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

Theorem ply1coe 19487
 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 2610 . . 3 (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
2 psr1baslem 19376 . . 3 (ℕ0𝑚 1𝑜) = {𝑑 ∈ (ℕ0𝑚 1𝑜) ∣ (𝑑 “ ℕ) ∈ Fin}
3 eqid 2610 . . 3 (0g𝑅) = (0g𝑅)
4 eqid 2610 . . 3 (1r𝑅) = (1r𝑅)
5 1onn 7606 . . . 4 1𝑜 ∈ ω
65a1i 11 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 1𝑜 ∈ ω)
7 ply1coe.p . . . 4 𝑃 = (Poly1𝑅)
8 eqid 2610 . . . 4 (PwSer1𝑅) = (PwSer1𝑅)
9 ply1coe.b . . . 4 𝐵 = (Base‘𝑃)
107, 8, 9ply1bas 19386 . . 3 𝐵 = (Base‘(1𝑜 mPoly 𝑅))
11 ply1coe.n . . . 4 · = ( ·𝑠𝑃)
127, 1, 11ply1vsca 19417 . . 3 · = ( ·𝑠 ‘(1𝑜 mPoly 𝑅))
13 simpl 472 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝑅 ∈ Ring)
14 simpr 476 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐾𝐵)
151, 2, 3, 4, 6, 10, 12, 13, 14mplcoe1 19286 . 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 19398 . . . . . 6 ((𝐾𝐵𝑎 ∈ (ℕ0𝑚 1𝑜)) → (𝐾𝑎) = (𝐴‘(𝑎‘∅)))
1817adantll 746 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → (𝐾𝑎) = (𝐴‘(𝑎‘∅)))
195a1i 11 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 1𝑜 ∈ ω)
20 eqid 2610 . . . . . . 7 (mulGrp‘(1𝑜 mPoly 𝑅)) = (mulGrp‘(1𝑜 mPoly 𝑅))
21 eqid 2610 . . . . . . 7 (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
22 eqid 2610 . . . . . . 7 (1𝑜 mVar 𝑅) = (1𝑜 mVar 𝑅)
23 simpll 786 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 𝑅 ∈ Ring)
24 simpr 476 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 𝑎 ∈ (ℕ0𝑚 1𝑜))
25 eqidd 2611 . . . . . . . . . 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 4718 . . . . . . . . . . 11 ∅ ∈ V
27 fveq2 6103 . . . . . . . . . . . . 13 (𝑏 = ∅ → ((1𝑜 mVar 𝑅)‘𝑏) = ((1𝑜 mVar 𝑅)‘∅))
2827oveq1d 6564 . . . . . . . . . . . 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 6565 . . . . . . . . . . . 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 2625 . . . . . . . . . . 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 4169 . . . . . . . . . 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 223 . . . . . . . . 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 6103 . . . . . . . . . . . . 13 (𝑥 = ∅ → ((1𝑜 mVar 𝑅)‘𝑥) = ((1𝑜 mVar 𝑅)‘∅))
3433oveq2d 6565 . . . . . . . . . . . 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 6564 . . . . . . . . . . . 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 2625 . . . . . . . . . . 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 2969 . . . . . . . . . 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 4169 . . . . . . . . 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 223 . . . . . . . 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 7459 . . . . . . . . 9 1𝑜 = {∅}
4140raleqi 3119 . . . . . . . . 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 2973 . . . . . . . 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 223 . . . . . . 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 19289 . . . . . 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 4665 . . . . . . . . 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 6560 . . . . . . 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 19273 . . . . . . . . . . 11 ((1𝑜 ∈ ω ∧ 𝑅 ∈ Ring) → (1𝑜 mPoly 𝑅) ∈ Ring)
495, 48mpan 702 . . . . . . . . . 10 (𝑅 ∈ Ring → (1𝑜 mPoly 𝑅) ∈ Ring)
5020ringmgp 18376 . . . . . . . . . 10 ((1𝑜 mPoly 𝑅) ∈ Ring → (mulGrp‘(1𝑜 mPoly 𝑅)) ∈ Mnd)
5149, 50syl 17 . . . . . . . . 9 (𝑅 ∈ Ring → (mulGrp‘(1𝑜 mPoly 𝑅)) ∈ Mnd)
5251ad2antrr 758 . . . . . . . 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 18318 . . . . . . . . . . . . 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 18318 . . . . . . . . . . . . 13 𝐵 = (Base‘𝑀)
5958a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐵 = (Base‘𝑀))
60 ssv 3588 . . . . . . . . . . . . 13 𝐵 ⊆ V
6160a1i 11 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐵 ⊆ V)
62 ovex 6577 . . . . . . . . . . . . 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 2610 . . . . . . . . . . . . . . . . 17 (.r𝑃) = (.r𝑃)
657, 1, 64ply1mulr 19418 . . . . . . . . . . . . . . . 16 (.r𝑃) = (.r‘(1𝑜 mPoly 𝑅))
6620, 65mgpplusg 18316 . . . . . . . . . . . . . . 15 (.r𝑃) = (+g‘(mulGrp‘(1𝑜 mPoly 𝑅)))
6757, 64mgpplusg 18316 . . . . . . . . . . . . . . 15 (.r𝑃) = (+g𝑀)
6866, 67eqtr3i 2634 . . . . . . . . . . . . . 14 (+g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = (+g𝑀)
6968oveqi 6562 . . . . . . . . . . . . 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 17407 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (.g‘(mulGrp‘(1𝑜 mPoly 𝑅))) = )
7271oveqd 6566 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) = ((𝑎‘∅) 𝑋))
7372adantr 480 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) = ((𝑎‘∅) 𝑋))
747ply1ring 19439 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
7557ringmgp 18376 . . . . . . . . . . . 12 (𝑃 ∈ Ring → 𝑀 ∈ Mnd)
7674, 75syl 17 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑀 ∈ Mnd)
7776ad2antrr 758 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 𝑀 ∈ Mnd)
78 elmapi 7765 . . . . . . . . . . . 12 (𝑎 ∈ (ℕ0𝑚 1𝑜) → 𝑎:1𝑜⟶ℕ0)
79 0lt1o 7471 . . . . . . . . . . . 12 ∅ ∈ 1𝑜
80 ffvelrn 6265 . . . . . . . . . . . 12 ((𝑎:1𝑜⟶ℕ0 ∧ ∅ ∈ 1𝑜) → (𝑎‘∅) ∈ ℕ0)
8178, 79, 80sylancl 693 . . . . . . . . . . 11 (𝑎 ∈ (ℕ0𝑚 1𝑜) → (𝑎‘∅) ∈ ℕ0)
8281adantl 481 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → (𝑎‘∅) ∈ ℕ0)
83 ply1coe.x . . . . . . . . . . . 12 𝑋 = (var1𝑅)
8483, 7, 9vr1cl 19408 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑋𝐵)
8584ad2antrr 758 . . . . . . . . . 10 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → 𝑋𝐵)
8658, 54mulgnn0cl 17381 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ (𝑎‘∅) ∈ ℕ0𝑋𝐵) → ((𝑎‘∅) 𝑋) ∈ 𝐵)
8777, 82, 85, 86syl3anc 1318 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((𝑎‘∅) 𝑋) ∈ 𝐵)
8873, 87eqeltrd 2688 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋) ∈ 𝐵)
89 fveq2 6103 . . . . . . . . . 10 (𝑐 = ∅ → (𝑎𝑐) = (𝑎‘∅))
90 fveq2 6103 . . . . . . . . . . 11 (𝑐 = ∅ → ((1𝑜 mVar 𝑅)‘𝑐) = ((1𝑜 mVar 𝑅)‘∅))
9183vr1val 19383 . . . . . . . . . . 11 𝑋 = ((1𝑜 mVar 𝑅)‘∅)
9290, 91syl6eqr 2662 . . . . . . . . . 10 (𝑐 = ∅ → ((1𝑜 mVar 𝑅)‘𝑐) = 𝑋)
9389, 92oveq12d 6567 . . . . . . . . 9 (𝑐 = ∅ → ((𝑎𝑐)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))((1𝑜 mVar 𝑅)‘𝑐)) = ((𝑎‘∅)(.g‘(mulGrp‘(1𝑜 mPoly 𝑅)))𝑋))
9455, 93gsumsn 18177 . . . . . . . 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 1318 . . . . . . 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 2656 . . . . . 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 2648 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅))) = ((𝑎‘∅) 𝑋))
9818, 97oveq12d 6567 . . . 4 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑎 ∈ (ℕ0𝑚 1𝑜)) → ((𝐾𝑎) · (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅)))) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋)))
9998mpteq2dva 4672 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐾𝑎) · (𝑏 ∈ (ℕ0𝑚 1𝑜) ↦ if(𝑏 = 𝑎, (1r𝑅), (0g𝑅))))) = (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋))))
10099oveq2d 6565 . 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 11175 . . . . . 6 0 ∈ V
102101mptex 6390 . . . . 5 (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∈ V
103102a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∈ V)
104 fvex 6113 . . . . . 6 (Poly1𝑅) ∈ V
1057, 104eqeltri 2684 . . . . 5 𝑃 ∈ V
106105a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝑃 ∈ V)
107 ovex 6577 . . . . 5 (1𝑜 mPoly 𝑅) ∈ V
108107a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (1𝑜 mPoly 𝑅) ∈ V)
1099, 10eqtr3i 2634 . . . . 5 (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅))
110109a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (Base‘𝑃) = (Base‘(1𝑜 mPoly 𝑅)))
111 eqid 2610 . . . . . 6 (+g𝑃) = (+g𝑃)
1127, 1, 111ply1plusg 19416 . . . . 5 (+g𝑃) = (+g‘(1𝑜 mPoly 𝑅))
113112a1i 11 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (+g𝑃) = (+g‘(1𝑜 mPoly 𝑅)))
114103, 106, 108, 110, 113gsumpropd 17095 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))) = ((1𝑜 mPoly 𝑅) Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))
115 eqid 2610 . . . . 5 (0g𝑃) = (0g𝑃)
1161, 7, 115ply1mpl0 19446 . . . 4 (0g𝑃) = (0g‘(1𝑜 mPoly 𝑅))
1171mpllmod 19272 . . . . . 6 ((1𝑜 ∈ ω ∧ 𝑅 ∈ Ring) → (1𝑜 mPoly 𝑅) ∈ LMod)
1185, 13, 117sylancr 694 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (1𝑜 mPoly 𝑅) ∈ LMod)
119 lmodcmn 18734 . . . . 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 19443 . . . . . . 7 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
123122ad2antrr 758 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑃 ∈ LMod)
124 eqid 2610 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
12516, 9, 7, 124coe1f 19402 . . . . . . . . 9 (𝐾𝐵𝐴:ℕ0⟶(Base‘𝑅))
126125adantl 481 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐴:ℕ0⟶(Base‘𝑅))
127126ffvelrnda 6267 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ (Base‘𝑅))
1287ply1sca 19444 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
129128eqcomd 2616 . . . . . . . . 9 (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
130129ad2antrr 758 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
131130fveq2d 6107 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
132127, 131eleqtrrd 2691 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ (Base‘(Scalar‘𝑃)))
13376ad2antrr 758 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ Mnd)
134 simpr 476 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
13584ad2antrr 758 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → 𝑋𝐵)
13658, 54mulgnn0cl 17381 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑘 ∈ ℕ0𝑋𝐵) → (𝑘 𝑋) ∈ 𝐵)
137133, 134, 135, 136syl3anc 1318 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → (𝑘 𝑋) ∈ 𝐵)
138 eqid 2610 . . . . . . 7 (Scalar‘𝑃) = (Scalar‘𝑃)
139 eqid 2610 . . . . . . 7 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
1409, 138, 11, 139lmodvscl 18703 . . . . . 6 ((𝑃 ∈ LMod ∧ (𝐴𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑘 𝑋) ∈ 𝐵) → ((𝐴𝑘) · (𝑘 𝑋)) ∈ 𝐵)
141123, 132, 137, 140syl3anc 1318 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐾𝐵) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) · (𝑘 𝑋)) ∈ 𝐵)
142 eqid 2610 . . . . 5 (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))
143141, 142fmptd 6292 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))):ℕ0𝐵)
1447, 83, 9, 11, 57, 54, 16ply1coefsupp 19486 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) finSupp (0g𝑃))
145 eqid 2610 . . . . . 6 (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅))
14640, 101, 26, 145mapsnf1o2 7791 . . . . 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 18140 . . 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 2611 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)) = (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅)))
150 eqidd 2611 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) = (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))))
151 fveq2 6103 . . . . . 6 (𝑘 = (𝑎‘∅) → (𝐴𝑘) = (𝐴‘(𝑎‘∅)))
152 oveq1 6556 . . . . . 6 (𝑘 = (𝑎‘∅) → (𝑘 𝑋) = ((𝑎‘∅) 𝑋))
153151, 152oveq12d 6567 . . . . 5 (𝑘 = (𝑎‘∅) → ((𝐴𝑘) · (𝑘 𝑋)) = ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋)))
15482, 149, 150, 153fmptco 6303 . . . 4 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → ((𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋))) ∘ (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ (𝑎‘∅))) = (𝑎 ∈ (ℕ0𝑚 1𝑜) ↦ ((𝐴‘(𝑎‘∅)) · ((𝑎‘∅) 𝑋))))
155154oveq2d 6565 . . 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 2649 . 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 2648 1 ((𝑅 ∈ Ring ∧ 𝐾𝐵) → 𝐾 = (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝐴𝑘) · (𝑘 𝑋)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125   ↦ cmpt 4643   ∘ ccom 5042  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ωcom 6957  1𝑜c1o 7440   ↑𝑚 cmap 7744  ℕ0cn0 11169  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923   Σg cgsu 15924  Mndcmnd 17117  .gcmg 17363  CMndccmn 18016  mulGrpcmgp 18312  1rcur 18324  Ringcrg 18370  LModclmod 18686   mVar cmvr 19173   mPoly cmpl 19174  PwSer1cps1 19366  var1cv1 19367  Poly1cpl1 19368  coe1cco1 19369 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-ple 15788  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-srg 18329  df-ring 18372  df-subrg 18601  df-lmod 18688  df-lss 18754  df-psr 19177  df-mvr 19178  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-vr1 19372  df-ply1 19373  df-coe1 19374 This theorem is referenced by:  eqcoe1ply1eq  19488  pmatcollpw1lem2  20399  mp2pm2mp  20435  plypf1  23772
 Copyright terms: Public domain W3C validator