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Theorem coe1tm 22185
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 2727 . . . 4 (Base‘𝑃) = (Base‘𝑃)
81, 2, 3, 4, 5, 6, 7ply1tmcl 22184 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ (Base‘𝑃))
9 eqid 2727 . . . 4 (coe1‘(𝐶 · (𝐷 𝑋))) = (coe1‘(𝐶 · (𝐷 𝑋)))
10 eqid 2727 . . . 4 (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1o × {𝑥}))
119, 7, 2, 10coe1fval2 22122 . . 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 6780 . . . . 5 (𝑥 ∈ ℕ0 → (1o × {𝑥}):1o⟶ℕ0)
14 nn0ex 12502 . . . . . 6 0 ∈ V
15 1oex 8490 . . . . . 6 1o ∈ V
1614, 15elmap 8883 . . . . 5 ((1o × {𝑥}) ∈ (ℕ0m 1o) ↔ (1o × {𝑥}):1o⟶ℕ0)
1713, 16sylibr 233 . . . 4 (𝑥 ∈ ℕ0 → (1o × {𝑥}) ∈ (ℕ0m 1o))
1817adantl 481 . . 3 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (1o × {𝑥}) ∈ (ℕ0m 1o))
19 eqidd 2728 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})))
20 eqid 2727 . . . . . . . 8 (.g‘(mulGrp‘(1o mPoly 𝑅))) = (.g‘(mulGrp‘(1o mPoly 𝑅)))
215, 7mgpbas 20073 . . . . . . . . 9 (Base‘𝑃) = (Base‘𝑁)
2221a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘𝑁))
23 eqid 2727 . . . . . . . . . 10 (mulGrp‘(1o mPoly 𝑅)) = (mulGrp‘(1o mPoly 𝑅))
24 eqid 2727 . . . . . . . . . . 11 (PwSer1𝑅) = (PwSer1𝑅)
252, 24, 7ply1bas 22107 . . . . . . . . . 10 (Base‘𝑃) = (Base‘(1o mPoly 𝑅))
2623, 25mgpbas 20073 . . . . . . . . 9 (Base‘𝑃) = (Base‘(mulGrp‘(1o mPoly 𝑅)))
2726a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘(mulGrp‘(1o mPoly 𝑅))))
28 ssv 4002 . . . . . . . . 9 (Base‘𝑃) ⊆ V
2928a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) ⊆ V)
30 ovexd 7449 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑁)𝑦) ∈ V)
31 eqid 2727 . . . . . . . . . . . 12 (.r𝑃) = (.r𝑃)
325, 31mgpplusg 20071 . . . . . . . . . . 11 (.r𝑃) = (+g𝑁)
33 eqid 2727 . . . . . . . . . . . . 13 (1o mPoly 𝑅) = (1o mPoly 𝑅)
342, 33, 31ply1mulr 22137 . . . . . . . . . . . 12 (.r𝑃) = (.r‘(1o mPoly 𝑅))
3523, 34mgpplusg 20071 . . . . . . . . . . 11 (.r𝑃) = (+g‘(mulGrp‘(1o mPoly 𝑅)))
3632, 35eqtr3i 2757 . . . . . . . . . 10 (+g𝑁) = (+g‘(mulGrp‘(1o mPoly 𝑅)))
3736a1i 11 . . . . . . . . 9 (𝑅 ∈ Ring → (+g𝑁) = (+g‘(mulGrp‘(1o mPoly 𝑅))))
3837oveqdr 7442 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑁)𝑦) = (𝑥(+g‘(mulGrp‘(1o mPoly 𝑅)))𝑦))
396, 20, 22, 27, 29, 30, 38mulgpropd 19064 . . . . . . 7 (𝑅 ∈ Ring → = (.g‘(mulGrp‘(1o mPoly 𝑅))))
40393ad2ant1 1131 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → = (.g‘(mulGrp‘(1o mPoly 𝑅))))
41 eqidd 2728 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐷 = 𝐷)
423vr1val 22104 . . . . . . 7 𝑋 = ((1o mVar 𝑅)‘∅)
4342a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝑋 = ((1o mVar 𝑅)‘∅))
4440, 41, 43oveq123d 7435 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐷 𝑋) = (𝐷(.g‘(mulGrp‘(1o mPoly 𝑅)))((1o mVar 𝑅)‘∅)))
4544oveq2d 7430 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) = (𝐶 · (𝐷(.g‘(mulGrp‘(1o mPoly 𝑅)))((1o mVar 𝑅)‘∅))))
46 psr1baslem 22097 . . . . . 6 (ℕ0m 1o) = {𝑎 ∈ (ℕ0m 1o) ∣ (𝑎 “ ℕ) ∈ Fin}
47 coe1tm.z . . . . . 6 0 = (0g𝑅)
48 eqid 2727 . . . . . 6 (1r𝑅) = (1r𝑅)
49 1on 8492 . . . . . . 7 1o ∈ On
5049a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 1o ∈ On)
51 eqid 2727 . . . . . 6 (1o mVar 𝑅) = (1o mVar 𝑅)
52 simp1 1134 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝑅 ∈ Ring)
53 0lt1o 8518 . . . . . . 7 ∅ ∈ 1o
5453a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ∅ ∈ 1o)
55 simp3 1136 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐷 ∈ ℕ0)
5633, 46, 47, 48, 50, 23, 20, 51, 52, 54, 55mplcoe3 21969 . . . . 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 7430 . . . 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, 33, 4ply1vsca 22136 . . . . 5 · = ( ·𝑠 ‘(1o mPoly 𝑅))
59 elsni 4641 . . . . . . . . . . 11 (𝑏 ∈ {∅} → 𝑏 = ∅)
60 df1o2 8487 . . . . . . . . . . 11 1o = {∅}
6159, 60eleq2s 2846 . . . . . . . . . 10 (𝑏 ∈ 1o𝑏 = ∅)
6261iftrued 4532 . . . . . . . . 9 (𝑏 ∈ 1o → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
6362adantl 481 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑏 ∈ 1o) → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
6463mpteq2dva 5242 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) = (𝑏 ∈ 1o𝐷))
65 fconstmpt 5734 . . . . . . 7 (1o × {𝐷}) = (𝑏 ∈ 1o𝐷)
6664, 65eqtr4di 2785 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) = (1o × {𝐷}))
67 fconst6g 6780 . . . . . . . 8 (𝐷 ∈ ℕ0 → (1o × {𝐷}):1o⟶ℕ0)
6814, 15elmap 8883 . . . . . . . 8 ((1o × {𝐷}) ∈ (ℕ0m 1o) ↔ (1o × {𝐷}):1o⟶ℕ0)
6967, 68sylibr 233 . . . . . . 7 (𝐷 ∈ ℕ0 → (1o × {𝐷}) ∈ (ℕ0m 1o))
70693ad2ant3 1133 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (1o × {𝐷}) ∈ (ℕ0m 1o))
7166, 70eqeltrd 2828 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ∈ (ℕ0m 1o))
72 simp2 1135 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐶𝐾)
7333, 58, 46, 48, 47, 1, 50, 52, 71, 72mplmon2 21998 . . . 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 )))
7445, 57, 733eqtr2d 2773 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) = (𝑦 ∈ (ℕ0m 1o) ↦ if(𝑦 = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )))
75 eqeq1 2731 . . . 4 (𝑦 = (1o × {𝑥}) → (𝑦 = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0))))
7675ifbid 4547 . . 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 ))
7718, 19, 74, 76fmptco 7132 . 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 2738 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1o × {𝑥}) = (1o × {𝐷})))
80 fveq1 6890 . . . . . . 7 ((1o × {𝑥}) = (1o × {𝐷}) → ((1o × {𝑥})‘∅) = ((1o × {𝐷})‘∅))
81 vex 3473 . . . . . . . . . 10 𝑥 ∈ V
8281fvconst2 7210 . . . . . . . . 9 (∅ ∈ 1o → ((1o × {𝑥})‘∅) = 𝑥)
8353, 82mp1i 13 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥})‘∅) = 𝑥)
84 simpl3 1191 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝐷 ∈ ℕ0)
85 fvconst2g 7208 . . . . . . . . 9 ((𝐷 ∈ ℕ0 ∧ ∅ ∈ 1o) → ((1o × {𝐷})‘∅) = 𝐷)
8684, 53, 85sylancl 585 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝐷})‘∅) = 𝐷)
8783, 86eqeq12d 2743 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (((1o × {𝑥})‘∅) = ((1o × {𝐷})‘∅) ↔ 𝑥 = 𝐷))
8880, 87imbitrid 243 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (1o × {𝐷}) → 𝑥 = 𝐷))
89 sneq 4634 . . . . . . 7 (𝑥 = 𝐷 → {𝑥} = {𝐷})
9089xpeq2d 5702 . . . . . 6 (𝑥 = 𝐷 → (1o × {𝑥}) = (1o × {𝐷}))
9188, 90impbid1 224 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (1o × {𝐷}) ↔ 𝑥 = 𝐷))
9279, 91bitrd 279 . . . 4 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ 𝑥 = 𝐷))
9392ifbid 4547 . . 3 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → if((1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 ) = if(𝑥 = 𝐷, 𝐶, 0 ))
9493mpteq2dva 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 )))
9512, 77, 943eqtrd 2771 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 1085   = wceq 1534  wcel 2099  Vcvv 3469  wss 3944  c0 4318  ifcif 4524  {csn 4624  cmpt 5225   × cxp 5670  ccom 5676  Oncon0 6363  wf 6538  cfv 6542  (class class class)co 7414  1oc1o 8473  m cmap 8838  0cc0 11132  0cn0 12496  Basecbs 17173  +gcplusg 17226  .rcmulr 17227   ·𝑠 cvsca 17230  0gc0g 17414  .gcmg 19016  mulGrpcmgp 20067  1rcur 20114  Ringcrg 20166   mVar cmvr 21831   mPoly cmpl 21832  PwSer1cps1 22087  var1cv1 22088  Poly1cpl1 22089  coe1cco1 22090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-ofr 7680  df-om 7865  df-1st 7987  df-2nd 7988  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-pm 8841  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9380  df-sup 9459  df-oi 9527  df-card 9956  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-nn 12237  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12497  df-z 12583  df-dec 12702  df-uz 12847  df-fz 13511  df-fzo 13654  df-seq 13993  df-hash 14316  df-struct 17109  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17174  df-ress 17203  df-plusg 17239  df-mulr 17240  df-sca 17242  df-vsca 17243  df-ip 17244  df-tset 17245  df-ple 17246  df-ds 17248  df-hom 17250  df-cco 17251  df-0g 17416  df-gsum 17417  df-prds 17422  df-pws 17424  df-mre 17559  df-mrc 17560  df-acs 17562  df-mgm 18593  df-sgrp 18672  df-mnd 18688  df-mhm 18733  df-submnd 18734  df-grp 18886  df-minusg 18887  df-sbg 18888  df-mulg 19017  df-subg 19071  df-ghm 19161  df-cntz 19261  df-cmn 19730  df-abl 19731  df-mgp 20068  df-rng 20086  df-ur 20115  df-ring 20168  df-subrng 20476  df-subrg 20501  df-lmod 20738  df-lss 20809  df-psr 21835  df-mvr 21836  df-mpl 21837  df-opsr 21839  df-psr1 22092  df-vr1 22093  df-ply1 22094  df-coe1 22095
This theorem is referenced by:  coe1tmfv1  22186  coe1tmfv2  22187  coe1scl  22199  gsummoncoe1  22220  decpmatid  22665  monmatcollpw  22674  mp2pm2mplem4  22704  coe1mon  33249
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