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Theorem aacllem 43078
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 𝑤 𝑥 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 11554 . . . . . 6 (𝜑𝑁 ∈ ℝ)
43ltp1d 11156 . . . . 5 (𝜑𝑁 < (𝑁 + 1))
5 peano2nn0 11535 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
62, 5syl 17 . . . . . . 7 (𝜑 → (𝑁 + 1) ∈ ℕ0)
76nn0red 11554 . . . . . 6 (𝜑 → (𝑁 + 1) ∈ ℝ)
83, 7ltnled 10386 . . . . 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 1111 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
12 eqid 2771 . . . . . . . . . . 11 (𝑛 ∈ (1...𝑁) ↦ 𝐶) = (𝑛 ∈ (1...𝑁) ↦ 𝐶)
1311, 12fmptd 6527 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶):(1...𝑁)⟶ℚ)
14 qex 12003 . . . . . . . . . . 11 ℚ ∈ V
15 ovex 6823 . . . . . . . . . . 11 (1...𝑁) ∈ V
1614, 15elmap 8038 . . . . . . . . . 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 2771 . . . . . . . . 9 (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
1917, 18fmptd 6527 . . . . . . . 8 (𝜑 → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)):(0...𝑁)⟶(ℚ ↑𝑚 (1...𝑁)))
20 eqid 2771 . . . . . . . . . . . 12 (ℂflds ℚ) = (ℂflds ℚ)
2120qdrng 25530 . . . . . . . . . . 11 (ℂflds ℚ) ∈ DivRing
22 drngring 18964 . . . . . . . . . . 11 ((ℂflds ℚ) ∈ DivRing → (ℂflds ℚ) ∈ Ring)
2321, 22ax-mp 5 . . . . . . . . . 10 (ℂflds ℚ) ∈ Ring
24 fzfi 12979 . . . . . . . . . 10 (1...𝑁) ∈ Fin
25 eqid 2771 . . . . . . . . . . 11 ((ℂflds ℚ) freeLMod (1...𝑁)) = ((ℂflds ℚ) freeLMod (1...𝑁))
2625frlmlmod 20310 . . . . . . . . . 10 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod)
2723, 24, 26mp2an 672 . . . . . . . . 9 ((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod
28 fzfi 12979 . . . . . . . . 9 (0...𝑁) ∈ Fin
2920qrngbas 25529 . . . . . . . . . . . 12 ℚ = (Base‘(ℂflds ℚ))
3025, 29frlmfibas 20322 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ DivRing ∧ (1...𝑁) ∈ Fin) → (ℚ ↑𝑚 (1...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (1...𝑁))))
3121, 24, 30mp2an 672 . . . . . . . . . 10 (ℚ ↑𝑚 (1...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (1...𝑁)))
3225frlmsca 20314 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ DivRing ∧ (1...𝑁) ∈ Fin) → (ℂflds ℚ) = (Scalar‘((ℂflds ℚ) freeLMod (1...𝑁))))
3321, 24, 32mp2an 672 . . . . . . . . . 10 (ℂflds ℚ) = (Scalar‘((ℂflds ℚ) freeLMod (1...𝑁)))
34 eqid 2771 . . . . . . . . . 10 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁))) = ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))
3520qrng0 25531 . . . . . . . . . . . 12 0 = (0g‘(ℂflds ℚ))
3625, 35frlm0 20315 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) × {0}) = (0g‘((ℂflds ℚ) freeLMod (1...𝑁))))
3723, 24, 36mp2an 672 . . . . . . . . . 10 ((1...𝑁) × {0}) = (0g‘((ℂflds ℚ) freeLMod (1...𝑁)))
38 eqid 2771 . . . . . . . . . . . 12 ((ℂflds ℚ) freeLMod (0...𝑁)) = ((ℂflds ℚ) freeLMod (0...𝑁))
3938, 29frlmfibas 20322 . . . . . . . . . . 11 (((ℂflds ℚ) ∈ DivRing ∧ (0...𝑁) ∈ Fin) → (ℚ ↑𝑚 (0...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (0...𝑁))))
4021, 28, 39mp2an 672 . . . . . . . . . 10 (ℚ ↑𝑚 (0...𝑁)) = (Base‘((ℂflds ℚ) freeLMod (0...𝑁)))
4131, 33, 34, 37, 35, 40islindf4 20394 . . . . . . . . 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 1562 . . . . . . . 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 8031 . . . . . . . . 9 (𝑤 ∈ (ℚ ↑𝑚 (0...𝑁)) → 𝑤:(0...𝑁)⟶ℚ)
45 fzfid 12980 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (0...𝑁) ∈ Fin)
46 fvexd 6344 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ V)
4715mptex 6630 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ V
4847a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ V)
49 simpr 471 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤:(0...𝑁)⟶ℚ) → 𝑤:(0...𝑁)⟶ℚ)
5049feqmptd 6391 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → 𝑤 = (𝑘 ∈ (0...𝑁) ↦ (𝑤𝑘)))
51 eqidd 2772 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
5245, 46, 48, 50, 51offval2 7061 . . . . . . . . . . . . . . . . 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 12980 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (1...𝑁) ∈ Fin)
54 ffvelrn 6500 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℚ)
5554adantll 693 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℚ)
5617adantlr 694 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) ∈ (ℚ ↑𝑚 (1...𝑁)))
57 cnfldmul 19967 . . . . . . . . . . . . . . . . . . . . . 22 · = (.r‘ℂfld)
5820, 57ressmulr 16214 . . . . . . . . . . . . . . . . . . . . 21 (ℚ ∈ V → · = (.r‘(ℂflds ℚ)))
5914, 58ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 · = (.r‘(ℂflds ℚ))
6025, 31, 29, 53, 55, 56, 34, 59frlmvscafval 20326 . . . . . . . . . . . . . . . . . . 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 6344 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑤𝑘) ∈ V)
6211adantllr 698 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℚ)
63 fconstmpt 5303 . . . . . . . . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ 𝐶) = (𝑛 ∈ (1...𝑁) ↦ 𝐶))
6653, 61, 62, 64, 65offval2 7061 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (((1...𝑁) × {(𝑤𝑘)}) ∘𝑓 · (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))
6760, 66eqtrd 2805 . . . . . . . . . . . . . . . . . 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 4878 . . . . . . . . . . . . . . . . 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 2805 . . . . . . . . . . . . . . . 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 6809 . . . . . . . . . . . . . . 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 12980 . . . . . . . . . . . . . . . 16 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (1...𝑁) ∈ Fin)
7223a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (ℂflds ℚ) ∈ Ring)
7355adantlr 694 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℚ)
7411an32s 631 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℚ)
7574adantllr 698 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℚ)
76 qmulcl 12009 . . . . . . . . . . . . . . . . . . . 20 (((𝑤𝑘) ∈ ℚ ∧ 𝐶 ∈ ℚ) → ((𝑤𝑘) · 𝐶) ∈ ℚ)
7773, 75, 76syl2anc 573 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · 𝐶) ∈ ℚ)
7877an32s 631 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑤𝑘) · 𝐶) ∈ ℚ)
79 eqid 2771 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))
8078, 79fmptd 6527 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
8114, 15elmap 8038 . . . . . . . . . . . . . . . . 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 2771 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ ((𝑤𝑘) · 𝐶)))
8415mptex 6630 . . . . . . . . . . . . . . . . . 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 5036 . . . . . . . . . . . . . . . . . . 19 {0} ∈ V
8715, 86xpex 7109 . . . . . . . . . . . . . . . . . 18 ((1...𝑁) × {0}) ∈ V
8887a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((1...𝑁) × {0}) ∈ V)
8983, 45, 85, 88fsuppmptdm 8442 . . . . . . . . . . . . . . . 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 20328 . . . . . . . . . . . . . . 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 19965 . . . . . . . . . . . . . . . . . 18 ℂ = (Base‘ℂfld)
92 cnfldadd 19966 . . . . . . . . . . . . . . . . . 18 + = (+g‘ℂfld)
93 cnfldex 19964 . . . . . . . . . . . . . . . . . . 19 fld ∈ V
9493a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ℂfld ∈ V)
95 fzfid 12980 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (0...𝑁) ∈ Fin)
96 qsscn 12002 . . . . . . . . . . . . . . . . . . 19 ℚ ⊆ ℂ
9796a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ℚ ⊆ ℂ)
98 eqid 2771 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶)) = (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))
9977, 98fmptd 6527 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶)):(0...𝑁)⟶ℚ)
100 0z 11590 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℤ
101 zq 11997 . . . . . . . . . . . . . . . . . . . 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 10421 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥)
105 addid1 10418 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ ℂ → (𝑥 + 0) = 𝑥)
106104, 105jca 501 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℂ → ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
107106adantl 467 . . . . . . . . . . . . . . . . . 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 17484 . . . . . . . . . . . . . . . . 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 752 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 𝑤:(0...𝑁)⟶ℚ)
110 qcn 12005 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤𝑘) ∈ ℚ → (𝑤𝑘) ∈ ℂ)
11154, 110syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℂ)
112109, 111sylan 569 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℂ)
113 qcn 12005 . . . . . . . . . . . . . . . . . . . . . 22 (𝐶 ∈ ℚ → 𝐶 ∈ ℂ)
11411, 113syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℂ)
115114an32s 631 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℂ)
116115adantllr 698 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → 𝐶 ∈ ℂ)
117112, 116mulcld 10262 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · 𝐶) ∈ ℂ)
11895, 117gsumfsum 20028 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (ℂfld Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
119108, 118eqtr3d 2807 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → ((ℂflds ℚ) Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑤𝑘) · 𝐶))) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
120119mpteq2dva 4878 . . . . . . . . . . . . . . 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 2809 . . . . . . . . . . . . . 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 12007 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ) → (𝑥 + 𝑦) ∈ ℚ)
123122adantl 467 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) ∧ (𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ)) → (𝑥 + 𝑦) ∈ ℚ)
12497, 123, 95, 77, 103fsumcllem 14671 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) ∈ ℚ)
125 eqid 2771 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)) = (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
126124, 125fmptd 6527 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶)):(1...𝑁)⟶ℚ)
12714, 15elmap 8038 . . . . . . . . . . . . . . 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 2850 . . . . . . . . . . . . 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 8031 . . . . . . . . . . . . 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 6185 . . . . . . . . . . . . 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 10236 . . . . . . . . . . . . 13 0 ∈ V
134 fnconstg 6233 . . . . . . . . . . . . 13 (0 ∈ V → ((1...𝑁) × {0}) Fn (1...𝑁))
135133, 134ax-mp 5 . . . . . . . . . . . 12 ((1...𝑁) × {0}) Fn (1...𝑁)
136 nfcv 2913 . . . . . . . . . . . . . 14 𝑛((ℂflds ℚ) freeLMod (1...𝑁))
137 nfcv 2913 . . . . . . . . . . . . . 14 𝑛 Σg
138 nfcv 2913 . . . . . . . . . . . . . . 15 𝑛𝑤
139 nfcv 2913 . . . . . . . . . . . . . . 15 𝑛𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))
140 nfcv 2913 . . . . . . . . . . . . . . . 16 𝑛(0...𝑁)
141 nfmpt1 4881 . . . . . . . . . . . . . . . 16 𝑛(𝑛 ∈ (1...𝑁) ↦ 𝐶)
142140, 141nfmpt 4880 . . . . . . . . . . . . . . 15 𝑛(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
143138, 139, 142nfov 6821 . . . . . . . . . . . . . 14 𝑛(𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)))
144136, 137, 143nfov 6821 . . . . . . . . . . . . 13 𝑛(((ℂflds ℚ) freeLMod (1...𝑁)) Σg (𝑤𝑓 ( ·𝑠 ‘((ℂflds ℚ) freeLMod (1...𝑁)))(𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))))
145 nfcv 2913 . . . . . . . . . . . . 13 𝑛((1...𝑁) × {0})
146144, 145eqfnfv2f 6458 . . . . . . . . . . . 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 574 . . . . . . . . . . 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 6334 . . . . . . . . . . . . . 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 14626 . . . . . . . . . . . . . . 15 Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) ∈ V
150125fvmpt2 6433 . . . . . . . . . . . . . . 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 671 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → ((𝑛 ∈ (1...𝑁) ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))‘𝑛) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶))
152148, 151sylan9eq 2825 . . . . . . . . . . . . 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 6613 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {0})‘𝑛) = 0)
154153adantl 467 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (((1...𝑁) × {0})‘𝑛) = 0)
155152, 154eqeq12d 2786 . . . . . . . . . . . 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 3134 . . . . . . . . . . 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 330 . . . . . . . . 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 580 . . . . . . . 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 3134 . . . . . . 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 19477 . . . . . . . . 9 ((ℂflds ℚ) ∈ DivRing → (ℂflds ℚ) ∈ NzRing)
16321, 162ax-mp 5 . . . . . . . 8 (ℂflds ℚ) ∈ NzRing
16433islindf3 20382 . . . . . . . 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 672 . . . . . . 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 6163 . . . . . . . . 9 dom (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) = (0...𝑁)
167 f1eq2 6237 . . . . . . . . 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 610 . . . . . . 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 305 . . . . . . . . 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 3047 . . . . . . . . . . 11 (𝑤 ∉ {((0...𝑁) × {0})} ↔ ¬ 𝑤 ∈ {((0...𝑁) × {0})})
173 velsn 4332 . . . . . . . . . . 11 (𝑤 ∈ {((0...𝑁) × {0})} ↔ 𝑤 = ((0...𝑁) × {0}))
174172, 173xchbinx 323 . . . . . . . . . 10 (𝑤 ∉ {((0...𝑁) × {0})} ↔ ¬ 𝑤 = ((0...𝑁) × {0}))
175174imbi1i 338 . . . . . . . . 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 3129 . . . . . . 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 3901 . . . . . . 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 3141 . . . . . . 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 2771 . . . . . . . . . . . . 13 (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))
18331, 182lssmre 19179 . . . . . . . . . . . 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 2771 . . . . . . . . . . . 12 (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))
187 eqid 2771 . . . . . . . . . . . 12 (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) = (mrCls‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))
188182, 186, 187mrclsp 19202 . . . . . . . . . . 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 2771 . . . . . . . . . 10 (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁)))) = (mrInd‘(LSubSp‘((ℂflds ℚ) freeLMod (1...𝑁))))
19133islvec 19317 . . . . . . . . . . . . 13 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LVec ↔ (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂflds ℚ) ∈ DivRing))
19227, 21, 191mpbir2an 690 . . . . . . . . . . . 12 ((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LVec
193182, 189, 31lssacsex 19358 . . . . . . . . . . . . 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 483 . . . . . . . . . . . 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 6193 . . . . . . . . . . . . 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 4097 . . . . . . . . . . . 12 ((ℚ ↑𝑚 (1...𝑁)) ∖ ∅) = (ℚ ↑𝑚 (1...𝑁))
200198, 199syl6sseqr 3801 . . . . . . . . . . 11 (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((ℚ ↑𝑚 (1...𝑁)) ∖ ∅))
201200adantr 466 . . . . . . . . . 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 2771 . . . . . . . . . . . . . . 15 ((ℂflds ℚ) unitVec (1...𝑁)) = ((ℂflds ℚ) unitVec (1...𝑁))
203202, 25, 31uvcff 20347 . . . . . . . . . . . . . 14 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ((ℂflds ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑𝑚 (1...𝑁)))
20423, 24, 203mp2an 672 . . . . . . . . . . . . 13 ((ℂflds ℚ) unitVec (1...𝑁)):(1...𝑁)⟶(ℚ ↑𝑚 (1...𝑁))
205 frn 6193 . . . . . . . . . . . . 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 3787 . . . . . . . . . . 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 4111 . . . . . . . . . . . . . 14 (ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅) = ran ((ℂflds ℚ) unitVec (1...𝑁))
210209fveq2i 6335 . . . . . . . . . . . . 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 2771 . . . . . . . . . . . . . . . 16 (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁))) = (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁)))
21225, 202, 211frlmlbs 20353 . . . . . . . . . . . . . . 15 (((ℂflds ℚ) ∈ Ring ∧ (1...𝑁) ∈ Fin) → ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁))))
21323, 24, 212mp2an 672 . . . . . . . . . . . . . 14 ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ (LBasis‘((ℂflds ℚ) freeLMod (1...𝑁)))
21431, 211, 186lbssp 19292 . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . 12 ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅)) = (ℚ ↑𝑚 (1...𝑁))
217198, 216syl6sseqr 3801 . . . . . . . . . . 11 (𝜑 → ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ⊆ ((LSpan‘((ℂflds ℚ) freeLMod (1...𝑁)))‘(ran ((ℂflds ℚ) unitVec (1...𝑁)) ∪ ∅)))
218217adantr 466 . . . . . . . . . 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 4111 . . . . . . . . . . 11 (ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∪ ∅) = ran (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶))
22027, 163pm3.2i 447 . . . . . . . . . . . . . 14 (((ℂflds ℚ) freeLMod (1...𝑁)) ∈ LMod ∧ (ℂflds ℚ) ∈ NzRing)
221186, 33lindsind2 20375 . . . . . . . . . . . . . 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 1559 . . . . . . . . . . . . 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 3115 . . . . . . . . . . . 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 16500 . . . . . . . . . . . . . 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 575 . . . . . . . . . . . . 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 463 . . . . . . . . . . . 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 580 . . . . . . . . . . 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 2854 . . . . . . . . . 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 8421 . . . . . . . . . . . . 13 ((0...𝑁) ∈ Fin → (𝑘 ∈ (0...𝑁) ↦ (𝑛 ∈ (1...𝑁) ↦ 𝐶)) ∈ Fin)
230 rnfi 8405 . . . . . . . . . . . . 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 852 . . . . . . . . . . 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 16516 . . . . . . . . 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 397 . . . . . . . 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 6823 . . . . . . . . . . . . 13 ((ℂflds ℚ) unitVec (1...𝑁)) ∈ V
237236rnex 7247 . . . . . . . . . . . 12 ran ((ℂflds ℚ) unitVec (1...𝑁)) ∈ V
238 elpwi 4307 . . . . . . . . . . . 12 (𝑣 ∈ 𝒫 ran ((ℂflds ℚ) unitVec (1...𝑁)) → 𝑣 ⊆ ran ((ℂflds ℚ) unitVec (1...𝑁)))
239 ssdomg 8155 . . . . . . . . . . . 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 8167 . . . . . . . . . . . . . 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 455 . . . . . . . . . . . . 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 6289 . . . . . . . . . . . . . . . 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 6823 . . . . . . . . . . . . . . . . 17 (0...𝑁) ∈ V
245244f1oen 8130 . . . . . . . . . . . . . . . 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 8167 . . . . . . . . . . . . . . . . 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 20403 . . . . . . . . . . . . . . . . . . . 20 (((ℂflds ℚ) ∈ NzRing ∧ (1...𝑁) ∈ Fin) → (1...𝑁) ≈ ran ((ℂflds ℚ) unitVec (1...𝑁)))
249163, 24, 248mp2an 672 . . . . . . . . . . . . . . . . . . 19 (1...𝑁) ≈ ran ((ℂflds ℚ) unitVec (1...𝑁))
250249ensymi 8159 . . . . . . . . . . . . . . . . . 18 ran ((ℂflds ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)
251 domentr 8168 . . . . . . . . . . . . . . . . . . 19 (((0...𝑁) ≼ ran ((ℂflds ℚ) unitVec (1...𝑁)) ∧ ran ((ℂflds ℚ) unitVec (1...𝑁)) ≈ (1...𝑁)) → (0...𝑁) ≼ (1...𝑁))
252 hashdom 13370 . . . . . . . . . . . . . . . . . . . . 21 (((0...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((♯‘(0...𝑁)) ≤ (♯‘(1...𝑁)) ↔ (0...𝑁) ≼ (1...𝑁)))
25328, 24, 252mp2an 672 . . . . . . . . . . . . . . . . . . . 20 ((♯‘(0...𝑁)) ≤ (♯‘(1...𝑁)) ↔ (0...𝑁) ≼ (1...𝑁))
254 hashfz0 13421 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (♯‘(0...𝑁)) = (𝑁 + 1))
2552, 254syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘(0...𝑁)) = (𝑁 + 1))
256 hashfz1 13338 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
2572, 256syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (♯‘(1...𝑁)) = 𝑁)
258255, 257breq12d 4799 . . . . . . . . . . . . . . . . . . . 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 677 . . . . . . . . . . . . . . . . 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 400 . . . . . . . . . . . . . . 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 440 . . . . . . . . . . 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 580 . . . . . . . . . 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 479 . . . . . . . . 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 3179 . . . . . . . 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 396 . . . . . 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 456 . . . . 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 142 . . 3 (𝜑 → ∃𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})})∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
276 eldifsn 4453 . . . . 5 (𝑤 ∈ ((ℚ ↑𝑚 (0...𝑁)) ∖ {((0...𝑁) × {0})}) ↔ (𝑤 ∈ (ℚ ↑𝑚 (0...𝑁)) ∧ 𝑤 ≠ ((0...𝑁) × {0})))
27744anim1i 602 . . . . 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 466 . . . . . . . . 9 ((𝜑𝑤:(0...𝑁)⟶ℚ) → 𝑁 ∈ ℕ0)
281279, 280, 55elplyd 24178 . . . . . . . 8 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ (Poly‘ℚ))
282281adantrr 696 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ (Poly‘ℚ))
283 uzdisj 12620 . . . . . . . . . . . . . . . . . 18 ((0...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ∅
2842nn0cnd 11555 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ ℂ)
285 pncan1 10656 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁)
286284, 285syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
287286oveq2d 6809 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
288287ineq1d 3964 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((0...((𝑁 + 1) − 1)) ∩ (ℤ‘(𝑁 + 1))) = ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))
289283, 288syl5eqr 2819 . . . . . . . . . . . . . . . . 17 (𝜑 → ∅ = ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))))
290289eqcomd 2777 . . . . . . . . . . . . . . . 16 (𝜑 → ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅)
291133fconst 6231 . . . . . . . . . . . . . . . . . 18 ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶{0}
292 snssi 4474 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ ℚ → {0} ⊆ ℚ)
293100, 101, 292mp2b 10 . . . . . . . . . . . . . . . . . . 19 {0} ⊆ ℚ
294293, 96sstri 3761 . . . . . . . . . . . . . . . . . 18 {0} ⊆ ℂ
295 fss 6196 . . . . . . . . . . . . . . . . . 18 ((((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶{0} ∧ {0} ⊆ ℂ) → ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℂ)
296291, 294, 295mp2an 672 . . . . . . . . . . . . . . . . 17 ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℂ
297 fun 6206 . . . . . . . . . . . . . . . . 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 681 . . . . . . . . . . . . . . . 16 ((𝑤:(0...𝑁)⟶ℚ ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
299290, 298sylan2 580 . . . . . . . . . . . . . . 15 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝜑) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
300299ancoms 455 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ))
301 nn0uz 11924 . . . . . . . . . . . . . . . . . 18 0 = (ℤ‘0)
3026, 301syl6eleq 2860 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 + 1) ∈ (ℤ‘0))
303 uzsplit 12619 . . . . . . . . . . . . . . . . . . 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 2817 . . . . . . . . . . . . . . . . 17 (𝜑 → ℕ0 = ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))))
306287uneq1d 3917 . . . . . . . . . . . . . . . . 17 (𝜑 → ((0...((𝑁 + 1) − 1)) ∪ (ℤ‘(𝑁 + 1))) = ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))))
307305, 306eqtr2d 2806 . . . . . . . . . . . . . . . 16 (𝜑 → ((0...𝑁) ∪ (ℤ‘(𝑁 + 1))) = ℕ0)
308 ssequn1 3934 . . . . . . . . . . . . . . . . . 18 (ℚ ⊆ ℂ ↔ (ℚ ∪ ℂ) = ℂ)
30996, 308mpbi 220 . . . . . . . . . . . . . . . . 17 (ℚ ∪ ℂ) = ℂ
310309a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → (ℚ ∪ ℂ) = ℂ)
311307, 310feq23d 6180 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):((0...𝑁) ∪ (ℤ‘(𝑁 + 1)))⟶(ℚ ∪ ℂ) ↔ (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})):ℕ0⟶ℂ))
312311adantr 466 . . . . . . . . . . . . . 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 6185 . . . . . . . . . . . . . . . 16 (𝑤:(0...𝑁)⟶ℚ → 𝑤 Fn (0...𝑁))
315 fnimadisj 6152 . . . . . . . . . . . . . . . 16 ((𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) → (𝑤 “ (ℤ‘(𝑁 + 1))) = ∅)
316314, 290, 315syl2anr 584 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 “ (ℤ‘(𝑁 + 1))) = ∅)
3172nn0zd 11682 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℤ)
318317peano2zd 11687 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 + 1) ∈ ℤ)
319 uzid 11903 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) ∈ ℤ → (𝑁 + 1) ∈ (ℤ‘(𝑁 + 1)))
320 ne0i 4069 . . . . . . . . . . . . . . . . . . 19 ((𝑁 + 1) ∈ (ℤ‘(𝑁 + 1)) → (ℤ‘(𝑁 + 1)) ≠ ∅)
321318, 319, 3203syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ℤ‘(𝑁 + 1)) ≠ ∅)
322 inidm 3971 . . . . . . . . . . . . . . . . . . 19 ((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) = (ℤ‘(𝑁 + 1))
323322neeq1i 3007 . . . . . . . . . . . . . . . . . 18 (((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) ≠ ∅ ↔ (ℤ‘(𝑁 + 1)) ≠ ∅)
324321, 323sylibr 224 . . . . . . . . . . . . . . . . 17 (𝜑 → ((ℤ‘(𝑁 + 1)) ∩ (ℤ‘(𝑁 + 1))) ≠ ∅)
325 xpima2 5719 . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . 15 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))) = {0})
328316, 327uneq12d 3919 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 “ (ℤ‘(𝑁 + 1))) ∪ (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1)))) = (∅ ∪ {0}))
329 imaundir 5687 . . . . . . . . . . . . . 14 ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) “ (ℤ‘(𝑁 + 1))) = ((𝑤 “ (ℤ‘(𝑁 + 1))) ∪ (((ℤ‘(𝑁 + 1)) × {0}) “ (ℤ‘(𝑁 + 1))))
330 uncom 3908 . . . . . . . . . . . . . . 15 (∅ ∪ {0}) = ({0} ∪ ∅)
331 un0 4111 . . . . . . . . . . . . . . 15 ({0} ∪ ∅) = {0}
332330, 331eqtr2i 2794 . . . . . . . . . . . . . 14 {0} = (∅ ∪ {0})
333328, 329, 3323eqtr4g 2830 . . . . . . . . . . . . 13 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) “ (ℤ‘(𝑁 + 1))) = {0})
334290, 314anim12ci 601 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅))
335 fnconstg 6233 . . . . . . . . . . . . . . . . . . . . 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 6411 . . . . . . . . . . . . . . . . . . . 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 1560 . . . . . . . . . . . . . . . . . . 19 ((𝑤 Fn (0...𝑁) ∧ (((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅ ∧ 𝑘 ∈ (0...𝑁))) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
339338anassrs 458 . . . . . . . . . . . . . . . . . 18 (((𝑤 Fn (0...𝑁) ∧ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1))) = ∅) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
340334, 339sylan 569 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) = (𝑤𝑘))
341340eqcomd 2777 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) = ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘))
342341oveq1d 6808 . . . . . . . . . . . . . . 15 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝑦𝑘)) = (((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦𝑘)))
343342sumeq2dv 14641 . . . . . . . . . . . . . 14 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)) = Σ𝑘 ∈ (0...𝑁)(((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0}))‘𝑘) · (𝑦𝑘)))
344343mpteq2dv 4879 . . . . . . . . . . . . 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 24203 . . . . . . . . . . . 12 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) = (𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})))
346345reseq1d 5533 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)))
347 res0 5538 . . . . . . . . . . . . . 14 (𝑤 ↾ ∅) = ∅
348289reseq2d 5534 . . . . . . . . . . . . . 14 (𝜑 → (𝑤 ↾ ∅) = (𝑤 ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
349 res0 5538 . . . . . . . . . . . . . . 15 (((ℤ‘(𝑁 + 1)) × {0}) ↾ ∅) = ∅
350289reseq2d 5534 . . . . . . . . . . . . . . 15 (𝜑 → (((ℤ‘(𝑁 + 1)) × {0}) ↾ ∅) = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
351349, 350syl5eqr 2819 . . . . . . . . . . . . . 14 (𝜑 → ∅ = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
352347, 348, 3513eqtr3a 2829 . . . . . . . . . . . . 13 (𝜑 → (𝑤 ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))) = (((ℤ‘(𝑁 + 1)) × {0}) ↾ ((0...𝑁) ∩ (ℤ‘(𝑁 + 1)))))
353 fss 6196 . . . . . . . . . . . . . . 15 ((((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶{0} ∧ {0} ⊆ ℚ) → ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℚ)
354291, 293, 353mp2an 672 . . . . . . . . . . . . . 14 ((ℤ‘(𝑁 + 1)) × {0}):(ℤ‘(𝑁 + 1))⟶ℚ
355 fresaunres1 6217 . . . . . . . . . . . . . 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 1560 . . . . . . . . . . . . 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 580 . . . . . . . . . . . 12 ((𝑤:(0...𝑁)⟶ℚ ∧ 𝜑) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
358357ancoms 455 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((𝑤 ∪ ((ℤ‘(𝑁 + 1)) × {0})) ↾ (0...𝑁)) = 𝑤)
359346, 358eqtrd 2805 . . . . . . . . . 10 ((𝜑𝑤:(0...𝑁)⟶ℚ) → ((coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) ↾ (0...𝑁)) = 𝑤)
360 fveq2 6332 . . . . . . . . . . 11 ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = 0𝑝 → (coeff‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))) = (coeff‘0𝑝))
361360reseq1d 5533 . . . . . . . . . 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 2791 . . . . . . . . . . . 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 24232 . . . . . . . . . . . . . 14 (coeff‘0𝑝) = (ℕ0 × {0})
364363reseq1i 5530 . . . . . . . . . . . . 13 ((coeff‘0𝑝) ↾ (0...𝑁)) = ((ℕ0 × {0}) ↾ (0...𝑁))
365 elfznn0 12640 . . . . . . . . . . . . . . 15 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℕ0)
366365ssriv 3756 . . . . . . . . . . . . . 14 (0...𝑁) ⊆ ℕ0
367 xpssres 5575 . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . 12 ((coeff‘0𝑝) ↾ (0...𝑁)) = ((0...𝑁) × {0})
370362, 369syl6eq 2821 . . . . . . . . . . 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 397 . . . . . . . . . 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 2964 . . . . . . . 8 ((𝜑𝑤:(0...𝑁)⟶ℚ) → (𝑤 ≠ ((0...𝑁) × {0}) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ≠ 0𝑝))
374373impr 442 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ≠ 0𝑝)
375 eldifsn 4453 . . . . . . 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 572 . . . . . 6 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ 𝑤 ≠ ((0...𝑁) × {0}))) → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) ∈ ((Poly‘ℚ) ∖ {0𝑝}))
377376adantrr 696 . . . . 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 6800 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝑦𝑘) = (𝐴𝑘))
379378oveq2d 6809 . . . . . . . . . . 11 (𝑦 = 𝐴 → ((𝑤𝑘) · (𝑦𝑘)) = ((𝑤𝑘) · (𝐴𝑘)))
380379sumeq2sdv 14643 . . . . . . . . . 10 (𝑦 = 𝐴 → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)))
381 eqid 2771 . . . . . . . . . 10 (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))
382 sumex 14626 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) ∈ V
383380, 381, 382fvmpt 6424 . . . . . . . . 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 466 . . . . . . 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 693 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → (𝑤𝑘) ∈ ℂ)
387 aacllem.2 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
388387adantlr 694 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
389114, 388mulcld 10262 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝐶 · 𝑋) ∈ ℂ)
390389adantllr 698 . . . . . . . . . . . . . 14 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝐶 · 𝑋) ∈ ℂ)
39153, 386, 390fsummulc2 14723 . . . . . . . . . . . . 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 6809 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝐴𝑘)) = ((𝑤𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)))
394393adantlr 694 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝐴𝑘)) = ((𝑤𝑘) · Σ𝑛 ∈ (1...𝑁)(𝐶 · 𝑋)))
395386adantr 466 . . . . . . . . . . . . . . 15 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (𝑤𝑘) ∈ ℂ)
396114adantllr 698 . . . . . . . . . . . . . . 15 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝐶 ∈ ℂ)
397 simpll 750 . . . . . . . . . . . . . . . 16 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → 𝜑)
398397, 387sylan 569 . . . . . . . . . . . . . . 15 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
399395, 396, 398mulassd 10265 . . . . . . . . . . . . . 14 ((((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑛 ∈ (1...𝑁)) → (((𝑤𝑘) · 𝐶) · 𝑋) = ((𝑤𝑘) · (𝐶 · 𝑋)))
400399sumeq2dv 14641 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → Σ𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁)((𝑤𝑘) · (𝐶 · 𝑋)))
401391, 394, 4003eqtr4d 2815 . . . . . . . . . . . 12 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑘 ∈ (0...𝑁)) → ((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
402401sumeq2dv 14641 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑘 ∈ (0...𝑁𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
403111ad2ant2lr 742 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → (𝑤𝑘) ∈ ℂ)
404114anasss 457 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝐶 ∈ ℂ)
405404adantlr 694 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝐶 ∈ ℂ)
406403, 405mulcld 10262 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → ((𝑤𝑘) · 𝐶) ∈ ℂ)
407387ad2ant2rl 743 . . . . . . . . . . . . 13 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → 𝑋 ∈ ℂ)
408406, 407mulcld 10262 . . . . . . . . . . . 12 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ (𝑘 ∈ (0...𝑁) ∧ 𝑛 ∈ (1...𝑁))) → (((𝑤𝑘) · 𝐶) · 𝑋) ∈ ℂ)
40945, 71, 408fsumcom 14714 . . . . . . . . . . 11 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁𝑛 ∈ (1...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
410402, 409eqtrd 2805 . . . . . . . . . 10 ((𝜑𝑤:(0...𝑁)⟶ℚ) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
411410adantrr 696 . . . . . . . . 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 1995 . . . . . . . . . . . 12 𝑛𝜑
413 nfv 1995 . . . . . . . . . . . . 13 𝑛 𝑤:(0...𝑁)⟶ℚ
414 nfra1 3090 . . . . . . . . . . . . 13 𝑛𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0
415413, 414nfan 1980 . . . . . . . . . . . 12 𝑛(𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
416412, 415nfan 1980 . . . . . . . . . . 11 𝑛(𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0))
417 rspa 3079 . . . . . . . . . . . . . . . 16 ((∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 ∧ 𝑛 ∈ (1...𝑁)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)
418417oveq1d 6808 . . . . . . . . . . . . . . 15 ((∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0 ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = (0 · 𝑋))
419418adantll 693 . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . 15 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → 𝑋 ∈ ℂ)
42295, 421, 117fsummulc1 14724 . . . . . . . . . . . . . 14 (((𝜑𝑤:(0...𝑁)⟶ℚ) ∧ 𝑛 ∈ (1...𝑁)) → (Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) · 𝑋) = Σ𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋))
423422adantlrr 700 . . . . . . . . . . . . 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 10436 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...𝑁)) → (0 · 𝑋) = 0)
425424adantlr 694 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) ∧ 𝑛 ∈ (1...𝑁)) → (0 · 𝑋) = 0)
426420, 423, 4253eqtr3d 2813 . . . . . . . . . . . 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 397 . . . . . . . . . . 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 3106 . . . . . . . . . 10 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)(((𝑤𝑘) · 𝐶) · 𝑋) = 0)
429428sumeq2d 14640 . . . . . . . . 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 2805 . . . . . . . 8 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = Σ𝑛 ∈ (1...𝑁)0)
43124olci 853 . . . . . . . . 9 ((1...𝑁) ⊆ (ℤ𝐵) ∨ (1...𝑁) ∈ Fin)
432 sumz 14661 . . . . . . . . 9 (((1...𝑁) ⊆ (ℤ𝐵) ∨ (1...𝑁) ∈ Fin) → Σ𝑛 ∈ (1...𝑁)0 = 0)
433431, 432ax-mp 5 . . . . . . . 8 Σ𝑛 ∈ (1...𝑁)0 = 0
434430, 433syl6eq 2821 . . . . . . 7 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝐴𝑘)) = 0)
435385, 434eqtrd 2805 . . . . . 6 ((𝜑 ∧ (𝑤:(0...𝑁)⟶ℚ ∧ ∀𝑛 ∈ (1...𝑁𝑘 ∈ (0...𝑁)((𝑤𝑘) · 𝐶) = 0)) → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = 0)
436435adantrlr 702 . . . . 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 6331 . . . . . . 7 (𝑥 = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) → (𝑥𝐴) = ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴))
438437eqeq1d 2773 . . . . . 6 (𝑥 = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘))) → ((𝑥𝐴) = 0 ↔ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝑤𝑘) · (𝑦𝑘)))‘𝐴) = 0))
439438rspcev 3460 . . . . 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 573 . . . 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 661 . . 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 3183 . 2 (𝜑 → ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0)
443 elqaa 24297 . 2 (𝐴 ∈ 𝔸 ↔ (𝐴 ∈ ℂ ∧ ∃𝑥 ∈ ((Poly‘ℚ) ∖ {0𝑝})(𝑥𝐴) = 0))
4441, 442, 443sylanbrc 572 1 (𝜑𝐴 ∈ 𝔸)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 834  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wnel 3046  wral 3061  wrex 3062  Vcvv 3351  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297  {csn 4316   class class class wbr 4786  cmpt 4863   × cxp 5247  dom cdm 5249  ran crn 5250  cres 5251  cima 5252   Fn wfn 6026  wf 6027  1-1wf1 6028  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  𝑓 cof 7042  𝑚 cmap 8009  cen 8106  cdom 8107  Fincfn 8109  cc 10136  0cc0 10138  1c1 10139   + caddc 10141   · cmul 10143   < clt 10276  cle 10277  cmin 10468  0cn0 11494  cz 11579  cuz 11888  cq 11991  ...cfz 12533  cexp 13067  chash 13321  Σcsu 14624  Basecbs 16064  s cress 16065  .rcmulr 16150  Scalarcsca 16152   ·𝑠 cvsca 16153  0gc0g 16308   Σg cgsu 16309  Moorecmre 16450  mrClscmrc 16451  mrIndcmri 16452  ACScacs 16453  Ringcrg 18755  DivRingcdr 18957  LModclmod 19073  LSubSpclss 19142  LSpanclspn 19184  LBasisclbs 19287  LVecclvec 19315  NzRingcnzr 19472  fldccnfld 19961   freeLMod cfrlm 20307   unitVec cuvc 20338   LIndF clindf 20360  LIndSclinds 20361  0𝑝c0p 23656  Polycply 24160  coeffccoe 24162  𝔸caa 24289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216  ax-addf 10217  ax-mulf 10218
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-tpos 7504  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-sup 8504  df-inf 8505  df-oi 8571  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-xnn0 11566  df-z 11580  df-dec 11696  df-uz 11889  df-q 11992  df-rp 12036  df-fz 12534  df-fzo 12674  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-abs 14184  df-clim 14427  df-rlim 14428  df-sum 14625  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-starv 16164  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16172  df-unif 16173  df-hom 16174  df-cco 16175  df-0g 16310  df-gsum 16311  df-prds 16316  df-pws 16318  df-mre 16454  df-mrc 16455  df-mri 16456  df-acs 16457  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-submnd 17544  df-grp 17633  df-minusg 17634  df-sbg 17635  df-mulg 17749  df-subg 17799  df-ghm 17866  df-cntz 17957  df-cmn 18402  df-abl 18403  df-mgp 18698  df-ur 18710  df-ring 18757  df-cring 18758  df-oppr 18831  df-dvdsr 18849  df-unit 18850  df-invr 18880  df-dvr 18891  df-drng 18959  df-subrg 18988  df-lmod 19075  df-lss 19143  df-lsp 19185  df-lmhm 19235  df-lbs 19288  df-lvec 19316  df-sra 19387  df-rgmod 19388  df-nzr 19473  df-cnfld 19962  df-dsmm 20293  df-frlm 20308  df-uvc 20339  df-lindf 20362  df-linds 20363  df-0p 23657  df-ply 24164  df-coe 24166  df-dgr 24167  df-aa 24290
This theorem is referenced by: (None)
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