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Theorem coe1tm 20441
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 2824 . . . 4 (Base‘𝑃) = (Base‘𝑃)
81, 2, 3, 4, 5, 6, 7ply1tmcl 20440 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) ∈ (Base‘𝑃))
9 eqid 2824 . . . 4 (coe1‘(𝐶 · (𝐷 𝑋))) = (coe1‘(𝐶 · (𝐷 𝑋)))
10 eqid 2824 . . . 4 (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1o × {𝑥}))
119, 7, 2, 10coe1fval2 20378 . . 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 6558 . . . . 5 (𝑥 ∈ ℕ0 → (1o × {𝑥}):1o⟶ℕ0)
14 nn0ex 11900 . . . . . 6 0 ∈ V
15 1oex 8106 . . . . . 6 1o ∈ V
1614, 15elmap 8431 . . . . 5 ((1o × {𝑥}) ∈ (ℕ0m 1o) ↔ (1o × {𝑥}):1o⟶ℕ0)
1713, 16sylibr 237 . . . 4 (𝑥 ∈ ℕ0 → (1o × {𝑥}) ∈ (ℕ0m 1o))
1817adantl 485 . . 3 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (1o × {𝑥}) ∈ (ℕ0m 1o))
19 eqidd 2825 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})) = (𝑥 ∈ ℕ0 ↦ (1o × {𝑥})))
20 eqid 2824 . . . . . . . 8 (.g‘(mulGrp‘(1o mPoly 𝑅))) = (.g‘(mulGrp‘(1o mPoly 𝑅)))
215, 7mgpbas 19245 . . . . . . . . 9 (Base‘𝑃) = (Base‘𝑁)
2221a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘𝑁))
23 eqid 2824 . . . . . . . . . 10 (mulGrp‘(1o mPoly 𝑅)) = (mulGrp‘(1o mPoly 𝑅))
24 eqid 2824 . . . . . . . . . . 11 (PwSer1𝑅) = (PwSer1𝑅)
252, 24, 7ply1bas 20363 . . . . . . . . . 10 (Base‘𝑃) = (Base‘(1o mPoly 𝑅))
2623, 25mgpbas 19245 . . . . . . . . 9 (Base‘𝑃) = (Base‘(mulGrp‘(1o mPoly 𝑅)))
2726a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) = (Base‘(mulGrp‘(1o mPoly 𝑅))))
28 ssv 3977 . . . . . . . . 9 (Base‘𝑃) ⊆ V
2928a1i 11 . . . . . . . 8 (𝑅 ∈ Ring → (Base‘𝑃) ⊆ V)
30 ovexd 7184 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑁)𝑦) ∈ V)
31 eqid 2824 . . . . . . . . . . . 12 (.r𝑃) = (.r𝑃)
325, 31mgpplusg 19243 . . . . . . . . . . 11 (.r𝑃) = (+g𝑁)
33 eqid 2824 . . . . . . . . . . . . 13 (1o mPoly 𝑅) = (1o mPoly 𝑅)
342, 33, 31ply1mulr 20395 . . . . . . . . . . . 12 (.r𝑃) = (.r‘(1o mPoly 𝑅))
3523, 34mgpplusg 19243 . . . . . . . . . . 11 (.r𝑃) = (+g‘(mulGrp‘(1o mPoly 𝑅)))
3632, 35eqtr3i 2849 . . . . . . . . . 10 (+g𝑁) = (+g‘(mulGrp‘(1o mPoly 𝑅)))
3736a1i 11 . . . . . . . . 9 (𝑅 ∈ Ring → (+g𝑁) = (+g‘(mulGrp‘(1o mPoly 𝑅))))
3837oveqdr 7177 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑁)𝑦) = (𝑥(+g‘(mulGrp‘(1o mPoly 𝑅)))𝑦))
396, 20, 22, 27, 29, 30, 38mulgpropd 18269 . . . . . . 7 (𝑅 ∈ Ring → = (.g‘(mulGrp‘(1o mPoly 𝑅))))
40393ad2ant1 1130 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → = (.g‘(mulGrp‘(1o mPoly 𝑅))))
41 eqidd 2825 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐷 = 𝐷)
423vr1val 20360 . . . . . . 7 𝑋 = ((1o mVar 𝑅)‘∅)
4342a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝑋 = ((1o mVar 𝑅)‘∅))
4440, 41, 43oveq123d 7170 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐷 𝑋) = (𝐷(.g‘(mulGrp‘(1o mPoly 𝑅)))((1o mVar 𝑅)‘∅)))
4544oveq2d 7165 . . . 4 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) = (𝐶 · (𝐷(.g‘(mulGrp‘(1o mPoly 𝑅)))((1o mVar 𝑅)‘∅))))
46 psr1baslem 20353 . . . . . 6 (ℕ0m 1o) = {𝑎 ∈ (ℕ0m 1o) ∣ (𝑎 “ ℕ) ∈ Fin}
47 coe1tm.z . . . . . 6 0 = (0g𝑅)
48 eqid 2824 . . . . . 6 (1r𝑅) = (1r𝑅)
49 1on 8105 . . . . . . 7 1o ∈ On
5049a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 1o ∈ On)
51 eqid 2824 . . . . . 6 (1o mVar 𝑅) = (1o mVar 𝑅)
52 simp1 1133 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝑅 ∈ Ring)
53 0lt1o 8125 . . . . . . 7 ∅ ∈ 1o
5453a1i 11 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → ∅ ∈ 1o)
55 simp3 1135 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐷 ∈ ℕ0)
5633, 46, 47, 48, 50, 23, 20, 51, 52, 54, 55mplcoe3 20247 . . . . 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 7165 . . . 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 20394 . . . . 5 · = ( ·𝑠 ‘(1o mPoly 𝑅))
59 elsni 4567 . . . . . . . . . . 11 (𝑏 ∈ {∅} → 𝑏 = ∅)
60 df1o2 8112 . . . . . . . . . . 11 1o = {∅}
6159, 60eleq2s 2934 . . . . . . . . . 10 (𝑏 ∈ 1o𝑏 = ∅)
6261iftrued 4458 . . . . . . . . 9 (𝑏 ∈ 1o → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
6362adantl 485 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑏 ∈ 1o) → if(𝑏 = ∅, 𝐷, 0) = 𝐷)
6463mpteq2dva 5147 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) = (𝑏 ∈ 1o𝐷))
65 fconstmpt 5601 . . . . . . 7 (1o × {𝐷}) = (𝑏 ∈ 1o𝐷)
6664, 65syl6eqr 2877 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) = (1o × {𝐷}))
67 fconst6g 6558 . . . . . . . 8 (𝐷 ∈ ℕ0 → (1o × {𝐷}):1o⟶ℕ0)
6814, 15elmap 8431 . . . . . . . 8 ((1o × {𝐷}) ∈ (ℕ0m 1o) ↔ (1o × {𝐷}):1o⟶ℕ0)
6967, 68sylibr 237 . . . . . . 7 (𝐷 ∈ ℕ0 → (1o × {𝐷}) ∈ (ℕ0m 1o))
70693ad2ant3 1132 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (1o × {𝐷}) ∈ (ℕ0m 1o))
7166, 70eqeltrd 2916 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ∈ (ℕ0m 1o))
72 simp2 1134 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → 𝐶𝐾)
7333, 58, 46, 48, 47, 1, 50, 52, 71, 72mplmon2 20273 . . . 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 2865 . . 3 ((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) → (𝐶 · (𝐷 𝑋)) = (𝑦 ∈ (ℕ0m 1o) ↦ if(𝑦 = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 )))
75 eqeq1 2828 . . . 4 (𝑦 = (1o × {𝑥}) → (𝑦 = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0))))
7675ifbid 4472 . . 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 6882 . 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 484 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) = (1o × {𝐷}))
7978eqeq2d 2835 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ (1o × {𝑥}) = (1o × {𝐷})))
80 fveq1 6660 . . . . . . 7 ((1o × {𝑥}) = (1o × {𝐷}) → ((1o × {𝑥})‘∅) = ((1o × {𝐷})‘∅))
81 vex 3483 . . . . . . . . . 10 𝑥 ∈ V
8281fvconst2 6957 . . . . . . . . 9 (∅ ∈ 1o → ((1o × {𝑥})‘∅) = 𝑥)
8353, 82mp1i 13 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥})‘∅) = 𝑥)
84 simpl3 1190 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝐷 ∈ ℕ0)
85 fvconst2g 6955 . . . . . . . . 9 ((𝐷 ∈ ℕ0 ∧ ∅ ∈ 1o) → ((1o × {𝐷})‘∅) = 𝐷)
8684, 53, 85sylancl 589 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝐷})‘∅) = 𝐷)
8783, 86eqeq12d 2840 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (((1o × {𝑥})‘∅) = ((1o × {𝐷})‘∅) ↔ 𝑥 = 𝐷))
8880, 87syl5ib 247 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (1o × {𝐷}) → 𝑥 = 𝐷))
89 sneq 4560 . . . . . . 7 (𝑥 = 𝐷 → {𝑥} = {𝐷})
9089xpeq2d 5572 . . . . . 6 (𝑥 = 𝐷 → (1o × {𝑥}) = (1o × {𝐷}))
9188, 90impbid1 228 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (1o × {𝐷}) ↔ 𝑥 = 𝐷))
9279, 91bitrd 282 . . . 4 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)) ↔ 𝑥 = 𝐷))
9392ifbid 4472 . . 3 (((𝑅 ∈ Ring ∧ 𝐶𝐾𝐷 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → if((1o × {𝑥}) = (𝑏 ∈ 1o ↦ if(𝑏 = ∅, 𝐷, 0)), 𝐶, 0 ) = if(𝑥 = 𝐷, 𝐶, 0 ))
9493mpteq2dva 5147 . 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 2863 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 399  w3a 1084   = wceq 1538  wcel 2115  Vcvv 3480  wss 3919  c0 4276  ifcif 4450  {csn 4550  cmpt 5132   × cxp 5540  ccom 5546  Oncon0 6178  wf 6339  cfv 6343  (class class class)co 7149  1oc1o 8091  m cmap 8402  0cc0 10535  0cn0 11894  Basecbs 16483  +gcplusg 16565  .rcmulr 16566   ·𝑠 cvsca 16569  0gc0g 16713  .gcmg 18224  mulGrpcmgp 19239  1rcur 19251  Ringcrg 19297   mVar cmvr 20132   mPoly cmpl 20133  PwSer1cps1 20343  var1cv1 20344  Poly1cpl1 20345  coe1cco1 20346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-ofr 7404  df-om 7575  df-1st 7684  df-2nd 7685  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fsupp 8831  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-dec 12096  df-uz 12241  df-fz 12895  df-fzo 13038  df-seq 13374  df-hash 13696  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-tset 16584  df-ple 16585  df-0g 16715  df-gsum 16716  df-mre 16857  df-mrc 16858  df-acs 16860  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-submnd 17957  df-grp 18106  df-minusg 18107  df-sbg 18108  df-mulg 18225  df-subg 18276  df-ghm 18356  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-subrg 19533  df-lmod 19636  df-lss 19704  df-psr 20136  df-mvr 20137  df-mpl 20138  df-opsr 20140  df-psr1 20348  df-vr1 20349  df-ply1 20350  df-coe1 20351
This theorem is referenced by:  coe1tmfv1  20442  coe1tmfv2  20443  coe1scl  20455  gsummoncoe1  20472  decpmatid  21378  monmatcollpw  21387  mp2pm2mplem4  21417
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