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Theorem coe1tm 19410
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 2609 . . . 4 (Base‘𝑃) = (Base‘𝑃)
81, 2, 3, 4, 5, 6, 7ply1tmcl 19409 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ (Base‘𝑃))
9 eqid 2609 . . . 4 (coe1‘(𝐶 · (𝐷 𝑋))) = (coe1‘(𝐶 · (𝐷 𝑋)))
10 eqid 2609 . . . 4 (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥}))
119, 7, 2, 10coe1fval2 19347 . . 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 5992 . . . . 5 (𝑥 ∈ ℕ0 → (1𝑜 × {𝑥}):1𝑜⟶ℕ0)
14 nn0ex 11145 . . . . . 6 0 ∈ V
15 1on 7431 . . . . . . 7 1𝑜 ∈ On
1615elexi 3185 . . . . . 6 1𝑜 ∈ V
1714, 16elmap 7749 . . . . 5 ((1𝑜 × {𝑥}) ∈ (ℕ0𝑚 1𝑜) ↔ (1𝑜 × {𝑥}):1𝑜⟶ℕ0)
1813, 17sylibr 222 . . . 4 (𝑥 ∈ ℕ0 → (1𝑜 × {𝑥}) ∈ (ℕ0𝑚 1𝑜))
1918adantl 480 . . 3 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (1𝑜 × {𝑥}) ∈ (ℕ0𝑚 1𝑜))
20 eqidd 2610 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1𝑜 × {𝑥})))
21 eqid 2609 . . . . . . . 8 (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
225, 7mgpbas 18264 . . . . . . . . 9 (Base‘𝑃) = (Base‘𝑁)
2322a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘𝑁))
24 eqid 2609 . . . . . . . . . 10 (mulGrp‘(1𝑜 mPoly 𝑅)) = (mulGrp‘(1𝑜 mPoly 𝑅))
25 eqid 2609 . . . . . . . . . . 11 (PwSer1𝑅) = (PwSer1𝑅)
262, 25, 7ply1bas 19332 . . . . . . . . . 10 (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅))
2724, 26mgpbas 18264 . . . . . . . . 9 (Base‘𝑃) = (Base‘(mulGrp‘(1𝑜 mPoly 𝑅)))
2827a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘(mulGrp‘(1𝑜 mPoly 𝑅))))
29 ssv 3587 . . . . . . . . 9 (Base‘𝑃) ⊆ V
3029a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) ⊆ V)
31 ovex 6555 . . . . . . . . 9 (𝑥(+g𝑁)𝑦) ∈ V
3231a1i 11 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑁)𝑦) ∈ V)
33 eqid 2609 . . . . . . . . . . . 12 (.r𝑃) = (.r𝑃)
345, 33mgpplusg 18262 . . . . . . . . . . 11 (.r𝑃) = (+g𝑁)
35 eqid 2609 . . . . . . . . . . . . 13 (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
362, 35, 33ply1mulr 19364 . . . . . . . . . . . 12 (.r𝑃) = (.r‘(1𝑜 mPoly 𝑅))
3724, 36mgpplusg 18262 . . . . . . . . . . 11 (.r𝑃) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
3834, 37eqtr3i 2633 . . . . . . . . . 10 (+g𝑁) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
3938a1i 11 . . . . . . . . 9 (𝑅 ∈ Ring → (+g𝑁) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅))))
4039oveqdr 6551 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑁)𝑦) = (𝑥(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑦))
416, 21, 23, 28, 30, 32, 40mulgpropd 17353 . . . . . . 7 (𝑅 ∈ Ring → = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))))
42413ad2ant1 1074 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))))
43 eqidd 2610 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐷 = 𝐷)
443vr1val 19329 . . . . . . 7 𝑋 = ((1𝑜 mVar 𝑅)‘∅)
4544a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝑋 = ((1𝑜 mVar 𝑅)‘∅))
4642, 43, 45oveq123d 6548 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐷 𝑋) = (𝐷(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
4746oveq2d 6543 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) = (𝐶 · (𝐷(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅))))
48 psr1baslem 19322 . . . . . 6 (ℕ0𝑚 1𝑜) = {𝑎 ∈ (ℕ0𝑚 1𝑜) ∣ (𝑎 “ ℕ) ∈ Fin}
49 coe1tm.z . . . . . 6 0 = (0g𝑅)
50 eqid 2609 . . . . . 6 (1r𝑅) = (1r𝑅)
5115a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 1𝑜 ∈ On)
52 eqid 2609 . . . . . 6 (1𝑜 mVar 𝑅) = (1𝑜 mVar 𝑅)
53 simp1 1053 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝑅 ∈ Ring)
54 0lt1o 7448 . . . . . . 7 ∅ ∈ 1𝑜
5554a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ∅ ∈ 1𝑜)
56 simp3 1055 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐷 ∈ ℕ0)
5735, 48, 49, 50, 51, 24, 21, 52, 53, 55, 56mplcoe3 19233 . . . . 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)‘∅)))
5857oveq2d 6543 . . . 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)‘∅))))
592, 35, 4ply1vsca 19363 . . . . 5 · = ( ·𝑠 ‘(1𝑜 mPoly 𝑅))
60 elsni 4141 . . . . . . . . . . 11 (𝑏 ∈ {∅} → 𝑏 = ∅)
61 df1o2 7436 . . . . . . . . . . 11 1𝑜 = {∅}
6260, 61eleq2s 2705 . . . . . . . . . 10 (𝑏 ∈ 1𝑜𝑏 = ∅)
6362iftrued 4043 . . . . . . . . 9 (𝑏 ∈ 1𝑜 → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
6463adantl 480 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑏 ∈ 1𝑜) → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
6564mpteq2dva 4666 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) = (𝑏 ∈ 1𝑜𝐷))
66 fconstmpt 5075 . . . . . . 7 (1𝑜 × {𝐷}) = (𝑏 ∈ 1𝑜𝐷)
6765, 66syl6eqr 2661 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) = (1𝑜 × {𝐷}))
68 fconst6g 5992 . . . . . . . 8 (𝐷 ∈ ℕ0 → (1𝑜 × {𝐷}):1𝑜⟶ℕ0)
6914, 16elmap 7749 . . . . . . . 8 ((1𝑜 × {𝐷}) ∈ (ℕ0𝑚 1𝑜) ↔ (1𝑜 × {𝐷}):1𝑜⟶ℕ0)
7068, 69sylibr 222 . . . . . . 7 (𝐷 ∈ ℕ0 → (1𝑜 × {𝐷}) ∈ (ℕ0𝑚 1𝑜))
71703ad2ant3 1076 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (1𝑜 × {𝐷}) ∈ (ℕ0𝑚 1𝑜))
7267, 71eqeltrd 2687 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ∈ (ℕ0𝑚 1𝑜))
73 simp2 1054 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐶𝐾)
7435, 59, 48, 50, 49, 1, 51, 53, 72, 73mplmon2 19260 . . . 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 )))
7547, 58, 743eqtr2d 2649 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) = (𝑦 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )))
76 eqeq1 2613 . . . 4 (𝑦 = (1𝑜 × {𝑥}) → (𝑦 = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0))))
7776ifbid 4057 . . 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 ))
7819, 20, 75, 77fmptco 6288 . 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 )))
7967adantr 479 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) = (1𝑜 × {𝐷}))
8079eqeq2d 2619 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1𝑜 × {𝑥}) = (1𝑜 × {𝐷})))
81 fveq1 6087 . . . . . . 7 ((1𝑜 × {𝑥}) = (1𝑜 × {𝐷}) → ((1𝑜 × {𝑥})‘∅) = ((1𝑜 × {𝐷})‘∅))
82 vex 3175 . . . . . . . . . 10 𝑥 ∈ V
8382fvconst2 6352 . . . . . . . . 9 (∅ ∈ 1𝑜 → ((1𝑜 × {𝑥})‘∅) = 𝑥)
8454, 83mp1i 13 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥})‘∅) = 𝑥)
85 simpl3 1058 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝐷 ∈ ℕ0)
86 fvconst2g 6350 . . . . . . . . 9 ((𝐷 ∈ ℕ0 ∧ ∅ ∈ 1𝑜) → ((1𝑜 × {𝐷})‘∅) = 𝐷)
8785, 54, 86sylancl 692 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝐷})‘∅) = 𝐷)
8884, 87eqeq12d 2624 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (((1𝑜 × {𝑥})‘∅) = ((1𝑜 × {𝐷})‘∅) ↔ 𝑥 = 𝐷))
8981, 88syl5ib 232 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (1𝑜 × {𝐷}) → 𝑥 = 𝐷))
90 sneq 4134 . . . . . . 7 (𝑥 = 𝐷 → {𝑥} = {𝐷})
9190xpeq2d 5053 . . . . . 6 (𝑥 = 𝐷 → (1𝑜 × {𝑥}) = (1𝑜 × {𝐷}))
9289, 91impbid1 213 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (1𝑜 × {𝐷}) ↔ 𝑥 = 𝐷))
9380, 92bitrd 266 . . . 4 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ 𝑥 = 𝐷))
9493ifbid 4057 . . 3 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → if((1𝑜 × {𝑥}) = (𝑏 ∈ 1𝑜 ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 ) = if(𝑥 = 𝐷, 𝐶, 0 ))
9594mpteq2dva 4666 . 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 )))
9612, 78, 953eqtrd 2647 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 382  w3a 1030   = wceq 1474  wcel 1976  Vcvv 3172  wss 3539  c0 3873  ifcif 4035  {csn 4124  cmpt 4637   × cxp 5026  ccom 5032  Oncon0 5626  wf 5786  cfv 5790  (class class class)co 6527  1𝑜c1o 7417  𝑚 cmap 7721  0cc0 9792  0cn0 11139  Basecbs 15641  +gcplusg 15714  .rcmulr 15715   ·𝑠 cvsca 15718  0gc0g 15869  .gcmg 17309  mulGrpcmgp 18258  1rcur 18270  Ringcrg 18316   mVar cmvr 19119   mPoly cmpl 19120  PwSer1cps1 19312  var1cv1 19313  Poly1cpl1 19314  coe1cco1 19315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-ofr 6773  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-oi 8275  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-fz 12153  df-fzo 12290  df-seq 12619  df-hash 12935  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-mulr 15728  df-sca 15730  df-vsca 15731  df-tset 15733  df-ple 15734  df-0g 15871  df-gsum 15872  df-mre 16015  df-mrc 16016  df-acs 16018  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-mhm 17104  df-submnd 17105  df-grp 17194  df-minusg 17195  df-sbg 17196  df-mulg 17310  df-subg 17360  df-ghm 17427  df-cntz 17519  df-cmn 17964  df-abl 17965  df-mgp 18259  df-ur 18271  df-ring 18318  df-subrg 18547  df-lmod 18634  df-lss 18700  df-psr 19123  df-mvr 19124  df-mpl 19125  df-opsr 19127  df-psr1 19317  df-vr1 19318  df-ply1 19319  df-coe1 19320
This theorem is referenced by:  coe1tmfv1  19411  coe1tmfv2  19412  coe1scl  19424  gsummoncoe1  19441  decpmatid  20336  monmatcollpw  20345  mp2pm2mplem4  20375
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