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Theorem pmatcollpwlem 20404
 Description: Lemma for pmatcollpw 20405. (Contributed by AV, 26-Oct-2019.) (Revised by AV, 4-Dec-2019.)
Hypotheses
Ref Expression
pmatcollpw.p 𝑃 = (Poly1𝑅)
pmatcollpw.c 𝐶 = (𝑁 Mat 𝑃)
pmatcollpw.b 𝐵 = (Base‘𝐶)
pmatcollpw.m = ( ·𝑠𝐶)
pmatcollpw.e = (.g‘(mulGrp‘𝑃))
pmatcollpw.x 𝑋 = (var1𝑅)
pmatcollpw.t 𝑇 = (𝑁 matToPolyMat 𝑅)
Assertion
Ref Expression
pmatcollpwlem ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)) = (𝑎((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))

Proof of Theorem pmatcollpwlem
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pmatcollpw.p . . . . . . . 8 𝑃 = (Poly1𝑅)
21ply1assa 19390 . . . . . . 7 (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
323ad2ant2 1076 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑃 ∈ AssAlg)
43adantr 480 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑃 ∈ AssAlg)
543ad2ant1 1075 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → 𝑃 ∈ AssAlg)
6 eqid 2610 . . . . . 6 (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
7 eqid 2610 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
8 eqid 2610 . . . . . 6 (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
9 simp2 1055 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → 𝑎𝑁)
10 simp3 1056 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → 𝑏𝑁)
11 simp2 1055 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ CRing)
1211adantr 480 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ CRing)
13 simp3 1056 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀𝐵)
1413adantr 480 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑀𝐵)
15 simpr 476 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
16 pmatcollpw.c . . . . . . . . 9 𝐶 = (𝑁 Mat 𝑃)
17 pmatcollpw.b . . . . . . . . 9 𝐵 = (Base‘𝐶)
181, 16, 17, 6, 8decpmatcl 20391 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝑀𝐵𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
1912, 14, 15, 18syl3anc 1318 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
20193ad2ant1 1075 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
216, 7, 8, 9, 10, 20matecld 20051 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅))
22 crngring 18381 . . . . . . . . . . . 12 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
23223ad2ant2 1076 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
241ply1sca 19444 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
2523, 24syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 = (Scalar‘𝑃))
2625eqcomd 2616 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Scalar‘𝑃) = 𝑅)
2726fveq2d 6107 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
2827eleq2d 2673 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅)))
2928adantr 480 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅)))
30293ad2ant1 1075 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅)))
3121, 30mpbird 246 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)))
32 pmatcollpw.x . . . . . . 7 𝑋 = (var1𝑅)
33 eqid 2610 . . . . . . 7 (mulGrp‘𝑃) = (mulGrp‘𝑃)
34 pmatcollpw.e . . . . . . 7 = (.g‘(mulGrp‘𝑃))
35 eqid 2610 . . . . . . 7 (Base‘𝑃) = (Base‘𝑃)
361, 32, 33, 34, 35ply1moncl 19462 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ (Base‘𝑃))
3723, 36sylan 487 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ (Base‘𝑃))
38373ad2ant1 1075 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → (𝑛 𝑋) ∈ (Base‘𝑃))
39 eqid 2610 . . . . 5 (algSc‘𝑃) = (algSc‘𝑃)
40 eqid 2610 . . . . 5 (Scalar‘𝑃) = (Scalar‘𝑃)
41 eqid 2610 . . . . 5 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
42 eqid 2610 . . . . 5 (.r𝑃) = (.r𝑃)
43 eqid 2610 . . . . 5 ( ·𝑠𝑃) = ( ·𝑠𝑃)
4439, 40, 41, 35, 42, 43asclmul2 19161 . . . 4 ((𝑃 ∈ AssAlg ∧ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑛 𝑋) ∈ (Base‘𝑃)) → ((𝑛 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏))) = ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)))
455, 31, 38, 44syl3anc 1318 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → ((𝑛 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏))) = ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)))
46 eqidd 2611 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))
47 oveq12 6558 . . . . . . . 8 ((𝑖 = 𝑎𝑗 = 𝑏) → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑎(𝑀 decompPMat 𝑛)𝑏))
4847fveq2d 6107 . . . . . . 7 ((𝑖 = 𝑎𝑗 = 𝑏) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)) = ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)))
4948adantl 481 . . . . . 6 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) ∧ (𝑖 = 𝑎𝑗 = 𝑏)) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)) = ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)))
50 fvex 6113 . . . . . . 7 ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)) ∈ V
5150a1i 11 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)) ∈ V)
5246, 49, 9, 10, 51ovmpt2d 6686 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏) = ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)))
5352eqcomd 2616 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏))
5453oveq2d 6565 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → ((𝑛 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏))) = ((𝑛 𝑋)(.r𝑃)(𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏)))
5545, 54eqtr3d 2646 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)) = ((𝑛 𝑋)(.r𝑃)(𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏)))
561ply1ring 19439 . . . . . . 7 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5722, 56syl 17 . . . . . 6 (𝑅 ∈ CRing → 𝑃 ∈ Ring)
58573ad2ant2 1076 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑃 ∈ Ring)
5958adantr 480 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑃 ∈ Ring)
60593ad2ant1 1075 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → 𝑃 ∈ Ring)
61 simpl1 1057 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
6212, 22syl 17 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
63623ad2ant1 1075 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
64 simp2 1055 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
65 simp3 1056 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
66193ad2ant1 1075 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
676, 7, 8, 64, 65, 66matecld 20051 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑀 decompPMat 𝑛)𝑗) ∈ (Base‘𝑅))
681, 39, 7, 35ply1sclcl 19477 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑖(𝑀 decompPMat 𝑛)𝑗) ∈ (Base‘𝑅)) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)) ∈ (Base‘𝑃))
6963, 67, 68syl2anc 691 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)) ∈ (Base‘𝑃))
7016, 35, 17, 61, 59, 69matbas2d 20048 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) ∈ 𝐵)
7137, 70jca 553 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑋) ∈ (Base‘𝑃) ∧ (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) ∈ 𝐵))
72713ad2ant1 1075 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → ((𝑛 𝑋) ∈ (Base‘𝑃) ∧ (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) ∈ 𝐵))
739, 10jca 553 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → (𝑎𝑁𝑏𝑁))
74 pmatcollpw.m . . . 4 = ( ·𝑠𝐶)
7516, 17, 35, 74, 42matvscacell 20061 . . 3 ((𝑃 ∈ Ring ∧ ((𝑛 𝑋) ∈ (Base‘𝑃) ∧ (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) ∈ 𝐵) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎((𝑛 𝑋) (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))𝑏) = ((𝑛 𝑋)(.r𝑃)(𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏)))
7660, 72, 73, 75syl3anc 1318 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → (𝑎((𝑛 𝑋) (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))𝑏) = ((𝑛 𝑋)(.r𝑃)(𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏)))
7723adantr 480 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
78 pmatcollpw.t . . . . . . . 8 𝑇 = (𝑁 matToPolyMat 𝑅)
7978, 6, 8, 1, 39mat2pmatval 20348 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅))) → (𝑇‘(𝑀 decompPMat 𝑛)) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))
8061, 77, 19, 79syl3anc 1318 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑀 decompPMat 𝑛)) = (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))
8180eqcomd 2616 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) = (𝑇‘(𝑀 decompPMat 𝑛)))
8281oveq2d 6565 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑛 𝑋) (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))) = ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))
8382oveqd 6566 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑎((𝑛 𝑋) (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))𝑏) = (𝑎((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
84833ad2ant1 1075 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → (𝑎((𝑛 𝑋) (𝑖𝑁, 𝑗𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))𝑏) = (𝑎((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
8555, 76, 843eqtr2d 2650 1 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)) = (𝑎((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  ℕ0cn0 11169  Basecbs 15695  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  .gcmg 17363  mulGrpcmgp 18312  Ringcrg 18370  CRingccrg 18371  AssAlgcasa 19130  algSccascl 19132  var1cv1 19367  Poly1cpl1 19368   Mat cmat 20032   matToPolyMat cmat2pmat 20328   decompPMat cdecpmat 20386 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-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-ot 4134  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-sup 8231  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-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  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-ring 18372  df-cring 18373  df-subrg 18601  df-lmod 18688  df-lss 18754  df-sra 18993  df-rgmod 18994  df-assa 19133  df-ascl 19135  df-psr 19177  df-mvr 19178  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-vr1 19372  df-ply1 19373  df-coe1 19374  df-dsmm 19895  df-frlm 19910  df-mat 20033  df-mat2pmat 20331  df-decpmat 20387 This theorem is referenced by:  pmatcollpw  20405
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