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Theorem ply1coe 20182
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 2771 . . 3 (1o mPoly 𝑅) = (1o mPoly 𝑅)
2 psr1baslem 20071 . . 3 (ℕ0𝑚 1o) = {𝑑 ∈ (ℕ0𝑚 1o) ∣ (𝑑 “ ℕ) ∈ Fin}
3 eqid 2771 . . 3 (0g𝑅) = (0g𝑅)
4 eqid 2771 . . 3 (1r𝑅) = (1r𝑅)
5 1onn 8064 . . . 4 1o ∈ ω
65a1i 11 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 1o ∈ ω)
7 ply1coe.p . . . 4 𝑃 = (Poly1𝑅)
8 eqid 2771 . . . 4 (PwSer1𝑅) = (PwSer1𝑅)
9 ply1coe.b . . . 4 𝐵 = (Base‘𝑃)
107, 8, 9ply1bas 20081 . . 3 𝐵 = (Base‘(1o mPoly 𝑅))
11 ply1coe.n . . . 4 · = ( ·𝑠𝑃)
127, 1, 11ply1vsca 20112 . . 3 · = ( ·𝑠 ‘(1o mPoly 𝑅))
13 simpl 475 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝑅 ∈ Ring)
14 simpr 477 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐾𝐵)
151, 2, 3, 4, 6, 10, 12, 13, 14mplcoe1 19971 . 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 20093 . . . . . 6 ((𝐾𝐵𝑎 ∈ (ℕ0𝑚 1o)) → (𝐾𝑎) = (𝐴‘(𝑎‘∅)))
1817adantll 702 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → (𝐾𝑎) = (𝐴‘(𝑎‘∅)))
195a1i 11 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → 1o ∈ ω)
20 eqid 2771 . . . . . . 7 (mulGrp‘(1o mPoly 𝑅)) = (mulGrp‘(1o mPoly 𝑅))
21 eqid 2771 . . . . . . 7 (.g‘(mulGrp‘(1o mPoly 𝑅))) = (.g‘(mulGrp‘(1o mPoly 𝑅)))
22 eqid 2771 . . . . . . 7 (1o mVar 𝑅) = (1o mVar 𝑅)
23 simpll 755 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → 𝑅 ∈ Ring)
24 simpr 477 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → 𝑎 ∈ (ℕ0𝑚 1o))
25 eqidd 2772 . . . . . . . . . 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 5064 . . . . . . . . . . 11 ∅ ∈ V
27 fveq2 6496 . . . . . . . . . . . . 13 (𝑏 = ∅ → ((1o mVar 𝑅)‘𝑏) = ((1o mVar 𝑅)‘∅))
2827oveq1d 6989 . . . . . . . . . . . 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 6990 . . . . . . . . . . . 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 2786 . . . . . . . . . . 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 4489 . . . . . . . . . 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 226 . . . . . . . . 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 6496 . . . . . . . . . . . . 13 (𝑥 = ∅ → ((1o mVar 𝑅)‘𝑥) = ((1o mVar 𝑅)‘∅))
3433oveq2d 6990 . . . . . . . . . . . 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 6989 . . . . . . . . . . . 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 2786 . . . . . . . . . . 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 3140 . . . . . . . . . 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 4489 . . . . . . . . 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 226 . . . . . . . 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 7916 . . . . . . . . 9 1o = {∅}
4140raleqi 3346 . . . . . . . . 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 3174 . . . . . . . 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 226 . . . . . . 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 19974 . . . . . 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)))))
4540mpteq1i 5013 . . . . . . . 8 (𝑐 ∈ 1o ↦ ((𝑎𝑐)(.g‘(mulGrp‘(1o mPoly 𝑅)))((1o mVar 𝑅)‘𝑐))) = (𝑐 ∈ {∅} ↦ ((𝑎𝑐)(.g‘(mulGrp‘(1o mPoly 𝑅)))((1o mVar 𝑅)‘𝑐)))
4645oveq2i 6985 . . . . . . 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))))
471mplring 19958 . . . . . . . . . . 11 ((1o ∈ ω ∧ 𝑅 ∈ Ring) → (1o mPoly 𝑅) ∈ Ring)
485, 47mpan 678 . . . . . . . . . 10 (𝑅 ∈ Ring → (1o mPoly 𝑅) ∈ Ring)
4920ringmgp 19038 . . . . . . . . . 10 ((1o mPoly 𝑅) ∈ Ring → (mulGrp‘(1o mPoly 𝑅)) ∈ Mnd)
5048, 49syl 17 . . . . . . . . 9 (𝑅 ∈ Ring → (mulGrp‘(1o mPoly 𝑅)) ∈ Mnd)
5150ad2antrr 714 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → (mulGrp‘(1o mPoly 𝑅)) ∈ Mnd)
5226a1i 11 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → ∅ ∈ V)
53 ply1coe.e . . . . . . . . . . . 12 = (.g𝑀)
5420, 10mgpbas 18980 . . . . . . . . . . . . 13 𝐵 = (Base‘(mulGrp‘(1o mPoly 𝑅)))
5554a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐵 = (Base‘(mulGrp‘(1o mPoly 𝑅))))
56 ply1coe.m . . . . . . . . . . . . . 14 𝑀 = (mulGrp‘𝑃)
5756, 9mgpbas 18980 . . . . . . . . . . . . 13 𝐵 = (Base‘𝑀)
5857a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐵 = (Base‘𝑀))
59 ssv 3874 . . . . . . . . . . . . 13 𝐵 ⊆ V
6059a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐵 ⊆ V)
61 ovexd 7008 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+g‘(mulGrp‘(1o mPoly 𝑅)))𝑏) ∈ V)
62 eqid 2771 . . . . . . . . . . . . . . . . 17 (.r𝑃) = (.r𝑃)
637, 1, 62ply1mulr 20113 . . . . . . . . . . . . . . . 16 (.r𝑃) = (.r‘(1o mPoly 𝑅))
6420, 63mgpplusg 18978 . . . . . . . . . . . . . . 15 (.r𝑃) = (+g‘(mulGrp‘(1o mPoly 𝑅)))
6556, 62mgpplusg 18978 . . . . . . . . . . . . . . 15 (.r𝑃) = (+g𝑀)
6664, 65eqtr3i 2797 . . . . . . . . . . . . . 14 (+g‘(mulGrp‘(1o mPoly 𝑅))) = (+g𝑀)
6766oveqi 6987 . . . . . . . . . . . . 13 (𝑎(+g‘(mulGrp‘(1o mPoly 𝑅)))𝑏) = (𝑎(+g𝑀)𝑏)
6867a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+g‘(mulGrp‘(1o mPoly 𝑅)))𝑏) = (𝑎(+g𝑀)𝑏))
6921, 53, 55, 58, 60, 61, 68mulgpropd 18065 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (.g‘(mulGrp‘(1o mPoly 𝑅))) = )
7069oveqd 6991 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((𝑎‘∅)(.g‘(mulGrp‘(1o mPoly 𝑅)))𝑋) = ((𝑎‘∅) 𝑋))
7170adantr 473 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → ((𝑎‘∅)(.g‘(mulGrp‘(1o mPoly 𝑅)))𝑋) = ((𝑎‘∅) 𝑋))
727ply1ring 20134 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
7356ringmgp 19038 . . . . . . . . . . . 12 (𝑃 ∈ Ring → 𝑀 ∈ Mnd)
7472, 73syl 17 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
7574ad2antrr 714 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → 𝑀 ∈ Mnd)
76 elmapi 8226 . . . . . . . . . . . 12 (𝑎 ∈ (ℕ0𝑚 1o) → 𝑎:1o⟶ℕ0)
77 0lt1o 7929 . . . . . . . . . . . 12 ∅ ∈ 1o
78 ffvelrn 6672 . . . . . . . . . . . 12 ((𝑎:1o⟶ℕ0 ∧ ∅ ∈ 1o) → (𝑎‘∅) ∈ ℕ0)
7976, 77, 78sylancl 578 . . . . . . . . . . 11 (𝑎 ∈ (ℕ0𝑚 1o) → (𝑎‘∅) ∈ ℕ0)
8079adantl 474 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → (𝑎‘∅) ∈ ℕ0)
81 ply1coe.x . . . . . . . . . . . 12 𝑋 = (var1𝑅)
8281, 7, 9vr1cl 20103 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑋𝐵)
8382ad2antrr 714 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → 𝑋𝐵)
8457, 53mulgnn0cl 18041 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ (𝑎‘∅) ∈ ℕ0𝑋𝐵) → ((𝑎‘∅) 𝑋) ∈ 𝐵)
8575, 80, 83, 84syl3anc 1352 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → ((𝑎‘∅) 𝑋) ∈ 𝐵)
8671, 85eqeltrd 2859 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → ((𝑎‘∅)(.g‘(mulGrp‘(1o mPoly 𝑅)))𝑋) ∈ 𝐵)
87 fveq2 6496 . . . . . . . . . 10 (𝑐 = ∅ → (𝑎𝑐) = (𝑎‘∅))
88 fveq2 6496 . . . . . . . . . . 11 (𝑐 = ∅ → ((1o mVar 𝑅)‘𝑐) = ((1o mVar 𝑅)‘∅))
8981vr1val 20078 . . . . . . . . . . 11 𝑋 = ((1o mVar 𝑅)‘∅)
9088, 89syl6eqr 2825 . . . . . . . . . 10 (𝑐 = ∅ → ((1o mVar 𝑅)‘𝑐) = 𝑋)
9187, 90oveq12d 6992 . . . . . . . . 9 (𝑐 = ∅ → ((𝑎𝑐)(.g‘(mulGrp‘(1o mPoly 𝑅)))((1o mVar 𝑅)‘𝑐)) = ((𝑎‘∅)(.g‘(mulGrp‘(1o mPoly 𝑅)))𝑋))
9254, 91gsumsn 18839 . . . . . . . 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))
9351, 52, 86, 92syl3anc 1352 . . . . . . 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))
9446, 93syl5eq 2819 . . . . . 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))
9544, 94, 713eqtrd 2811 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → (𝑏 ∈ (ℕ0𝑚 1o) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅))) = ((𝑎‘∅) 𝑋))
9618, 95oveq12d 6992 . . . 4 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1o)) → ((𝐾𝑎) · (𝑏 ∈ (ℕ0𝑚 1o) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅)))) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋)))
9796mpteq2dva 5018 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑎 ∈ (ℕ0𝑚 1o) ↦ ((𝐾𝑎) · (𝑏 ∈ (ℕ0𝑚 1o) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅))))) = (𝑎 ∈ (ℕ0𝑚 1o) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋))))
9897oveq2d 6990 . 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)))))
99 nn0ex 11712 . . . . . 6 0 ∈ V
10099mptex 6810 . . . . 5 (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∈ V
101100a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∈ V)
1027fvexi 6510 . . . . 5 𝑃 ∈ V
103102a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝑃 ∈ V)
104 ovexd 7008 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (1o mPoly 𝑅) ∈ V)
1059, 10eqtr3i 2797 . . . . 5 (Base‘𝑃) = (Base‘(1o mPoly 𝑅))
106105a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (Base‘𝑃) = (Base‘(1o mPoly 𝑅)))
107 eqid 2771 . . . . . 6 (+g𝑃) = (+g𝑃)
1087, 1, 107ply1plusg 20111 . . . . 5 (+g𝑃) = (+g‘(1o mPoly 𝑅))
109108a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (+g𝑃) = (+g‘(1o mPoly 𝑅)))
110101, 103, 104, 106, 109gsumpropd 17752 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))) = ((1o mPoly 𝑅) Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))
111 eqid 2771 . . . . 5 (0g𝑃) = (0g𝑃)
1121, 7, 111ply1mpl0 20141 . . . 4 (0g𝑃) = (0g‘(1o mPoly 𝑅))
1131mpllmod 19957 . . . . . 6 ((1o ∈ ω ∧ 𝑅 ∈ Ring) → (1o mPoly 𝑅) ∈ LMod)
1145, 13, 113sylancr 579 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (1o mPoly 𝑅) ∈ LMod)
115 lmodcmn 19416 . . . . 5 ((1o mPoly 𝑅) ∈ LMod → (1o mPoly 𝑅) ∈ CMnd)
116114, 115syl 17 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (1o mPoly 𝑅) ∈ CMnd)
11799a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ℕ0 ∈ V)
1187ply1lmod 20138 . . . . . . 7 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
119118ad2antrr 714 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ LMod)
120 eqid 2771 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
12116, 9, 7, 120coe1f 20097 . . . . . . . . 9 (𝐾𝐵𝐴:ℕ0⟶(Base‘𝑅))
122121adantl 474 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐴:ℕ0⟶(Base‘𝑅))
123122ffvelrnda 6674 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ (Base‘𝑅))
1247ply1sca 20139 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
125124eqcomd 2777 . . . . . . . . 9 (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
126125ad2antrr 714 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
127126fveq2d 6500 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
128123, 127eleqtrrd 2862 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ (Base‘(Scalar‘𝑃)))
12974ad2antrr 714 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ Mnd)
130 simpr 477 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
13182ad2antrr 714 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑋𝐵)
13257, 53mulgnn0cl 18041 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑘 ∈ ℕ0𝑋𝐵) → (𝑘 𝑋) ∈ 𝐵)
133129, 130, 131, 132syl3anc 1352 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑘 𝑋) ∈ 𝐵)
134 eqid 2771 . . . . . . 7 (Scalar‘𝑃) = (Scalar‘𝑃)
135 eqid 2771 . . . . . . 7 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
1369, 134, 11, 135lmodvscl 19385 . . . . . 6 ((𝑃 ∈ LMod ∧ (𝐴𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑘 𝑋) ∈ 𝐵) → ((𝐴𝑘) · (𝑘 𝑋)) ∈ 𝐵)
137119, 128, 133, 136syl3anc 1352 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) · (𝑘 𝑋)) ∈ 𝐵)
138137fmpttd 6700 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))):ℕ0𝐵)
1397, 81, 9, 11, 56, 53, 16ply1coefsupp 20181 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) finSupp (0g𝑃))
140 eqid 2771 . . . . . 6 (𝑎 ∈ (ℕ0𝑚 1o) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0𝑚 1o) ↦ (𝑎‘∅))
14140, 99, 26, 140mapsnf1o2 8254 . . . . 5 (𝑎 ∈ (ℕ0𝑚 1o) ↦ (𝑎‘∅)):(ℕ0𝑚 1o)–1-1-onto→ℕ0
142141a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑎 ∈ (ℕ0𝑚 1o) ↦ (𝑎‘∅)):(ℕ0𝑚 1o)–1-1-onto→ℕ0)
14310, 112, 116, 117, 138, 139, 142gsumf1o 18802 . . 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‘∅)))))
144 eqidd 2772 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑎 ∈ (ℕ0𝑚 1o) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0𝑚 1o) ↦ (𝑎‘∅)))
145 eqidd 2772 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))))
146 fveq2 6496 . . . . . 6 (𝑘 = (𝑎‘∅) → (𝐴𝑘) = (𝐴‘(𝑎‘∅)))
147 oveq1 6981 . . . . . 6 (𝑘 = (𝑎‘∅) → (𝑘 𝑋) = ((𝑎‘∅) 𝑋))
148146, 147oveq12d 6992 . . . . 5 (𝑘 = (𝑎‘∅) → ((𝐴𝑘) · (𝑘 𝑋)) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋)))
14980, 144, 145, 148fmptco 6712 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∘ (𝑎 ∈ (ℕ0𝑚 1o) ↦ (𝑎‘∅))) = (𝑎 ∈ (ℕ0𝑚 1o) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋))))
150149oveq2d 6990 . . 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)))))
151110, 143, 1503eqtrrd 2812 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((1o mPoly 𝑅) Σg (𝑎 ∈ (ℕ0𝑚 1o) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋)))) = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))
15215, 98, 1513eqtrd 2811 1 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wcel 2051  wral 3081  Vcvv 3408  wss 3822  c0 4172  ifcif 4344  {csn 4435  cmpt 5004  ccom 5407  wf 6181  1-1-ontowf1o 6184  cfv 6185  (class class class)co 6974  ωcom 7394  1oc1o 7896  𝑚 cmap 8204  0cn0 11705  Basecbs 16337  +gcplusg 16419  .rcmulr 16420  Scalarcsca 16422   ·𝑠 cvsca 16423  0gc0g 16567   Σg cgsu 16568  Mndcmnd 17774  .gcmg 18023  CMndccmn 18678  mulGrpcmgp 18974  1rcur 18986  Ringcrg 19032  LModclmod 19368   mVar cmvr 19858   mPoly cmpl 19859  PwSer1cps1 20061  var1cv1 20062  Poly1cpl1 20063  coe1cco1 20064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-iin 4791  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-se 5363  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-isom 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-of 7225  df-ofr 7226  df-om 7395  df-1st 7499  df-2nd 7500  df-supp 7632  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-2o 7904  df-oadd 7907  df-er 8087  df-map 8206  df-pm 8207  df-ixp 8258  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-fsupp 8627  df-oi 8767  df-card 9160  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-nn 11438  df-2 11501  df-3 11502  df-4 11503  df-5 11504  df-6 11505  df-7 11506  df-8 11507  df-9 11508  df-n0 11706  df-z 11792  df-dec 11910  df-uz 12057  df-fz 12707  df-fzo 12848  df-seq 13183  df-hash 13504  df-struct 16339  df-ndx 16340  df-slot 16341  df-base 16343  df-sets 16344  df-ress 16345  df-plusg 16432  df-mulr 16433  df-sca 16435  df-vsca 16436  df-tset 16438  df-ple 16439  df-0g 16569  df-gsum 16570  df-mre 16727  df-mrc 16728  df-acs 16730  df-mgm 17722  df-sgrp 17764  df-mnd 17775  df-mhm 17815  df-submnd 17816  df-grp 17906  df-minusg 17907  df-sbg 17908  df-mulg 18024  df-subg 18072  df-ghm 18139  df-cntz 18230  df-cmn 18680  df-abl 18681  df-mgp 18975  df-ur 18987  df-srg 18991  df-ring 19034  df-subrg 19268  df-lmod 19370  df-lss 19438  df-psr 19862  df-mvr 19863  df-mpl 19864  df-opsr 19866  df-psr1 20066  df-vr1 20067  df-ply1 20068  df-coe1 20069
This theorem is referenced by:  eqcoe1ply1eq  20183  pmatcollpw1lem2  21102  mp2pm2mp  21138  plypf1  24520
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