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Theorem coe1tm 19691
 Description: Coefficient vector of a polynomial term. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Hypotheses
Ref Expression
coe1tm.z 0 = (0g𝑅)
coe1tm.k 𝐾 = (Base‘𝑅)
coe1tm.p 𝑃 = (Poly1𝑅)
coe1tm.x 𝑋 = (var1𝑅)
coe1tm.m · = ( ·𝑠𝑃)
coe1tm.n 𝑁 = (mulGrp‘𝑃)
coe1tm.e = (.g𝑁)
Assertion
Ref Expression
coe1tm ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 𝑋))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )))
Distinct variable groups:   𝑥, 0   𝑥,𝐶   𝑥,𝐷   𝑥,𝐾   𝑥,   𝑥,𝑁   𝑥,𝑃   𝑥,𝑋   𝑥,𝑅   𝑥, ·

Proof of Theorem coe1tm
Dummy variables 𝑎 𝑏 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coe1tm.k . . . 4 𝐾 = (Base‘𝑅)
2 coe1tm.p . . . 4 𝑃 = (Poly1𝑅)
3 coe1tm.x . . . 4 𝑋 = (var1𝑅)
4 coe1tm.m . . . 4 · = ( ·𝑠𝑃)
5 coe1tm.n . . . 4 𝑁 = (mulGrp‘𝑃)
6 coe1tm.e . . . 4 = (.g𝑁)
7 eqid 2651 . . . 4 (Base‘𝑃) = (Base‘𝑃)
81, 2, 3, 4, 5, 6, 7ply1tmcl 19690 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ (Base‘𝑃))
9 eqid 2651 . . . 4 (coe1‘(𝐶 · (𝐷 𝑋))) = (coe1‘(𝐶 · (𝐷 𝑋)))
10 eqid 2651 . . . 4 (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥}))
119, 7, 2, 10coe1fval2 19628 . . 3 ((𝐶 · (𝐷 𝑋)) ∈ (Base‘𝑃) → (coe1‘(𝐶 · (𝐷 𝑋))) = ((𝐶 · (𝐷 𝑋)) ∘ (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥}))))
128, 11syl 17 . 2 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 𝑋))) = ((𝐶 · (𝐷 𝑋)) ∘ (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥}))))
13 fconst6g 6132 . . . . 5 (𝑥 ∈ ℕ0 → (1𝑜 × {𝑥}):1𝑜⟶ℕ0)
14 nn0ex 11336 . . . . . 6 0 ∈ V
15 1on 7612 . . . . . . 7 1𝑜 ∈ On
1615elexi 3244 . . . . . 6 1𝑜 ∈ V
1714, 16elmap 7928 . . . . 5 ((1𝑜 × {𝑥}) ∈ (ℕ0𝑚 1𝑜) ↔ (1𝑜 × {𝑥}):1𝑜⟶ℕ0)
1813, 17sylibr 224 . . . 4 (𝑥 ∈ ℕ0 → (1𝑜 × {𝑥}) ∈ (ℕ0𝑚 1𝑜))
1918adantl 481 . . 3 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (1𝑜 × {𝑥}) ∈ (ℕ0𝑚 1𝑜))
20 eqidd 2652 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥})))
21 eqid 2651 . . . . . . . 8 (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
225, 7mgpbas 18541 . . . . . . . . 9 (Base‘𝑃) = (Base‘𝑁)
2322a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘𝑁))
24 eqid 2651 . . . . . . . . . 10 (mulGrp‘(1𝑜 mPoly 𝑅)) = (mulGrp‘(1𝑜 mPoly 𝑅))
25 eqid 2651 . . . . . . . . . . 11 (PwSer1𝑅) = (PwSer1𝑅)
262, 25, 7ply1bas 19613 . . . . . . . . . 10 (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅))
2724, 26mgpbas 18541 . . . . . . . . 9 (Base‘𝑃) = (Base‘(mulGrp‘(1𝑜 mPoly 𝑅)))
2827a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘(mulGrp‘(1𝑜 mPoly 𝑅))))
29 ssv 3658 . . . . . . . . 9 (Base‘𝑃) ⊆ V
3029a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) ⊆ V)
31 ovexd 6720 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑁)𝑦) ∈ V)
32 eqid 2651 . . . . . . . . . . . 12 (.r𝑃) = (.r𝑃)
335, 32mgpplusg 18539 . . . . . . . . . . 11 (.r𝑃) = (+g𝑁)
34 eqid 2651 . . . . . . . . . . . . 13 (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
352, 34, 32ply1mulr 19645 . . . . . . . . . . . 12 (.r𝑃) = (.r‘(1𝑜 mPoly 𝑅))
3624, 35mgpplusg 18539 . . . . . . . . . . 11 (.r𝑃) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
3733, 36eqtr3i 2675 . . . . . . . . . 10 (+g𝑁) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
3837a1i 11 . . . . . . . . 9 (𝑅 ∈ Ring → (+g𝑁) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅))))
3938oveqdr 6714 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑁)𝑦) = (𝑥(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑦))
406, 21, 23, 28, 30, 31, 39mulgpropd 17631 . . . . . . 7 (𝑅 ∈ Ring → = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))))
41403ad2ant1 1102 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))))
42 eqidd 2652 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐷 = 𝐷)
433vr1val 19610 . . . . . . 7 𝑋 = ((1𝑜 mVar 𝑅)‘∅)
4443a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝑋 = ((1𝑜 mVar 𝑅)‘∅))
4541, 42, 44oveq123d 6711 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐷 𝑋) = (𝐷(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
4645oveq2d 6706 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) = (𝐶 · (𝐷(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅))))
47 psr1baslem 19603 . . . . . 6 (ℕ0𝑚 1𝑜) = {𝑎 ∈ (ℕ0𝑚 1𝑜) ∣ (𝑎 “ ℕ) ∈ Fin}
48 coe1tm.z . . . . . 6 0 = (0g𝑅)
49 eqid 2651 . . . . . 6 (1r𝑅) = (1r𝑅)
5015a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 1𝑜 ∈ On)
51 eqid 2651 . . . . . 6 (1𝑜 mVar 𝑅) = (1𝑜 mVar 𝑅)
52 simp1 1081 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝑅 ∈ Ring)
53 0lt1o 7629 . . . . . . 7 ∅ ∈ 1𝑜
5453a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ∅ ∈ 1𝑜)
55 simp3 1083 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐷 ∈ ℕ0)
5634, 47, 48, 49, 50, 24, 21, 51, 52, 54, 55mplcoe3 19514 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑦 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), (1r𝑅), 0 )) = (𝐷(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
5756oveq2d 6706 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝑦 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), (1r𝑅), 0 ))) = (𝐶 · (𝐷(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅))))
582, 34, 4ply1vsca 19644 . . . . 5 · = ( ·𝑠 ‘(1𝑜 mPoly 𝑅))
59 elsni 4227 . . . . . . . . . . 11 (𝑏 ∈ {∅} → 𝑏 = ∅)
60 df1o2 7617 . . . . . . . . . . 11 1𝑜 = {∅}
6159, 60eleq2s 2748 . . . . . . . . . 10 (𝑏 ∈ 1𝑜𝑏 = ∅)
6261iftrued 4127 . . . . . . . . 9 (𝑏 ∈ 1𝑜 → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
6362adantl 481 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑏 ∈ 1𝑜) → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
6463mpteq2dva 4777 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) = (𝑏 ∈ 1𝑜𝐷))
65 fconstmpt 5197 . . . . . . 7 (1𝑜 × {𝐷}) = (𝑏 ∈ 1𝑜𝐷)
6664, 65syl6eqr 2703 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) = (1𝑜 × {𝐷}))
67 fconst6g 6132 . . . . . . . 8 (𝐷 ∈ ℕ0 → (1𝑜 × {𝐷}):1𝑜⟶ℕ0)
6814, 16elmap 7928 . . . . . . . 8 ((1𝑜 × {𝐷}) ∈ (ℕ0𝑚 1𝑜) ↔ (1𝑜 × {𝐷}):1𝑜⟶ℕ0)
6967, 68sylibr 224 . . . . . . 7 (𝐷 ∈ ℕ0 → (1𝑜 × {𝐷}) ∈ (ℕ0𝑚 1𝑜))
70693ad2ant3 1104 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (1𝑜 × {𝐷}) ∈ (ℕ0𝑚 1𝑜))
7166, 70eqeltrd 2730 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ∈ (ℕ0𝑚 1𝑜))
72 simp2 1082 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐶𝐾)
7334, 58, 47, 49, 48, 1, 50, 52, 71, 72mplmon2 19541 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝑦 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), (1r𝑅), 0 ))) = (𝑦 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )))
7446, 57, 733eqtr2d 2691 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) = (𝑦 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )))
75 eqeq1 2655 . . . 4 (𝑦 = (1𝑜 × {𝑥}) → (𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0))))
7675ifbid 4141 . . 3 (𝑦 = (1𝑜 × {𝑥}) → if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 ) = if((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 ))
7719, 20, 74, 76fmptco 6436 . 2 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ((𝐶 · (𝐷 𝑋)) ∘ (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥}))) = (𝑥 ∈ ℕ0 ↦ if((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )))
7866adantr 480 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) = (1𝑜 × {𝐷}))
7978eqeq2d 2661 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1𝑜 × {𝑥}) = (1𝑜 × {𝐷})))
80 fveq1 6228 . . . . . . 7 ((1𝑜 × {𝑥}) = (1𝑜 × {𝐷}) → ((1𝑜 × {𝑥})‘∅) = ((1𝑜 × {𝐷})‘∅))
81 vex 3234 . . . . . . . . . 10 𝑥 ∈ V
8281fvconst2 6510 . . . . . . . . 9 (∅ ∈ 1𝑜 → ((1𝑜 × {𝑥})‘∅) = 𝑥)
8353, 82mp1i 13 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥})‘∅) = 𝑥)
84 simpl3 1086 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝐷 ∈ ℕ0)
85 fvconst2g 6508 . . . . . . . . 9 ((𝐷 ∈ ℕ0 ∧ ∅ ∈ 1𝑜) → ((1𝑜 × {𝐷})‘∅) = 𝐷)
8684, 53, 85sylancl 695 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝐷})‘∅) = 𝐷)
8783, 86eqeq12d 2666 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (((1𝑜 × {𝑥})‘∅) = ((1𝑜 × {𝐷})‘∅) ↔ 𝑥 = 𝐷))
8880, 87syl5ib 234 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (1𝑜 × {𝐷}) → 𝑥 = 𝐷))
89 sneq 4220 . . . . . . 7 (𝑥 = 𝐷 → {𝑥} = {𝐷})
9089xpeq2d 5173 . . . . . 6 (𝑥 = 𝐷 → (1𝑜 × {𝑥}) = (1𝑜 × {𝐷}))
9188, 90impbid1 215 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (1𝑜 × {𝐷}) ↔ 𝑥 = 𝐷))
9279, 91bitrd 268 . . . 4 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ 𝑥 = 𝐷))
9392ifbid 4141 . . 3 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → if((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 ) = if(𝑥 = 𝐷, 𝐶, 0 ))
9493mpteq2dva 4777 . 2 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑥 ∈ ℕ0 ↦ if((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )))
9512, 77, 943eqtrd 2689 1 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (coe1‘(𝐶 · (𝐷 𝑋))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 𝐷, 𝐶, 0 )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607  ∅c0 3948  ifcif 4119  {csn 4210   ↦ cmpt 4762   × cxp 5141   ∘ ccom 5147  Oncon0 5761  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  1𝑜c1o 7598   ↑𝑚 cmap 7899  0cc0 9974  ℕ0cn0 11330  Basecbs 15904  +gcplusg 15988  .rcmulr 15989   ·𝑠 cvsca 15992  0gc0g 16147  .gcmg 17587  mulGrpcmgp 18535  1rcur 18547  Ringcrg 18593   mVar cmvr 19400   mPoly cmpl 19401  PwSer1cps1 19593  var1cv1 19594  Poly1cpl1 19595  coe1cco1 19596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-tset 16007  df-ple 16008  df-0g 16149  df-gsum 16150  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-subrg 18826  df-lmod 18913  df-lss 18981  df-psr 19404  df-mvr 19405  df-mpl 19406  df-opsr 19408  df-psr1 19598  df-vr1 19599  df-ply1 19600  df-coe1 19601 This theorem is referenced by:  coe1tmfv1  19692  coe1tmfv2  19693  coe1scl  19705  gsummoncoe1  19722  decpmatid  20623  monmatcollpw  20632  mp2pm2mplem4  20662
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