MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ply1coe Structured version   Visualization version   GIF version

Theorem ply1coe 19939
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 2765 . . 3 (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
2 psr1baslem 19828 . . 3 (ℕ0𝑚 1𝑜) = {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ (𝑑 “ ℕ) ∈ Fin}
3 eqid 2765 . . 3 (0g𝑅) = (0g𝑅)
4 eqid 2765 . . 3 (1r𝑅) = (1r𝑅)
5 1onn 7924 . . . 4 1𝑜 ∈ ω
65a1i 11 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 1𝑜 ∈ ω)
7 ply1coe.p . . . 4 𝑃 = (Poly1𝑅)
8 eqid 2765 . . . 4 (PwSer1𝑅) = (PwSer1𝑅)
9 ply1coe.b . . . 4 𝐵 = (Base‘𝑃)
107, 8, 9ply1bas 19838 . . 3 𝐵 = (Base‘(1𝑜 mPoly 𝑅))
11 ply1coe.n . . . 4 · = ( ·𝑠𝑃)
127, 1, 11ply1vsca 19869 . . 3 · = ( ·𝑠 ‘(1𝑜 mPoly 𝑅))
13 simpl 474 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝑅 ∈ Ring)
14 simpr 477 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐾𝐵)
151, 2, 3, 4, 6, 10, 12, 13, 14mplcoe1 19739 . 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 19850 . . . . . 6 ((𝐾𝐵𝑎 ∈ (ℕ0𝑚 1𝑜)) → (𝐾𝑎) = (𝐴‘(𝑎‘∅)))
1817adantll 705 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → (𝐾𝑎) = (𝐴‘(𝑎‘∅)))
195a1i 11 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 1𝑜 ∈ ω)
20 eqid 2765 . . . . . . 7 (mulGrp‘(1𝑜 mPoly 𝑅)) = (mulGrp‘(1𝑜 mPoly 𝑅))
21 eqid 2765 . . . . . . 7 (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
22 eqid 2765 . . . . . . 7 (1𝑜 mVar 𝑅) = (1𝑜 mVar 𝑅)
23 simpll 783 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 𝑅 ∈ Ring)
24 simpr 477 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 𝑎 ∈ (ℕ0𝑚 1𝑜))
25 eqidd 2766 . . . . . . . . . 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 4950 . . . . . . . . . . 11 ∅ ∈ V
27 fveq2 6375 . . . . . . . . . . . . 13 (𝑏 = ∅ → ((1𝑜 mVar 𝑅)‘𝑏) = ((1𝑜 mVar 𝑅)‘∅))
2827oveq1d 6857 . . . . . . . . . . . 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 6858 . . . . . . . . . . . 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 2780 . . . . . . . . . . 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 4379 . . . . . . . . . 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 225 . . . . . . . . 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 6375 . . . . . . . . . . . . 13 (𝑥 = ∅ → ((1𝑜 mVar 𝑅)‘𝑥) = ((1𝑜 mVar 𝑅)‘∅))
3433oveq2d 6858 . . . . . . . . . . . 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 6857 . . . . . . . . . . . 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 2780 . . . . . . . . . . 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 3133 . . . . . . . . . 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 4379 . . . . . . . . 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 225 . . . . . . . 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 7777 . . . . . . . . 9 1𝑜 = {∅}
4140raleqi 3290 . . . . . . . . 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 3137 . . . . . . . 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 225 . . . . . . 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 19742 . . . . . 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 4896 . . . . . . . . 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 6853 . . . . . . 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 19726 . . . . . . . . . . 11 ((1𝑜 ∈ ω ∧ 𝑅 ∈ Ring) → (1𝑜 mPoly 𝑅) ∈ Ring)
495, 48mpan 681 . . . . . . . . . 10 (𝑅 ∈ Ring → (1𝑜 mPoly 𝑅) ∈ Ring)
5020ringmgp 18820 . . . . . . . . . 10 ((1𝑜 mPoly 𝑅) ∈ Ring → (mulGrp‘(1𝑜 mPoly 𝑅)) ∈ Mnd)
5149, 50syl 17 . . . . . . . . 9 (𝑅 ∈ Ring → (mulGrp‘(1𝑜 mPoly 𝑅)) ∈ Mnd)
5251ad2antrr 717 . . . . . . . 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 18762 . . . . . . . . . . . . 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 18762 . . . . . . . . . . . . 13 𝐵 = (Base‘𝑀)
5958a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐵 = (Base‘𝑀))
60 ssv 3785 . . . . . . . . . . . . 13 𝐵 ⊆ V
6160a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐵 ⊆ V)
62 ovexd 6876 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) ∈ V)
63 eqid 2765 . . . . . . . . . . . . . . . . 17 (.r𝑃) = (.r𝑃)
647, 1, 63ply1mulr 19870 . . . . . . . . . . . . . . . 16 (.r𝑃) = (.r‘(1𝑜 mPoly 𝑅))
6520, 64mgpplusg 18760 . . . . . . . . . . . . . . 15 (.r𝑃) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
6657, 63mgpplusg 18760 . . . . . . . . . . . . . . 15 (.r𝑃) = (+g𝑀)
6765, 66eqtr3i 2789 . . . . . . . . . . . . . 14 (+g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (+g𝑀)
6867oveqi 6855 . . . . . . . . . . . . 13 (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) = (𝑎(+g𝑀)𝑏)
6968a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ (𝑎 ∈ V ∧ 𝑏 ∈ V)) → (𝑎(+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑏) = (𝑎(+g𝑀)𝑏))
7021, 54, 56, 59, 61, 62, 69mulgpropd 17848 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = )
7170oveqd 6859 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) = ((𝑎‘∅) 𝑋))
7271adantr 472 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) = ((𝑎‘∅) 𝑋))
737ply1ring 19891 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
7457ringmgp 18820 . . . . . . . . . . . 12 (𝑃 ∈ Ring → 𝑀 ∈ Mnd)
7573, 74syl 17 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
7675ad2antrr 717 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 𝑀 ∈ Mnd)
77 elmapi 8082 . . . . . . . . . . . 12 (𝑎 ∈ (ℕ0𝑚 1𝑜) → 𝑎:1𝑜⟶ℕ0)
78 0lt1o 7789 . . . . . . . . . . . 12 ∅ ∈ 1𝑜
79 ffvelrn 6547 . . . . . . . . . . . 12 ((𝑎:1𝑜⟶ℕ0 ∧ ∅ ∈ 1𝑜) → (𝑎‘∅) ∈ ℕ0)
8077, 78, 79sylancl 580 . . . . . . . . . . 11 (𝑎 ∈ (ℕ0𝑚 1𝑜) → (𝑎‘∅) ∈ ℕ0)
8180adantl 473 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → (𝑎‘∅) ∈ ℕ0)
82 ply1coe.x . . . . . . . . . . . 12 𝑋 = (var1𝑅)
8382, 7, 9vr1cl 19860 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑋𝐵)
8483ad2antrr 717 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 𝑋𝐵)
8558, 54mulgnn0cl 17824 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ (𝑎‘∅) ∈ ℕ0𝑋𝐵) → ((𝑎‘∅) 𝑋) ∈ 𝐵)
8676, 81, 84, 85syl3anc 1490 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((𝑎‘∅) 𝑋) ∈ 𝐵)
8772, 86eqeltrd 2844 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) ∈ 𝐵)
88 fveq2 6375 . . . . . . . . . 10 (𝑐 = ∅ → (𝑎𝑐) = (𝑎‘∅))
89 fveq2 6375 . . . . . . . . . . 11 (𝑐 = ∅ → ((1𝑜 mVar 𝑅)‘𝑐) = ((1𝑜 mVar 𝑅)‘∅))
9082vr1val 19835 . . . . . . . . . . 11 𝑋 = ((1𝑜 mVar 𝑅)‘∅)
9189, 90syl6eqr 2817 . . . . . . . . . 10 (𝑐 = ∅ → ((1𝑜 mVar 𝑅)‘𝑐) = 𝑋)
9288, 91oveq12d 6860 . . . . . . . . 9 (𝑐 = ∅ → ((𝑎𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)) = ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋))
9355, 92gsumsn 18620 . . . . . . . 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))
9452, 53, 87, 93syl3anc 1490 . . . . . . 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))
9547, 94syl5eq 2811 . . . . . 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))
9644, 95, 723eqtrd 2803 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅))) = ((𝑎‘∅) 𝑋))
9718, 96oveq12d 6860 . . . 4 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((𝐾𝑎) · (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅)))) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋)))
9897mpteq2dva 4903 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐾𝑎) · (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅))))) = (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋))))
9998oveq2d 6858 . 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)))))
100 nn0ex 11545 . . . . . 6 0 ∈ V
101100mptex 6679 . . . . 5 (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∈ V
102101a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∈ V)
1037fvexi 6389 . . . . 5 𝑃 ∈ V
104103a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝑃 ∈ V)
105 ovexd 6876 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (1𝑜 mPoly 𝑅) ∈ V)
1069, 10eqtr3i 2789 . . . . 5 (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅))
107106a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅)))
108 eqid 2765 . . . . . 6 (+g𝑃) = (+g𝑃)
1097, 1, 108ply1plusg 19868 . . . . 5 (+g𝑃) = (+g‘(1𝑜 mPoly 𝑅))
110109a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (+g𝑃) = (+g‘(1𝑜 mPoly 𝑅)))
111102, 104, 105, 107, 110gsumpropd 17538 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))) = ((1𝑜 mPoly 𝑅) Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))
112 eqid 2765 . . . . 5 (0g𝑃) = (0g𝑃)
1131, 7, 112ply1mpl0 19898 . . . 4 (0g𝑃) = (0g‘(1𝑜 mPoly 𝑅))
1141mpllmod 19725 . . . . . 6 ((1𝑜 ∈ ω ∧ 𝑅 ∈ Ring) → (1𝑜 mPoly 𝑅) ∈ LMod)
1155, 13, 114sylancr 581 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (1𝑜 mPoly 𝑅) ∈ LMod)
116 lmodcmn 19180 . . . . 5 ((1𝑜 mPoly 𝑅) ∈ LMod → (1𝑜 mPoly 𝑅) ∈ CMnd)
117115, 116syl 17 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (1𝑜 mPoly 𝑅) ∈ CMnd)
118100a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ℕ0 ∈ V)
1197ply1lmod 19895 . . . . . . 7 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
120119ad2antrr 717 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ LMod)
121 eqid 2765 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
12216, 9, 7, 121coe1f 19854 . . . . . . . . 9 (𝐾𝐵𝐴:ℕ0⟶(Base‘𝑅))
123122adantl 473 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐴:ℕ0⟶(Base‘𝑅))
124123ffvelrnda 6549 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ (Base‘𝑅))
1257ply1sca 19896 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
126125eqcomd 2771 . . . . . . . . 9 (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
127126ad2antrr 717 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
128127fveq2d 6379 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
129124, 128eleqtrrd 2847 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ (Base‘(Scalar‘𝑃)))
13075ad2antrr 717 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ Mnd)
131 simpr 477 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
13283ad2antrr 717 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑋𝐵)
13358, 54mulgnn0cl 17824 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑘 ∈ ℕ0𝑋𝐵) → (𝑘 𝑋) ∈ 𝐵)
134130, 131, 132, 133syl3anc 1490 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑘 𝑋) ∈ 𝐵)
135 eqid 2765 . . . . . . 7 (Scalar‘𝑃) = (Scalar‘𝑃)
136 eqid 2765 . . . . . . 7 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
1379, 135, 11, 136lmodvscl 19149 . . . . . 6 ((𝑃 ∈ LMod ∧ (𝐴𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑘 𝑋) ∈ 𝐵) → ((𝐴𝑘) · (𝑘 𝑋)) ∈ 𝐵)
138120, 129, 134, 137syl3anc 1490 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) · (𝑘 𝑋)) ∈ 𝐵)
139138fmpttd 6575 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))):ℕ0𝐵)
1407, 82, 9, 11, 57, 54, 16ply1coefsupp 19938 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) finSupp (0g𝑃))
141 eqid 2765 . . . . . 6 (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅))
14240, 100, 26, 141mapsnf1o2 8110 . . . . 5 (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)):(ℕ0𝑚 1𝑜)–1-1-onto→ℕ0
143142a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)):(ℕ0𝑚 1𝑜)–1-1-onto→ℕ0)
14410, 113, 117, 118, 139, 140, 143gsumf1o 18583 . . 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‘∅)))))
145 eqidd 2766 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)))
146 eqidd 2766 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))))
147 fveq2 6375 . . . . . 6 (𝑘 = (𝑎‘∅) → (𝐴𝑘) = (𝐴‘(𝑎‘∅)))
148 oveq1 6849 . . . . . 6 (𝑘 = (𝑎‘∅) → (𝑘 𝑋) = ((𝑎‘∅) 𝑋))
149147, 148oveq12d 6860 . . . . 5 (𝑘 = (𝑎‘∅) → ((𝐴𝑘) · (𝑘 𝑋)) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋)))
15081, 145, 146, 149fmptco 6587 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∘ (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅))) = (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋))))
151150oveq2d 6858 . . 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)))))
152111, 144, 1513eqtrrd 2804 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((1𝑜 mPoly 𝑅) Σg (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋)))) = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))
15315, 99, 1523eqtrd 2803 1 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  wss 3732  c0 4079  ifcif 4243  {csn 4334  cmpt 4888  ccom 5281  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  ωcom 7263  1𝑜c1o 7757  𝑚 cmap 8060  0cn0 11538  Basecbs 16130  +gcplusg 16214  .rcmulr 16215  Scalarcsca 16217   ·𝑠 cvsca 16218  0gc0g 16366   Σg cgsu 16367  Mndcmnd 17560  .gcmg 17807  CMndccmn 18459  mulGrpcmgp 18756  1rcur 18768  Ringcrg 18814  LModclmod 19132   mVar cmvr 19626   mPoly cmpl 19627  PwSer1cps1 19818  var1cv1 19819  Poly1cpl1 19820  coe1cco1 19821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-ofr 7096  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-sca 16230  df-vsca 16231  df-tset 16233  df-ple 16234  df-0g 16368  df-gsum 16369  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-mhm 17601  df-submnd 17602  df-grp 17692  df-minusg 17693  df-sbg 17694  df-mulg 17808  df-subg 17855  df-ghm 17922  df-cntz 18013  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-srg 18773  df-ring 18816  df-subrg 19047  df-lmod 19134  df-lss 19202  df-psr 19630  df-mvr 19631  df-mpl 19632  df-opsr 19634  df-psr1 19823  df-vr1 19824  df-ply1 19825  df-coe1 19826
This theorem is referenced by:  eqcoe1ply1eq  19940  pmatcollpw1lem2  20859  mp2pm2mp  20895  plypf1  24259
  Copyright terms: Public domain W3C validator