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Theorem pmatcollpwscmatlem2 20965
Description: Lemma 2 for pmatcollpwscmat 20966. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
Hypotheses
Ref Expression
pmatcollpwscmat.p 𝑃 = (Poly1𝑅)
pmatcollpwscmat.c 𝐶 = (𝑁 Mat 𝑃)
pmatcollpwscmat.b 𝐵 = (Base‘𝐶)
pmatcollpwscmat.m1 = ( ·𝑠𝐶)
pmatcollpwscmat.e1 = (.g‘(mulGrp‘𝑃))
pmatcollpwscmat.x 𝑋 = (var1𝑅)
pmatcollpwscmat.t 𝑇 = (𝑁 matToPolyMat 𝑅)
pmatcollpwscmat.a 𝐴 = (𝑁 Mat 𝑅)
pmatcollpwscmat.d 𝐷 = (Base‘𝐴)
pmatcollpwscmat.u 𝑈 = (algSc‘𝑃)
pmatcollpwscmat.k 𝐾 = (Base‘𝑅)
pmatcollpwscmat.e2 𝐸 = (Base‘𝑃)
pmatcollpwscmat.s 𝑆 = (algSc‘𝑃)
pmatcollpwscmat.1 1 = (1r𝐶)
pmatcollpwscmat.m2 𝑀 = (𝑄 1 )
Assertion
Ref Expression
pmatcollpwscmatlem2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))

Proof of Theorem pmatcollpwscmatlem2
Dummy variables 𝑎 𝑏 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 476 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
2 simpr 479 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
32adantr 474 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑅 ∈ Ring)
4 simpr 479 . . . . . . . 8 ((𝐿 ∈ ℕ0𝑄𝐸) → 𝑄𝐸)
54anim2i 612 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑄𝐸))
6 df-3an 1115 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑄𝐸))
75, 6sylibr 226 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸))
8 pmatcollpwscmat.m2 . . . . . . 7 𝑀 = (𝑄 1 )
9 pmatcollpwscmat.p . . . . . . . 8 𝑃 = (Poly1𝑅)
10 pmatcollpwscmat.c . . . . . . . 8 𝐶 = (𝑁 Mat 𝑃)
11 pmatcollpwscmat.b . . . . . . . 8 𝐵 = (Base‘𝐶)
12 pmatcollpwscmat.e2 . . . . . . . 8 𝐸 = (Base‘𝑃)
13 pmatcollpwscmat.m1 . . . . . . . 8 = ( ·𝑠𝐶)
14 pmatcollpwscmat.1 . . . . . . . 8 1 = (1r𝐶)
159, 10, 11, 12, 13, 141pmatscmul 20877 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) → (𝑄 1 ) ∈ 𝐵)
168, 15syl5eqel 2910 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) → 𝑀𝐵)
177, 16syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑀𝐵)
18 simprl 789 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝐿 ∈ ℕ0)
19 pmatcollpwscmat.a . . . . . 6 𝐴 = (𝑁 Mat 𝑅)
20 pmatcollpwscmat.d . . . . . 6 𝐷 = (Base‘𝐴)
219, 10, 11, 19, 20decpmatcl 20942 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0) → (𝑀 decompPMat 𝐿) ∈ 𝐷)
223, 17, 18, 21syl3anc 1496 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑀 decompPMat 𝐿) ∈ 𝐷)
23 df-3an 1115 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷))
241, 22, 23sylanbrc 580 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷))
25 pmatcollpwscmat.t . . . 4 𝑇 = (𝑁 matToPolyMat 𝑅)
26 eqid 2825 . . . 4 (algSc‘𝑃) = (algSc‘𝑃)
2725, 19, 20, 9, 26mat2pmatval 20899 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝐿) ∈ 𝐷) → (𝑇‘(𝑀 decompPMat 𝐿)) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗))))
2824, 27syl 17 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗))))
293, 17, 183jca 1164 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0))
30293ad2ant1 1169 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0))
31 3simpc 1188 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑁𝑗𝑁))
329, 10, 11decpmate 20941 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵𝐿 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝐿)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝐿))
3330, 31, 32syl2anc 581 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑀 decompPMat 𝐿)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝐿))
3433fveq2d 6437 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗)) = ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)))
3534mpt2eq3dva 6979 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝐿)𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿))))
36 simp1lr 1324 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
37 simp2 1173 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
38 simp3 1174 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
39173ad2ant1 1169 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝑀𝐵)
4010, 12, 11, 37, 38, 39matecld 20599 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ 𝐸)
41183ad2ant1 1169 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 𝐿 ∈ ℕ0)
42 eqid 2825 . . . . . . 7 (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑖𝑀𝑗))
43 pmatcollpwscmat.k . . . . . . 7 𝐾 = (Base‘𝑅)
4442, 12, 9, 43coe1fvalcl 19942 . . . . . 6 (((𝑖𝑀𝑗) ∈ 𝐸𝐿 ∈ ℕ0) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾)
4540, 41, 44syl2anc 581 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾)
46 eqid 2825 . . . . . 6 (var1𝑅) = (var1𝑅)
47 eqid 2825 . . . . . 6 ( ·𝑠𝑃) = ( ·𝑠𝑃)
48 eqid 2825 . . . . . 6 (mulGrp‘𝑃) = (mulGrp‘𝑃)
49 eqid 2825 . . . . . 6 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
5043, 9, 46, 47, 48, 49, 26ply1scltm 20011 . . . . 5 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾) → ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)) = (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
5136, 45, 50syl2anc 581 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿)) = (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
5251mpt2eq3dva 6979 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿))) = (𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))))
53 pmatcollpwscmat.e1 . . . . . . 7 = (.g‘(mulGrp‘𝑃))
54 pmatcollpwscmat.x . . . . . . 7 𝑋 = (var1𝑅)
55 pmatcollpwscmat.u . . . . . . 7 𝑈 = (algSc‘𝑃)
56 pmatcollpwscmat.s . . . . . . 7 𝑆 = (algSc‘𝑃)
579, 10, 11, 13, 53, 54, 25, 19, 20, 55, 43, 12, 56, 14, 8pmatcollpwscmatlem1 20964 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))
58 eqidd 2826 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = (𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))))
59 oveq12 6914 . . . . . . . . . . 11 ((𝑖 = 𝑎𝑗 = 𝑏) → (𝑖𝑀𝑗) = (𝑎𝑀𝑏))
6059fveq2d 6437 . . . . . . . . . 10 ((𝑖 = 𝑎𝑗 = 𝑏) → (coe1‘(𝑖𝑀𝑗)) = (coe1‘(𝑎𝑀𝑏)))
6160fveq1d 6435 . . . . . . . . 9 ((𝑖 = 𝑎𝑗 = 𝑏) → ((coe1‘(𝑖𝑀𝑗))‘𝐿) = ((coe1‘(𝑎𝑀𝑏))‘𝐿))
6261oveq1d 6920 . . . . . . . 8 ((𝑖 = 𝑎𝑗 = 𝑏) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
6362adantl 475 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) ∧ (𝑖 = 𝑎𝑗 = 𝑏)) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
64 simprl 789 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑎𝑁)
65 simprr 791 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑏𝑁)
66 ovexd 6939 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ V)
6758, 63, 64, 65, 66ovmpt2d 7048 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))𝑏) = (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
68 simpll 785 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑁 ∈ Fin)
699ply1ring 19978 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
7069adantl 475 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
7170adantr 474 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → 𝑃 ∈ Ring)
72 pm3.22 453 . . . . . . . . . . 11 ((𝐿 ∈ ℕ0𝑄𝐸) → (𝑄𝐸𝐿 ∈ ℕ0))
7372adantl 475 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑄𝐸𝐿 ∈ ℕ0))
74 eqid 2825 . . . . . . . . . . 11 (coe1𝑄) = (coe1𝑄)
7574, 12, 9, 43coe1fvalcl 19942 . . . . . . . . . 10 ((𝑄𝐸𝐿 ∈ ℕ0) → ((coe1𝑄)‘𝐿) ∈ 𝐾)
7673, 75syl 17 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((coe1𝑄)‘𝐿) ∈ 𝐾)
779, 55, 43, 12ply1sclcl 20016 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ ((coe1𝑄)‘𝐿) ∈ 𝐾) → (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸)
783, 76, 77syl2anc 581 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸)
7968, 71, 783jca 1164 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸))
80 eqid 2825 . . . . . . . 8 (0g𝑃) = (0g𝑃)
8110, 12, 80, 14, 13scmatscmide 20681 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring ∧ (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))
8279, 81sylan 577 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))
8357, 67, 823eqtr4d 2871 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))𝑏) = (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏))
8483ralrimivva 3180 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ∀𝑎𝑁𝑏𝑁 (𝑎(𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))𝑏) = (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏))
85 0nn0 11635 . . . . . . . 8 0 ∈ ℕ0
8685a1i 11 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → 0 ∈ ℕ0)
8743, 9, 46, 47, 48, 49, 12ply1tmcl 20002 . . . . . . 7 ((𝑅 ∈ Ring ∧ ((coe1‘(𝑖𝑀𝑗))‘𝐿) ∈ 𝐾 ∧ 0 ∈ ℕ0) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ 𝐸)
8836, 45, 86, 87syl3anc 1496 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ 𝑖𝑁𝑗𝑁) → (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ 𝐸)
8910, 12, 11, 68, 71, 88matbas2d 20596 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) ∈ 𝐵)
909, 10, 11, 12, 13, 141pmatscmul 20877 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑈‘((coe1𝑄)‘𝐿)) ∈ 𝐸) → ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ∈ 𝐵)
9168, 3, 78, 90syl3anc 1496 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ∈ 𝐵)
9210, 11eqmat 20597 . . . . 5 (((𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) ∈ 𝐵 ∧ ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ∈ 𝐵) → ((𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ↔ ∀𝑎𝑁𝑏𝑁 (𝑎(𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))𝑏) = (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏)))
9389, 91, 92syl2anc 581 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → ((𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ) ↔ ∀𝑎𝑁𝑏𝑁 (𝑎(𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))𝑏) = (𝑎((𝑈‘((coe1𝑄)‘𝐿)) 1 )𝑏)))
9484, 93mpbird 249 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ (((coe1‘(𝑖𝑀𝑗))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))
9552, 94eqtrd 2861 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘((coe1‘(𝑖𝑀𝑗))‘𝐿))) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))
9628, 35, 953eqtrd 2865 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3117  Vcvv 3414  ifcif 4306  cfv 6123  (class class class)co 6905  cmpt2 6907  Fincfn 8222  0cc0 10252  0cn0 11618  Basecbs 16222   ·𝑠 cvsca 16309  0gc0g 16453  .gcmg 17894  mulGrpcmgp 18843  1rcur 18855  Ringcrg 18901  algSccascl 19672  var1cv1 19906  Poly1cpl1 19907  coe1cco1 19908   Mat cmat 20580   matToPolyMat cmat2pmat 20879   decompPMat cdecpmat 20937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-ot 4406  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-of 7157  df-ofr 7158  df-om 7327  df-1st 7428  df-2nd 7429  df-supp 7560  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-ixp 8176  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-fsupp 8545  df-sup 8617  df-oi 8684  df-card 9078  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-z 11705  df-dec 11822  df-uz 11969  df-fz 12620  df-fzo 12761  df-seq 13096  df-hash 13411  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-sets 16229  df-ress 16230  df-plusg 16318  df-mulr 16319  df-sca 16321  df-vsca 16322  df-ip 16323  df-tset 16324  df-ple 16325  df-ds 16327  df-hom 16329  df-cco 16330  df-0g 16455  df-gsum 16456  df-prds 16461  df-pws 16463  df-mre 16599  df-mrc 16600  df-acs 16602  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-mhm 17688  df-submnd 17689  df-grp 17779  df-minusg 17780  df-sbg 17781  df-mulg 17895  df-subg 17942  df-ghm 18009  df-cntz 18100  df-cmn 18548  df-abl 18549  df-mgp 18844  df-ur 18856  df-ring 18903  df-subrg 19134  df-lmod 19221  df-lss 19289  df-sra 19533  df-rgmod 19534  df-ascl 19675  df-psr 19717  df-mvr 19718  df-mpl 19719  df-opsr 19721  df-psr1 19910  df-vr1 19911  df-ply1 19912  df-coe1 19913  df-dsmm 20439  df-frlm 20454  df-mamu 20557  df-mat 20581  df-mat2pmat 20882  df-decpmat 20938
This theorem is referenced by:  pmatcollpwscmat  20966
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