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Theorem ply1coe 19433
Description: Decompose a univariate polynomial as a sum of powers. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 7-Oct-2019.)
Hypotheses
Ref Expression
ply1coe.p 𝑃 = (Poly1𝑅)
ply1coe.x 𝑋 = (var1𝑅)
ply1coe.b 𝐵 = (Base‘𝑃)
ply1coe.n · = ( ·𝑠𝑃)
ply1coe.m 𝑀 = (mulGrp‘𝑃)
ply1coe.e = (.g𝑀)
ply1coe.a 𝐴 = (coe1𝐾)
Assertion
Ref Expression
ply1coe ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘   𝑘,𝐾   𝑘,𝑋   ,𝑘   𝑅,𝑘   · ,𝑘   𝑃,𝑘
Allowed substitution hint:   𝑀(𝑘)

Proof of Theorem ply1coe
Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2609 . . 3 (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
2 psr1baslem 19322 . . 3 (ℕ0𝑚 1𝑜) = {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ (𝑑 “ ℕ) ∈ Fin}
3 eqid 2609 . . 3 (0g𝑅) = (0g𝑅)
4 eqid 2609 . . 3 (1r𝑅) = (1r𝑅)
5 1onn 7583 . . . 4 1𝑜 ∈ ω
65a1i 11 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 1𝑜 ∈ ω)
7 ply1coe.p . . . 4 𝑃 = (Poly1𝑅)
8 eqid 2609 . . . 4 (PwSer1𝑅) = (PwSer1𝑅)
9 ply1coe.b . . . 4 𝐵 = (Base‘𝑃)
107, 8, 9ply1bas 19332 . . 3 𝐵 = (Base‘(1𝑜 mPoly 𝑅))
11 ply1coe.n . . . 4 · = ( ·𝑠𝑃)
127, 1, 11ply1vsca 19363 . . 3 · = ( ·𝑠 ‘(1𝑜 mPoly 𝑅))
13 simpl 471 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝑅 ∈ Ring)
14 simpr 475 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐾𝐵)
151, 2, 3, 4, 6, 10, 12, 13, 14mplcoe1 19232 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐾 = ((1𝑜 mPoly 𝑅) Σg (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐾𝑎) · (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅)))))))
16 ply1coe.a . . . . . . 7 𝐴 = (coe1𝐾)
1716fvcoe1 19344 . . . . . 6 ((𝐾𝐵𝑎 ∈ (ℕ0𝑚 1𝑜)) → (𝐾𝑎) = (𝐴‘(𝑎‘∅)))
1817adantll 745 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → (𝐾𝑎) = (𝐴‘(𝑎‘∅)))
195a1i 11 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 1𝑜 ∈ ω)
20 eqid 2609 . . . . . . 7 (mulGrp‘(1𝑜 mPoly 𝑅)) = (mulGrp‘(1𝑜 mPoly 𝑅))
21 eqid 2609 . . . . . . 7 (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
22 eqid 2609 . . . . . . 7 (1𝑜 mVar 𝑅) = (1𝑜 mVar 𝑅)
23 simpll 785 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 𝑅 ∈ Ring)
24 simpr 475 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 𝑎 ∈ (ℕ0𝑚 1𝑜))
25 eqidd 2610 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
26 0ex 4713 . . . . . . . . . . 11 ∅ ∈ V
27 fveq2 6088 . . . . . . . . . . . . 13 (𝑏 = ∅ → ((1𝑜 mVar 𝑅)‘𝑏) = ((1𝑜 mVar 𝑅)‘∅))
2827oveq1d 6542 . . . . . . . . . . . 12 (𝑏 = ∅ → (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
2927oveq2d 6543 . . . . . . . . . . . 12 (𝑏 = ∅ → (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
3028, 29eqeq12d 2624 . . . . . . . . . . 11 (𝑏 = ∅ → ((((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅))))
3126, 30ralsn 4168 . . . . . . . . . 10 (∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
3225, 31sylibr 222 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
33 fveq2 6088 . . . . . . . . . . . . 13 (𝑥 = ∅ → ((1𝑜 mVar 𝑅)‘𝑥) = ((1𝑜 mVar 𝑅)‘∅))
3433oveq2d 6543 . . . . . . . . . . . 12 (𝑥 = ∅ → (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)))
3533oveq1d 6542 . . . . . . . . . . . 12 (𝑥 = ∅ → (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
3634, 35eqeq12d 2624 . . . . . . . . . . 11 (𝑥 = ∅ → ((((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏))))
3736ralbidv 2968 . . . . . . . . . 10 (𝑥 = ∅ → (∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ ∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏))))
3826, 37ralsn 4168 . . . . . . . . 9 (∀𝑥 ∈ {∅}∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ ∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘∅)) = (((1𝑜 mVar 𝑅)‘∅)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
3932, 38sylibr 222 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ∀𝑥 ∈ {∅}∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
40 df1o2 7436 . . . . . . . . 9 1𝑜 = {∅}
4140raleqi 3118 . . . . . . . . 9 (∀𝑏 ∈ 1𝑜 (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ ∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
4240, 41raleqbii 2972 . . . . . . . 8 (∀𝑥 ∈ 1𝑜𝑏 ∈ 1𝑜 (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)) ↔ ∀𝑥 ∈ {∅}∀𝑏 ∈ {∅} (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
4339, 42sylibr 222 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ∀𝑥 ∈ 1𝑜𝑏 ∈ 1𝑜 (((1𝑜 mVar 𝑅)‘𝑏)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑥)) = (((1𝑜 mVar 𝑅)‘𝑥)(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑏)))
441, 2, 3, 4, 19, 20, 21, 22, 23, 24, 43mplcoe5 19235 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅))) = ((mulGrp‘(1𝑜 mPoly 𝑅)) Σg (𝑐 ∈ 1𝑜 ↦ ((𝑎𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)))))
45 mpteq1 4659 . . . . . . . . 9 (1𝑜 = {∅} → (𝑐 ∈ 1𝑜 ↦ ((𝑎𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐))) = (𝑐 ∈ {∅} ↦ ((𝑎𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐))))
4640, 45ax-mp 5 . . . . . . . 8 (𝑐 ∈ 1𝑜 ↦ ((𝑎𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐))) = (𝑐 ∈ {∅} ↦ ((𝑎𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)))
4746oveq2i 6538 . . . . . . 7 ((mulGrp‘(1𝑜 mPoly 𝑅)) Σg (𝑐 ∈ 1𝑜 ↦ ((𝑎𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)))) = ((mulGrp‘(1𝑜 mPoly 𝑅)) Σg (𝑐 ∈ {∅} ↦ ((𝑎𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐))))
481mplring 19219 . . . . . . . . . . 11 ((1𝑜 ∈ ω ∧ 𝑅 ∈ Ring) → (1𝑜 mPoly 𝑅) ∈ Ring)
495, 48mpan 701 . . . . . . . . . 10 (𝑅 ∈ Ring → (1𝑜 mPoly 𝑅) ∈ Ring)
5020ringmgp 18322 . . . . . . . . . 10 ((1𝑜 mPoly 𝑅) ∈ Ring → (mulGrp‘(1𝑜 mPoly 𝑅)) ∈ Mnd)
5149, 50syl 17 . . . . . . . . 9 (𝑅 ∈ Ring → (mulGrp‘(1𝑜 mPoly 𝑅)) ∈ Mnd)
5251ad2antrr 757 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → (mulGrp‘(1𝑜 mPoly 𝑅)) ∈ Mnd)
5326a1i 11 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ∅ ∈ V)
54 ply1coe.e . . . . . . . . . . . 12 = (.g𝑀)
5520, 10mgpbas 18264 . . . . . . . . . . . . 13 𝐵 = (Base‘(mulGrp‘(1𝑜 mPoly 𝑅)))
5655a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐵 = (Base‘(mulGrp‘(1𝑜 mPoly 𝑅))))
57 ply1coe.m . . . . . . . . . . . . . 14 𝑀 = (mulGrp‘𝑃)
5857, 9mgpbas 18264 . . . . . . . . . . . . 13 𝐵 = (Base‘𝑀)
5958a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐵 = (Base‘𝑀))
60 ssv 3587 . . . . . . . . . . . . 13 𝐵 ⊆ V
6160a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐵 ⊆ V)
62 ovex 6555 . . . . . . . . . . . . 13 (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) ∈ V
6362a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) ∈ V)
64 eqid 2609 . . . . . . . . . . . . . . . . 17 (.r𝑃) = (.r𝑃)
657, 1, 64ply1mulr 19364 . . . . . . . . . . . . . . . 16 (.r𝑃) = (.r‘(1𝑜 mPoly 𝑅))
6620, 65mgpplusg 18262 . . . . . . . . . . . . . . 15 (.r𝑃) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
6757, 64mgpplusg 18262 . . . . . . . . . . . . . . 15 (.r𝑃) = (+g𝑀)
6866, 67eqtr3i 2633 . . . . . . . . . . . . . 14 (+g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (+g𝑀)
6968oveqi 6540 . . . . . . . . . . . . 13 (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) = (𝑎(+g𝑀)𝑏)
7069a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) = (𝑎(+g𝑀)𝑏))
7121, 54, 56, 59, 61, 63, 70mulgpropd 17353 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = )
7271oveqd 6544 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) = ((𝑎‘∅) 𝑋))
7372adantr 479 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) = ((𝑎‘∅) 𝑋))
747ply1ring 19385 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
7557ringmgp 18322 . . . . . . . . . . . 12 (𝑃 ∈ Ring → 𝑀 ∈ Mnd)
7674, 75syl 17 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
7776ad2antrr 757 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 𝑀 ∈ Mnd)
78 elmapi 7742 . . . . . . . . . . . 12 (𝑎 ∈ (ℕ0𝑚 1𝑜) → 𝑎:1𝑜⟶ℕ0)
79 0lt1o 7448 . . . . . . . . . . . 12 ∅ ∈ 1𝑜
80 ffvelrn 6250 . . . . . . . . . . . 12 ((𝑎:1𝑜⟶ℕ0 ∧ ∅ ∈ 1𝑜) → (𝑎‘∅) ∈ ℕ0)
8178, 79, 80sylancl 692 . . . . . . . . . . 11 (𝑎 ∈ (ℕ0𝑚 1𝑜) → (𝑎‘∅) ∈ ℕ0)
8281adantl 480 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → (𝑎‘∅) ∈ ℕ0)
83 ply1coe.x . . . . . . . . . . . 12 𝑋 = (var1𝑅)
8483, 7, 9vr1cl 19354 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑋𝐵)
8584ad2antrr 757 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 𝑋𝐵)
8658, 54mulgnn0cl 17327 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ (𝑎‘∅) ∈ ℕ0𝑋𝐵) → ((𝑎‘∅) 𝑋) ∈ 𝐵)
8777, 82, 85, 86syl3anc 1317 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((𝑎‘∅) 𝑋) ∈ 𝐵)
8873, 87eqeltrd 2687 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) ∈ 𝐵)
89 fveq2 6088 . . . . . . . . . 10 (𝑐 = ∅ → (𝑎𝑐) = (𝑎‘∅))
90 fveq2 6088 . . . . . . . . . . 11 (𝑐 = ∅ → ((1𝑜 mVar 𝑅)‘𝑐) = ((1𝑜 mVar 𝑅)‘∅))
9183vr1val 19329 . . . . . . . . . . 11 𝑋 = ((1𝑜 mVar 𝑅)‘∅)
9290, 91syl6eqr 2661 . . . . . . . . . 10 (𝑐 = ∅ → ((1𝑜 mVar 𝑅)‘𝑐) = 𝑋)
9389, 92oveq12d 6545 . . . . . . . . 9 (𝑐 = ∅ → ((𝑎𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)) = ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋))
9455, 93gsumsn 18123 . . . . . . . 8 (((mulGrp‘(1𝑜 mPoly 𝑅)) ∈ Mnd ∧ ∅ ∈ V ∧ ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) ∈ 𝐵) → ((mulGrp‘(1𝑜 mPoly 𝑅)) Σg (𝑐 ∈ {∅} ↦ ((𝑎𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)))) = ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋))
9552, 53, 88, 94syl3anc 1317 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((mulGrp‘(1𝑜 mPoly 𝑅)) Σg (𝑐 ∈ {∅} ↦ ((𝑎𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)))) = ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋))
9647, 95syl5eq 2655 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((mulGrp‘(1𝑜 mPoly 𝑅)) Σg (𝑐 ∈ 1𝑜 ↦ ((𝑎𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)))) = ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋))
9744, 96, 733eqtrd 2647 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅))) = ((𝑎‘∅) 𝑋))
9818, 97oveq12d 6545 . . . 4 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((𝐾𝑎) · (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅)))) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋)))
9998mpteq2dva 4666 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐾𝑎) · (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅))))) = (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋))))
10099oveq2d 6543 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((1𝑜 mPoly 𝑅) Σg (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐾𝑎) · (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅)))))) = ((1𝑜 mPoly 𝑅) Σg (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋)))))
101 nn0ex 11145 . . . . . 6 0 ∈ V
102101mptex 6368 . . . . 5 (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∈ V
103102a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∈ V)
104 fvex 6098 . . . . . 6 (Poly1𝑅) ∈ V
1057, 104eqeltri 2683 . . . . 5 𝑃 ∈ V
106105a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝑃 ∈ V)
107 ovex 6555 . . . . 5 (1𝑜 mPoly 𝑅) ∈ V
108107a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (1𝑜 mPoly 𝑅) ∈ V)
1099, 10eqtr3i 2633 . . . . 5 (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅))
110109a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅)))
111 eqid 2609 . . . . . 6 (+g𝑃) = (+g𝑃)
1127, 1, 111ply1plusg 19362 . . . . 5 (+g𝑃) = (+g‘(1𝑜 mPoly 𝑅))
113112a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (+g𝑃) = (+g‘(1𝑜 mPoly 𝑅)))
114103, 106, 108, 110, 113gsumpropd 17041 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))) = ((1𝑜 mPoly 𝑅) Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))
115 eqid 2609 . . . . 5 (0g𝑃) = (0g𝑃)
1161, 7, 115ply1mpl0 19392 . . . 4 (0g𝑃) = (0g‘(1𝑜 mPoly 𝑅))
1171mpllmod 19218 . . . . . 6 ((1𝑜 ∈ ω ∧ 𝑅 ∈ Ring) → (1𝑜 mPoly 𝑅) ∈ LMod)
1185, 13, 117sylancr 693 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (1𝑜 mPoly 𝑅) ∈ LMod)
119 lmodcmn 18680 . . . . 5 ((1𝑜 mPoly 𝑅) ∈ LMod → (1𝑜 mPoly 𝑅) ∈ CMnd)
120118, 119syl 17 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (1𝑜 mPoly 𝑅) ∈ CMnd)
121101a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ℕ0 ∈ V)
1227ply1lmod 19389 . . . . . . 7 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
123122ad2antrr 757 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ LMod)
124 eqid 2609 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
12516, 9, 7, 124coe1f 19348 . . . . . . . . 9 (𝐾𝐵𝐴:ℕ0⟶(Base‘𝑅))
126125adantl 480 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐴:ℕ0⟶(Base‘𝑅))
127126ffvelrnda 6252 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ (Base‘𝑅))
1287ply1sca 19390 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
129128eqcomd 2615 . . . . . . . . 9 (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
130129ad2antrr 757 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
131130fveq2d 6092 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
132127, 131eleqtrrd 2690 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ (Base‘(Scalar‘𝑃)))
13376ad2antrr 757 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ Mnd)
134 simpr 475 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
13584ad2antrr 757 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑋𝐵)
13658, 54mulgnn0cl 17327 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑘 ∈ ℕ0𝑋𝐵) → (𝑘 𝑋) ∈ 𝐵)
137133, 134, 135, 136syl3anc 1317 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑘 𝑋) ∈ 𝐵)
138 eqid 2609 . . . . . . 7 (Scalar‘𝑃) = (Scalar‘𝑃)
139 eqid 2609 . . . . . . 7 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
1409, 138, 11, 139lmodvscl 18649 . . . . . 6 ((𝑃 ∈ LMod ∧ (𝐴𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑘 𝑋) ∈ 𝐵) → ((𝐴𝑘) · (𝑘 𝑋)) ∈ 𝐵)
141123, 132, 137, 140syl3anc 1317 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) · (𝑘 𝑋)) ∈ 𝐵)
142 eqid 2609 . . . . 5 (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))
143141, 142fmptd 6277 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))):ℕ0𝐵)
1447, 83, 9, 11, 57, 54, 16ply1coefsupp 19432 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) finSupp (0g𝑃))
145 eqid 2609 . . . . . 6 (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅))
14640, 101, 26, 145mapsnf1o2 7768 . . . . 5 (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)):(ℕ0𝑚 1𝑜)–1-1-onto→ℕ0
147146a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)):(ℕ0𝑚 1𝑜)–1-1-onto→ℕ0)
14810, 116, 120, 121, 143, 144, 147gsumf1o 18086 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((1𝑜 mPoly 𝑅) Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))) = ((1𝑜 mPoly 𝑅) Σg ((𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∘ (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)))))
149 eqidd 2610 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)))
150 eqidd 2610 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))))
151 fveq2 6088 . . . . . 6 (𝑘 = (𝑎‘∅) → (𝐴𝑘) = (𝐴‘(𝑎‘∅)))
152 oveq1 6534 . . . . . 6 (𝑘 = (𝑎‘∅) → (𝑘 𝑋) = ((𝑎‘∅) 𝑋))
153151, 152oveq12d 6545 . . . . 5 (𝑘 = (𝑎‘∅) → ((𝐴𝑘) · (𝑘 𝑋)) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋)))
15482, 149, 150, 153fmptco 6288 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∘ (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅))) = (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋))))
155154oveq2d 6543 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((1𝑜 mPoly 𝑅) Σg ((𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∘ (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)))) = ((1𝑜 mPoly 𝑅) Σg (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋)))))
156114, 148, 1553eqtrrd 2648 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((1𝑜 mPoly 𝑅) Σg (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋)))) = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))
15715, 100, 1563eqtrd 2647 1 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172  wss 3539  c0 3873  ifcif 4035  {csn 4124  cmpt 4637  ccom 5032  wf 5786  1-1-ontowf1o 5789  cfv 5790  (class class class)co 6527  ωcom 6934  1𝑜c1o 7417  𝑚 cmap 7721  0cn0 11139  Basecbs 15641  +gcplusg 15714  .rcmulr 15715  Scalarcsca 15717   ·𝑠 cvsca 15718  0gc0g 15869   Σg cgsu 15870  Mndcmnd 17063  .gcmg 17309  CMndccmn 17962  mulGrpcmgp 18258  1rcur 18270  Ringcrg 18316  LModclmod 18632   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-fal 1480  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-srg 18275  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:  eqcoe1ply1eq  19434  pmatcollpw1lem2  20341  mp2pm2mp  20377  plypf1  23689
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