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Theorem coe1tm 22276
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 2737 . . . 4 (Base‘𝑃) = (Base‘𝑃)
81, 2, 3, 4, 5, 6, 7ply1tmcl 22275 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ (Base‘𝑃))
9 eqid 2737 . . . 4 (coe1‘(𝐶 · (𝐷 𝑋))) = (coe1‘(𝐶 · (𝐷 𝑋)))
10 eqid 2737 . . . 4 (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1o × {𝑥}))
119, 7, 2, 10coe1fval2 22212 . . 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 6797 . . . . 5 (𝑥 ∈ ℕ0 → (1o × {𝑥}):1o⟶ℕ0)
14 nn0ex 12532 . . . . . 6 0 ∈ V
15 1oex 8516 . . . . . 6 1o ∈ V
1614, 15elmap 8911 . . . . 5 ((1o × {𝑥}) ∈ (ℕ0m 1o) ↔ (1o × {𝑥}):1o⟶ℕ0)
1713, 16sylibr 234 . . . 4 (𝑥 ∈ ℕ0 → (1o × {𝑥}) ∈ (ℕ0m 1o))
1817adantl 481 . . 3 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (1o × {𝑥}) ∈ (ℕ0m 1o))
19 eqidd 2738 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})))
20 eqid 2737 . . . . . . . 8 (.g‘(mulGrp‘(1o mPoly 𝑅))) = (.g‘(mulGrp‘(1o mPoly 𝑅)))
215, 7mgpbas 20142 . . . . . . . . 9 (Base‘𝑃) = (Base‘𝑁)
2221a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘𝑁))
23 eqid 2737 . . . . . . . . . 10 (mulGrp‘(1o mPoly 𝑅)) = (mulGrp‘(1o mPoly 𝑅))
242, 7ply1bas 22196 . . . . . . . . . 10 (Base‘𝑃) = (Base‘(1o mPoly 𝑅))
2523, 24mgpbas 20142 . . . . . . . . 9 (Base‘𝑃) = (Base‘(mulGrp‘(1o mPoly 𝑅)))
2625a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘(mulGrp‘(1o mPoly 𝑅))))
27 ssv 4008 . . . . . . . . 9 (Base‘𝑃) ⊆ V
2827a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) ⊆ V)
29 ovexd 7466 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑁)𝑦) ∈ V)
30 eqid 2737 . . . . . . . . . . . 12 (.r𝑃) = (.r𝑃)
315, 30mgpplusg 20141 . . . . . . . . . . 11 (.r𝑃) = (+g𝑁)
32 eqid 2737 . . . . . . . . . . . . 13 (1o mPoly 𝑅) = (1o mPoly 𝑅)
332, 32, 30ply1mulr 22227 . . . . . . . . . . . 12 (.r𝑃) = (.r‘(1o mPoly 𝑅))
3423, 33mgpplusg 20141 . . . . . . . . . . 11 (.r𝑃) = (+g‘(mulGrp‘(1o mPoly 𝑅)))
3531, 34eqtr3i 2767 . . . . . . . . . 10 (+g𝑁) = (+g‘(mulGrp‘(1o mPoly 𝑅)))
3635a1i 11 . . . . . . . . 9 (𝑅 ∈ Ring → (+g𝑁) = (+g‘(mulGrp‘(1o mPoly 𝑅))))
3736oveqdr 7459 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑁)𝑦) = (𝑥(+g‘(mulGrp‘(1o mPoly 𝑅)))𝑦))
386, 20, 22, 26, 28, 29, 37mulgpropd 19134 . . . . . . 7 (𝑅 ∈ Ring → = (.g‘(mulGrp‘(1o mPoly 𝑅))))
39383ad2ant1 1134 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → = (.g‘(mulGrp‘(1o mPoly 𝑅))))
40 eqidd 2738 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐷 = 𝐷)
413vr1val 22193 . . . . . . 7 𝑋 = ((1o mVar 𝑅)‘∅)
4241a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝑋 = ((1o mVar 𝑅)‘∅))
4339, 40, 42oveq123d 7452 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐷 𝑋) = (𝐷(.g‘(mulGrp‘(1o mPoly 𝑅)))((1o mVar 𝑅)‘∅)))
4443oveq2d 7447 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) = (𝐶 · (𝐷(.g‘(mulGrp‘(1o mPoly 𝑅)))((1o mVar 𝑅)‘∅))))
45 psr1baslem 22186 . . . . . 6 (ℕ0m 1o) = {𝑎 ∈ (ℕ0m 1o) ∣ (𝑎 “ ℕ) ∈ Fin}
46 coe1tm.z . . . . . 6 0 = (0g𝑅)
47 eqid 2737 . . . . . 6 (1r𝑅) = (1r𝑅)
48 1on 8518 . . . . . . 7 1o ∈ On
4948a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 1o ∈ On)
50 eqid 2737 . . . . . 6 (1o mVar 𝑅) = (1o mVar 𝑅)
51 simp1 1137 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝑅 ∈ Ring)
52 0lt1o 8542 . . . . . . 7 ∅ ∈ 1o
5352a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ∅ ∈ 1o)
54 simp3 1139 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐷 ∈ ℕ0)
5532, 45, 46, 47, 49, 23, 20, 50, 51, 53, 54mplcoe3 22056 . . . . 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)‘∅)))
5655oveq2d 7447 . . . 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)‘∅))))
572, 32, 4ply1vsca 22226 . . . . 5 · = ( ·𝑠 ‘(1o mPoly 𝑅))
58 elsni 4643 . . . . . . . . . . 11 (𝑏 ∈ {∅} → 𝑏 = ∅)
59 df1o2 8513 . . . . . . . . . . 11 1o = {∅}
6058, 59eleq2s 2859 . . . . . . . . . 10 (𝑏 ∈ 1o𝑏 = ∅)
6160iftrued 4533 . . . . . . . . 9 (𝑏 ∈ 1o → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
6261adantl 481 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑏 ∈ 1o) → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
6362mpteq2dva 5242 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) = (𝑏 ∈ 1o𝐷))
64 fconstmpt 5747 . . . . . . 7 (1o × {𝐷}) = (𝑏 ∈ 1o𝐷)
6563, 64eqtr4di 2795 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) = (1o × {𝐷}))
66 fconst6g 6797 . . . . . . . 8 (𝐷 ∈ ℕ0 → (1o × {𝐷}):1o⟶ℕ0)
6714, 15elmap 8911 . . . . . . . 8 ((1o × {𝐷}) ∈ (ℕ0m 1o) ↔ (1o × {𝐷}):1o⟶ℕ0)
6866, 67sylibr 234 . . . . . . 7 (𝐷 ∈ ℕ0 → (1o × {𝐷}) ∈ (ℕ0m 1o))
69683ad2ant3 1136 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (1o × {𝐷}) ∈ (ℕ0m 1o))
7065, 69eqeltrd 2841 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ∈ (ℕ0m 1o))
71 simp2 1138 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐶𝐾)
7232, 57, 45, 47, 46, 1, 49, 51, 70, 71mplmon2 22085 . . . 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 )))
7344, 56, 723eqtr2d 2783 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) = (𝑦 ∈ (ℕ0m 1o) ↦ if(𝑦 = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )))
74 eqeq1 2741 . . . 4 (𝑦 = (1o × {𝑥}) → (𝑦 = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0))))
7574ifbid 4549 . . 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 ))
7618, 19, 73, 75fmptco 7149 . 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 )))
7765adantr 480 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) = (1o × {𝐷}))
7877eqeq2d 2748 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1o × {𝑥}) = (1o × {𝐷})))
79 fveq1 6905 . . . . . . 7 ((1o × {𝑥}) = (1o × {𝐷}) → ((1o × {𝑥})‘∅) = ((1o × {𝐷})‘∅))
80 vex 3484 . . . . . . . . . 10 𝑥 ∈ V
8180fvconst2 7224 . . . . . . . . 9 (∅ ∈ 1o → ((1o × {𝑥})‘∅) = 𝑥)
8252, 81mp1i 13 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥})‘∅) = 𝑥)
83 simpl3 1194 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝐷 ∈ ℕ0)
84 fvconst2g 7222 . . . . . . . . 9 ((𝐷 ∈ ℕ0 ∧ ∅ ∈ 1o) → ((1o × {𝐷})‘∅) = 𝐷)
8583, 52, 84sylancl 586 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝐷})‘∅) = 𝐷)
8682, 85eqeq12d 2753 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (((1o × {𝑥})‘∅) = ((1o × {𝐷})‘∅) ↔ 𝑥 = 𝐷))
8779, 86imbitrid 244 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (1o × {𝐷}) → 𝑥 = 𝐷))
88 sneq 4636 . . . . . . 7 (𝑥 = 𝐷 → {𝑥} = {𝐷})
8988xpeq2d 5715 . . . . . 6 (𝑥 = 𝐷 → (1o × {𝑥}) = (1o × {𝐷}))
9087, 89impbid1 225 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (1o × {𝐷}) ↔ 𝑥 = 𝐷))
9178, 90bitrd 279 . . . 4 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ 𝑥 = 𝐷))
9291ifbid 4549 . . 3 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → if((1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 ) = if(𝑥 = 𝐷, 𝐶, 0 ))
9392mpteq2dva 5242 . 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 )))
9412, 76, 933eqtrd 2781 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 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  wss 3951  c0 4333  ifcif 4525  {csn 4626  cmpt 5225   × cxp 5683  ccom 5689  Oncon0 6384  wf 6557  cfv 6561  (class class class)co 7431  1oc1o 8499  m cmap 8866  0cc0 11155  0cn0 12526  Basecbs 17247  +gcplusg 17297  .rcmulr 17298   ·𝑠 cvsca 17301  0gc0g 17484  .gcmg 19085  mulGrpcmgp 20137  1rcur 20178  Ringcrg 20230   mVar cmvr 21925   mPoly cmpl 21926  var1cv1 22177  Poly1cpl1 22178  coe1cco1 22179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184
This theorem is referenced by:  coe1tmfv1  22277  coe1tmfv2  22278  coe1scl  22290  gsummoncoe1  22312  decpmatid  22776  monmatcollpw  22785  mp2pm2mplem4  22815  coe1mon  33610
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