ((𝑤:(0...𝑁)⟶ℚ ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
299290, 298sylan2 491 . . . . . . . . . . . . . . 15 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝜑) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
300299ancoms 469 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
301 nn0uz 11719 . . . . . . . . . . . . . . . . . 18 0 = (ℤ‘0)
3026, 301syl6eleq 2710 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 + 1) ∈ (ℤ‘0))
303 uzsplit 12408 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) ∈ (ℤ‘0) → (ℤ‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
304302, 303syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℤ‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
305301, 304syl5eq 2667 . . . . . . . . . . . . . . . . 17 (𝜑 → ℕ0 = ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
306287uneq1d 3764 . . . . . . . . . . . . . . . . 17 (𝜑 → ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))) = ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))))
307305, 306eqtr2d 2656 . . . . . . . . . . . . . . . 16 (𝜑 → ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))) = ℕ0)
308 ssequn1 3781 . . . . . . . . . . . . . . . . . 18 (ℚ ⊆ ℂ ↔ (ℚ ∪ ℂ) = ℂ)
30996, 308mpbi 220 . . . . . . . . . . . . . . . . 17 (ℚ ∪ ℂ) = ℂ
310309a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → (ℚ ∪ ℂ) = ℂ)
311307, 310feq23d 6038 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ) ↔ (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):ℕ0⟶ℂ))
312311adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ) ↔ (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):ℕ0⟶ℂ))
313300, 312mpbid 222 . . . . . . . . . . . . 13 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):ℕ0⟶ℂ)
314 ffn 6043 . . . . . . . . . . . . . . . 16 (𝑤:(0...𝑁)⟶ℚ → 𝑤 Fn (0...𝑁))
315 fnimadisj 6010 . . . . . . . . . . . . . . . 16 ((𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) → (𝑤 “ (ℤ‘(𝑁 + 1))) = ∅)
316314, 290, 315syl2anr 495 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 “ (ℤ‘(𝑁 + 1))) = ∅)
3172nn0zd 11477 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℤ)
318317peano2zd 11482 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 + 1) ∈ ℤ)
319 uzid 11699 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) ∈ ℤ → (𝑁 + 1) ∈ (ℤ‘(𝑁 + 1)))
320 ne0i 3919 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) ∈ (ℤ‘(𝑁 + 1)) → (ℤ‘(𝑁 + 1)) ≠ ∅)
321318, 319, 3203syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℤ‘(𝑁 + 1)) ≠ ∅)
322 inidm 3820 . . . . . . . . . . . . . . . . . . 19 ((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) = (ℤ‘(𝑁 + 1))
323322neeq1i 2857 . . . . . . . . . . . . . . . . . 18 (((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) ≠ ∅ ↔ (ℤ‘(𝑁 + 1)) ≠ ∅)
324321, 323sylibr 224 . . . . . . . . . . . . . . . . 17 (𝜑 → ((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) ≠ ∅)
325 xpima2 5576 . . . . . . . . . . . . . . . . 17 (((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) ≠ ∅ → (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))) = {0})
326324, 325syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))) = {0})
327326adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))) = {0})
328316, 327uneq12d 3766 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 “ (ℤ‘(𝑁 + 1))) ∪ (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1)))) = (∅ ∪ {0}))
329 imaundir 5544 . . . . . . . . . . . . . 14 ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) “ (ℤ‘(𝑁 + 1))) = ((𝑤 “ (ℤ‘(𝑁 + 1))) ∪ (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))))
330 uncom 3755 . . . . . . . . . . . . . . 15 (∅ ∪ {0}) = ({0} ∪ ∅)
331 un0 3965 . . . . . . . . . . . . . . 15 ({0} ∪ ∅) = {0}
332330, 331eqtr2i 2644 . . . . . . . . . . . . . 14 {0} = (∅ ∪ {0})
333328, 329, 3323eqtr4g 2680 . . . . . . . . . . . . 13 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) “ (ℤ‘(𝑁 + 1))) = {0})
334290, 314anim12ci 591 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅))
335 fnconstg 6091 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ V → ((ℤ‘(𝑁 + 1)) × {0}) Fn (ℤ‘(𝑁 + 1)))
336133, 335ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((ℤ‘(𝑁 + 1)) × {0}) Fn (ℤ‘(𝑁 + 1))
337 fvun1 6267 . . . . . . . . . . . . . . . . . . . 20 ((𝑤 Fn (0...𝑁) ∧ ((ℤ‘(𝑁 + 1)) × {0}) Fn (ℤ‘(𝑁 + 1)) ∧ (((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅ ∧ 𝑘 ∈ (0...𝑁))) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
338336, 337mp3an2 1411 . . . . . . . . . . . . . . . . . . 19 ((𝑤 Fn (0...𝑁) ∧ (((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅ ∧ 𝑘 ∈ (0...𝑁))) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
339338anassrs 680 . . . . . . . . . . . . . . . . . 18 (((𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
340334, 339sylan 488 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
341340eqcomd 2627 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) = ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘))
342341oveq1d 6662 . . . . . . . . . . . . . . 15 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝑦𝑘)) = (((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦𝑘)))
343342sumeq2dv 14427 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)) = Σ𝑘 ∈ (0...𝑁)(((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦𝑘)))
344343mpteq2dv 4743 . . . . . . . . . . . . 13 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦𝑘))))
345281, 280, 313, 333, 344coeeq 23977 . . . . . . . . . . . 12 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) = (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})))
346345reseq1d 5393 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)))
347 res0 5398 . . . . . . . . . . . . . 14 (𝑤 ↾ ∅) = ∅
348289reseq2d 5394 . . . . . . . . . . . . . 14 (𝜑 → (𝑤 ↾ ∅) = (𝑤 ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
349 res0 5398 . . . . . . . . . . . . . . 15 (((ℤ‘(𝑁 + 1)) × {0}) ↾ ∅) = ∅
350289reseq2d 5394 . . . . . . . . . . . . . . 15 (𝜑 → (((ℤ‘(𝑁 + 1)) × {0}) ↾ ∅) = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
351349, 350syl5eqr 2669 . . . . . . . . . . . . . 14 (𝜑 → ∅ = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
352347, 348, 3513eqtr3a 2679 . . . . . . . . . . . . 13 (𝜑 → (𝑤 ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))) = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
353 fss 6054 . . . . . . . . . . . . . . 15 ((((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶{0} ∧ {0} ⊆ ℚ) → ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℚ)
354291, 293, 353mp2an 708 . . . . . . . . . . . . . 14 ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℚ
355 fresaunres1 6075 . . . . . . . . . . . . . 14 ((𝑤:(0...𝑁)⟶ℚ ∧ ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℚ ∧ (𝑤 ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))) = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
356354, 355mp3an2 1411 . . . . . . . . . . . . 13 ((𝑤:(0...𝑁)⟶ℚ ∧ (𝑤 ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))) = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
357352, 356sylan2 491 . . . . . . . . . . . 12 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝜑) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
358357ancoms 469 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
359346, 358eqtrd 2655 . . . . . . . . . 10 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = 𝑤)
360 fveq2 6189 . . . . . . . . . . 11 ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = 0𝑝 → (coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) = (coeff‘0𝑝))
361360reseq1d 5393 . . . . . . . . . 10 ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = 0𝑝 → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((coeff‘0𝑝) ↾ (0...𝑁)))
362 eqtr2 2641 . . . . . . . . . . . 12 ((((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = 𝑤 ∧ ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((coeff‘0𝑝) ↾ (0...𝑁))) → 𝑤 = ((coeff‘0𝑝) ↾ (0...𝑁)))
363 coe0 24006 . . . . . . . . . . . . . 14 (coeff‘0𝑝) = (ℕ0 × {0})
364363reseq1i 5390 . . . . . . . . . . . . 13 ((coeff‘0𝑝) ↾ (0...𝑁)) = ((ℕ0 × {0}) ↾ (0...𝑁))
365 elfznn0 12429 . . . . . . . . . . . . . . 15 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℕ0)
366365ssriv 3605 . . . . . . . . . . . . . 14 (0...𝑁) ⊆ ℕ0
367 xpssres 5432 . . . . . . . . . . . . . 14 ((0...𝑁) ⊆ ℕ0 → ((ℕ0 × {0}) ↾ (0...𝑁)) = ((0...𝑁) × {0}))
368366, 367ax-mp 5 . . . . . . . . . . . . 13 ((ℕ0 × {0}) ↾ (0...𝑁)) = ((0...𝑁) × {0})
369364, 368eqtri 2643 . . . . . . . . . . . 12 ((coeff‘0𝑝) ↾ (0...𝑁)) = ((0...𝑁) × {0})
370362, 369syl6eq 2671 . . . . . . . . . . 11 ((((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = 𝑤 ∧ ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((coeff‘0𝑝) ↾ (0...𝑁))) → 𝑤 = ((0...𝑁) × {0}))
371370ex 450 . . . . . . . . . 10 (((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = 𝑤 → (((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((coeff‘0𝑝) ↾ (0...𝑁)) → 𝑤 = ((0...𝑁) × {0})))
372359, 361, 371syl2im 40 . . . . . . . . 9 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = 0𝑝𝑤 = ((0...𝑁) × {0})))
373372necon3d 2814 . . . . . . . 8 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 ≠ ((0...𝑁) × {0}) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ≠ 0𝑝))
374373impr 649 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ≠ 0𝑝)
375 eldifsn 4315 . . . . . . 7 ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ ((Poly‘ℚ) ∖ {0𝑝}) ↔ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ (Poly‘ℚ) ∧ (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ≠ 0𝑝))
376282, 374, 375sylanbrc 698 . . . . . 6 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ ((Poly‘ℚ) ∖ {0𝑝}))
377376adantrr 753 . . . . 5 ((𝜑 ∧ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})) ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ ((Poly‘ℚ) ∖ {0𝑝}))
378 oveq1 6654 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝑦𝑘) = (𝐴𝑘))
379378oveq2d 6663 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝑤𝑘) · (𝑦𝑘)) = ((𝑤𝑘) · (𝐴𝑘)))
380379sumeq2sdv 14429 . . . . . . . . . 10 (𝑦 = 𝐴 → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)))
381 eqid 2621 . . . . . . . . . 10 (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))
382 sumex 14412 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) ∈ V
383380, 381, 382fvmpt 6280 . . . . . . . . 9 (𝐴 ∈ ℂ → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)))
3841, 383syl 17 . . . . . . . 8 (𝜑 → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)))
385384adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)))
386111adantll 750 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℂ)
387 aacllem.2 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
388387adantlr 751 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
389114, 388mulcld 10057 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝐶 · 𝑋) ∈ ℂ)
390389adantllr 755 . . . . . . . . . . . . . 14 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝐶 · 𝑋) ∈ ℂ)
39153, 386, 390fsummulc2 14510 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)) = Σ𝑛 ∈ (1...𝑁)((𝑤𝑘) · (𝐶 · 𝑋)))
392 aacllem.4 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (0...𝑁)) → (𝐴𝑘) = Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋))
393392oveq2d 6663 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝐴𝑘)) = ((𝑤𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)))
394393adantlr 751 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝐴𝑘)) = ((𝑤𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)))
395386adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑤𝑘) ∈ ℂ)
396114adantllr 755 . . . . . . . . . . . . . . 15 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℂ)
397 simpll 790 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → 𝜑)
398397, 387sylan 488 . . . . . . . . . . . . . . 15 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
399395, 396, 398mulassd 10060 . . . . . . . . . . . . . 14 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (((𝑤𝑘) · 𝐶) · 𝑋) = ((𝑤𝑘) · (𝐶 · 𝑋)))
400399sumeq2dv 14427 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁)((𝑤𝑘) · (𝐶 · 𝑋)))
401391, 394, 4003eqtr4d 2665 . . . . . . . . . . . 12 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
402401sumeq2dv 14427 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑘 ∈ (0...𝑁𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
403111ad2ant2lr 784 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → (𝑤𝑘) ∈ ℂ)
404114anasss 679 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝐶 ∈ ℂ)
405404adantlr 751 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝐶 ∈ ℂ)
406403, 405mulcld 10057 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → ((𝑤𝑘) · 𝐶) ∈ ℂ)
407387ad2ant2rl 785 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝑋 ∈ ℂ)
408406, 407mulcld 10057 . . . . . . . . . . . 12 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → (((𝑤𝑘) · 𝐶) · 𝑋) ∈ ℂ)
40945, 71, 408fsumcom 14501 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
410402, 409eqtrd 2655 . . . . . . . . . 10 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
411410adantrr 753 . . . . . . . . 9 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
412 nfv 1842 . . . . . . . . . . . 12 𝑛𝜑
413 nfv 1842 . . . . . . . . . . . . 13 𝑛 𝑤:(0...𝑁)⟶ℚ
414 nfra1 2940 . . . . . . . . . . . . 13 𝑛𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0
415413, 414nfan 1827 . . . . . . . . . . . 12 𝑛(𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
416412, 415nfan 1827 . . . . . . . . . . 11 𝑛(𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
417 rspa 2929 . . . . . . . . . . . . . . . 16 ((∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
418417oveq1d 6662 . . . . . . . . . . . . . . 15 ((∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = (0 · 𝑋))
419418adantll 750 . . . . . . . . . . . . . 14 (((𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = (0 · 𝑋))
420419adantll 750 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = (0 · 𝑋))
421387adantlr 751 . . . . . . . . . . . . . . 15 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
42295, 421, 117fsummulc1 14511 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
423422adantlrr 757 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
424387mul02d 10231 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → (0 · 𝑋) = 0)
425424adantlr 751 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (0 · 𝑋) = 0)
426420, 423, 4253eqtr3d 2663 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = 0)
427426ex 450 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → (𝑛 ∈ (1...𝑁) → Σ𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = 0))
428416, 427ralrimi 2956 . . . . . . . . . 10 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = 0)
429428sumeq2d 14426 . . . . . . . . 9 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁)0)
430411, 429eqtrd 2655 . . . . . . . 8 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁)0)
43124olci 406 . . . . . . . . 9 ((1...𝑁) ⊆ (ℤ𝐵) ∨ (1...𝑁) ∈ Fin)
432 sumz 14447 . . . . . . . . 9 (((1...𝑁) ⊆ (ℤ𝐵) ∨ (1...𝑁) ∈ Fin) → Σ𝑛 ∈ (1...𝑁)0 = 0)
433431, 432ax-mp 5 . . . . . . . 8 Σ𝑛 ∈ (1...𝑁)0 = 0
434430, 433syl6eq 2671 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = 0)
435385, 434eqtrd 2655 . . . . . 6 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = 0)
436435adantrlr 759 . . . . 5 ((𝜑 ∧ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})) ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = 0)
437 fveq1 6188 . . . . . . 7 (𝑥 = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) → (𝑥𝐴) = ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴))
438437eqeq1d 2623 . . . . . 6 (𝑥 = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) → ((𝑥𝐴) = 0 ↔ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = 0))
439438rspcev 3307 . . . . 5 (((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ ((Poly‘ℚ) ∖ {0𝑝}) ∧ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = 0) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0)
440377, 436, 439syl2anc 693 . . . 4 ((𝜑 ∧ ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0})) ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0)
441278, 440sylanr1 684 . . 3 ((𝜑 ∧ (𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0)
442275, 441rexlimddv 3033 . 2 (𝜑 → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0)
443 elqaa 24071 . 2 (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0))
4441, 442, 443sylanbrc 698 1 (𝜑𝐴 ∈ 𝔸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1037   = wceq 1482  wcel 1989  wne 2793  wnel 2896  wral 2911  wrex 2912  Vcvv 3198  cdif 3569  cun 3570  cin 3571  wss 3572  c0 3913  𝒫 cpw 4156  {csn 4175   class class class wbr 4651  cmpt 4727   × cxp 5110  dom cdm 5112  ran crn 5113  cres 5114  cima 5115   Fn wfn 5881  wf 5882  1-1wf1 5883  1-1-ontowf1o 5885  cfv 5886  (class class class)co 6647  𝑓 cof 6892  𝑚 cmap 7854  cen 7949  cdom 7950  Fincfn 7952  cc 9931  0cc0 9933  1c1 9934   + caddc 9936   · cmul 9938   < clt 10071  cle 10072  cmin 10263  0cn0 11289  cz 11374  cuz 11684  cq 11785  ...cfz 12323  cexp 12855  #chash 13112  Σcsu 14410  Basecbs 15851  s cress 15852  .rcmulr 15936  Scalarcsca 15938   ·𝑠 cvsca 15939  0gc0g 16094   Σg cgsu 16095  Moorecmre 16236  mrClscmrc 16237  mrIndcmri 16238  ACScacs 16239  Ringcrg 18541  DivRingcdr 18741  LModclmod 18857  LSubSpclss 18926  LSpanclspn 18965  LBasisclbs 19068  LVecclvec 19096  NzRingcnzr 19251  fldccnfld 19740   freeLMod cfrlm 20084   unitVec cuvc 20115   LIndF clindf 20137  LIndSclinds 20138  0𝑝c0p 23430  Polycply 23934  coeffccoe 23936  𝔸caa 24063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-inf2 8535  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010  ax-pre-sup 10011  ax-addf 10012  ax-mulf 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-iin 4521  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-of 6894  df-om 7063  df-1st 7165  df-2nd 7166  df-supp 7293  df-tpos 7349  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-er 7739  df-map 7856  df-pm 7857  df-ixp 7906  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-fsupp 8273  df-sup 8345  df-inf 8346  df-oi 8412  df-card 8762  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-div 10682  df-nn 11018  df-2 11076  df-3 11077  df-4 11078  df-5 11079  df-6 11080  df-7 11081  df-8 11082  df-9 11083  df-n0 11290  df-xnn0 11361  df-z 11375  df-dec 11491  df-uz 11685  df-q 11786  df-rp 11830  df-fz 12324  df-fzo 12462  df-fl 12588  df-mod 12664  df-seq 12797  df-exp 12856  df-hash 13113  df-cj 13833  df-re 13834  df-im 13835  df-sqrt 13969  df-abs 13970  df-clim 14213  df-rlim 14214  df-sum 14411  df-struct 15853  df-ndx 15854  df-slot 15855  df-base 15857  df-sets 15858  df-ress 15859  df-plusg 15948  df-mulr 15949  df-starv 15950  df-sca 15951  df-vsca 15952  df-ip 15953  df-tset 15954  df-ple 15955  df-ds 15958  df-unif 15959  df-hom 15960  df-cco 15961  df-0g 16096  df-gsum 16097  df-prds 16102  df-pws 16104  df-mre 16240  df-mrc 16241  df-mri 16242  df-acs 16243  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-mhm 17329  df-submnd 17330  df-grp 17419  df-minusg 17420  df-sbg 17421  df-mulg 17535  df-subg 17585  df-ghm 17652  df-cntz 17744  df-cmn 18189  df-abl 18190  df-mgp 18484  df-ur 18496  df-ring 18543  df-cring 18544  df-oppr 18617  df-dvdsr 18635  df-unit 18636  df-invr 18666  df-dvr 18677  df-drng 18743  df-subrg 18772  df-lmod 18859  df-lss 18927  df-lsp 18966  df-lmhm 19016  df-lbs 19069  df-lvec 19097  df-sra 19166  df-rgmod 19167  df-nzr 19252  df-cnfld 19741  df-dsmm 20070  df-frlm 20085  df-uvc 20116  df-lindf 20139  df-linds 20140  df-0p 23431  df-ply 23938  df-coe 23940  df-dgr 23941  df-aa 24064
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator