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Theorem monmatcollpw 22250
Description: The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix having scaled monomials with the same power as entries is the matrix of the coefficients of the monomials or a zero matrix. Generalization of decpmatid 22241 (but requires 𝑅 to be commutative!). (Contributed by AV, 11-Nov-2019.) (Revised by AV, 4-Dec-2019.)
Hypotheses
Ref Expression
monmatcollpw.p 𝑃 = (Poly1𝑅)
monmatcollpw.c 𝐶 = (𝑁 Mat 𝑃)
monmatcollpw.a 𝐴 = (𝑁 Mat 𝑅)
monmatcollpw.k 𝐾 = (Base‘𝐴)
monmatcollpw.0 0 = (0g𝐴)
monmatcollpw.e = (.g‘(mulGrp‘𝑃))
monmatcollpw.x 𝑋 = (var1𝑅)
monmatcollpw.m · = ( ·𝑠𝐶)
monmatcollpw.t 𝑇 = (𝑁 matToPolyMat 𝑅)
Assertion
Ref Expression
monmatcollpw (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (((𝐿 𝑋) · (𝑇𝑀)) decompPMat 𝐼) = if(𝐼 = 𝐿, 𝑀, 0 ))

Proof of Theorem monmatcollpw
Dummy variables 𝑖 𝑗 𝑙 𝑥 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 766 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑁 ∈ Fin)
2 crngring 20050 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3 monmatcollpw.p . . . . . . 7 𝑃 = (Poly1𝑅)
43ply1ring 21741 . . . . . 6 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
52, 4syl 17 . . . . 5 (𝑅 ∈ CRing → 𝑃 ∈ Ring)
65ad2antlr 726 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑃 ∈ Ring)
72adantl 483 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
8 simp2 1138 . . . . . 6 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝐿 ∈ ℕ0)
9 monmatcollpw.x . . . . . . 7 𝑋 = (var1𝑅)
10 eqid 2733 . . . . . . 7 (mulGrp‘𝑃) = (mulGrp‘𝑃)
11 monmatcollpw.e . . . . . . 7 = (.g‘(mulGrp‘𝑃))
12 eqid 2733 . . . . . . 7 (Base‘𝑃) = (Base‘𝑃)
133, 9, 10, 11, 12ply1moncl 21764 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐿 ∈ ℕ0) → (𝐿 𝑋) ∈ (Base‘𝑃))
147, 8, 13syl2an 597 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝐿 𝑋) ∈ (Base‘𝑃))
152anim2i 618 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
16 simp1 1137 . . . . . . . 8 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝑀𝐾)
1715, 16anim12i 614 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀𝐾))
18 df-3an 1090 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀𝐾))
1917, 18sylibr 233 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐾))
20 monmatcollpw.t . . . . . . 7 𝑇 = (𝑁 matToPolyMat 𝑅)
21 monmatcollpw.a . . . . . . 7 𝐴 = (𝑁 Mat 𝑅)
22 monmatcollpw.k . . . . . . 7 𝐾 = (Base‘𝐴)
23 monmatcollpw.c . . . . . . 7 𝐶 = (𝑁 Mat 𝑃)
2420, 21, 22, 3, 23mat2pmatbas 22197 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐾) → (𝑇𝑀) ∈ (Base‘𝐶))
2519, 24syl 17 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑇𝑀) ∈ (Base‘𝐶))
2614, 25jca 513 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝐿 𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑀) ∈ (Base‘𝐶)))
27 eqid 2733 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
28 monmatcollpw.m . . . . 5 · = ( ·𝑠𝐶)
2912, 23, 27, 28matvscl 21902 . . . 4 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝐿 𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑀) ∈ (Base‘𝐶))) → ((𝐿 𝑋) · (𝑇𝑀)) ∈ (Base‘𝐶))
301, 6, 26, 29syl21anc 837 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝐿 𝑋) · (𝑇𝑀)) ∈ (Base‘𝐶))
31 simpr3 1197 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝐼 ∈ ℕ0)
3223, 27decpmatval 22236 . . 3 ((((𝐿 𝑋) · (𝑇𝑀)) ∈ (Base‘𝐶) ∧ 𝐼 ∈ ℕ0) → (((𝐿 𝑋) · (𝑇𝑀)) decompPMat 𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗))‘𝐼)))
3330, 31, 32syl2anc 585 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (((𝐿 𝑋) · (𝑇𝑀)) decompPMat 𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗))‘𝐼)))
3463ad2ant1 1134 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑃 ∈ Ring)
35263ad2ant1 1134 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑀) ∈ (Base‘𝐶)))
36 3simpc 1151 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑁𝑗𝑁))
37 eqid 2733 . . . . . . . 8 (.r𝑃) = (.r𝑃)
3823, 27, 12, 28, 37matvscacell 21907 . . . . . . 7 ((𝑃 ∈ Ring ∧ ((𝐿 𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑀) ∈ (Base‘𝐶)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗) = ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)))
3934, 35, 36, 38syl3anc 1372 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗) = ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)))
4039fveq2d 6885 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘(𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗)) = (coe1‘((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗))))
4140fveq1d 6883 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗))‘𝐼) = ((coe1‘((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)))‘𝐼))
4216anim2i 618 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀𝐾))
43 df-3an 1090 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀𝐾))
4442, 43sylibr 233 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾))
45443ad2ant1 1134 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾))
46 eqid 2733 . . . . . . . . . 10 (algSc‘𝑃) = (algSc‘𝑃)
4720, 21, 22, 3, 46mat2pmatvalel 22196 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇𝑀)𝑗) = ((algSc‘𝑃)‘(𝑖𝑀𝑗)))
4845, 36, 47syl2anc 585 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑇𝑀)𝑗) = ((algSc‘𝑃)‘(𝑖𝑀𝑗)))
4948oveq2d 7412 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)) = ((𝐿 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))))
503ply1assa 21692 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
5150ad2antlr 726 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑃 ∈ AssAlg)
52513ad2ant1 1134 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑃 ∈ AssAlg)
53 eqid 2733 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
54 eqid 2733 . . . . . . . . . 10 (Base‘𝐴) = (Base‘𝐴)
55 simp2 1138 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
56 simp3 1139 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
5722eleq2i 2826 . . . . . . . . . . . . . 14 (𝑀𝐾𝑀 ∈ (Base‘𝐴))
5857biimpi 215 . . . . . . . . . . . . 13 (𝑀𝐾𝑀 ∈ (Base‘𝐴))
59583ad2ant1 1134 . . . . . . . . . . . 12 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝑀 ∈ (Base‘𝐴))
6059adantl 483 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑀 ∈ (Base‘𝐴))
61603ad2ant1 1134 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑀 ∈ (Base‘𝐴))
6221, 53, 54, 55, 56, 61matecld 21897 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
633ply1sca 21746 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
6463adantl 483 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝑃))
6564eqcomd 2739 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝑃) = 𝑅)
6665fveq2d 6885 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
6766adantr 482 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
68673ad2ant1 1134 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
6962, 68eleqtrrd 2837 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Scalar‘𝑃)))
70143ad2ant1 1134 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝐿 𝑋) ∈ (Base‘𝑃))
71 eqid 2733 . . . . . . . . 9 (Scalar‘𝑃) = (Scalar‘𝑃)
72 eqid 2733 . . . . . . . . 9 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
73 eqid 2733 . . . . . . . . 9 ( ·𝑠𝑃) = ( ·𝑠𝑃)
7446, 71, 72, 12, 37, 73asclmul2 21412 . . . . . . . 8 ((𝑃 ∈ AssAlg ∧ (𝑖𝑀𝑗) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝐿 𝑋) ∈ (Base‘𝑃)) → ((𝐿 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))) = ((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))
7552, 69, 70, 74syl3anc 1372 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))) = ((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))
7649, 75eqtrd 2773 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)) = ((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))
7776fveq2d 6885 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗))) = (coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋))))
7877fveq1d 6883 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)))‘𝐼) = ((coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))‘𝐼))
792ad2antlr 726 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑅 ∈ Ring)
80793ad2ant1 1134 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
81 simp1r2 1271 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝐿 ∈ ℕ0)
82 eqid 2733 . . . . . . 7 (0g𝑅) = (0g𝑅)
8382, 53, 3, 9, 73, 10, 11coe1tm 21766 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑖𝑀𝑗) ∈ (Base‘𝑅) ∧ 𝐿 ∈ ℕ0) → (coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
8480, 62, 81, 83syl3anc 1372 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
8584fveq1d 6883 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))‘𝐼) = ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼))
8641, 78, 853eqtrd 2777 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗))‘𝐼) = ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼))
8786mpoeq3dva 7473 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗))‘𝐼)) = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼)))
88 monmatcollpw.0 . . . . . . . . 9 0 = (0g𝐴)
8915adantr 482 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
9089adantr 482 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
9121, 82mat0op 21890 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑧𝑁, 𝑤𝑁 ↦ (0g𝑅)))
9290, 91syl 17 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (0g𝐴) = (𝑧𝑁, 𝑤𝑁 ↦ (0g𝑅)))
9388, 92eqtrid 2785 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 0 = (𝑧𝑁, 𝑤𝑁 ↦ (0g𝑅)))
94 eqidd 2734 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) ∧ (𝑧 = 𝑥𝑤 = 𝑦)) → (0g𝑅) = (0g𝑅))
95 simprl 770 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 𝑥𝑁)
96 simprr 772 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 𝑦𝑁)
97 fvexd 6896 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (0g𝑅) ∈ V)
9893, 94, 95, 96, 97ovmpod 7547 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥 0 𝑦) = (0g𝑅))
9998eqcomd 2739 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (0g𝑅) = (𝑥 0 𝑦))
10099ifeq2d 4544 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (𝑥 0 𝑦)))
101 eqidd 2734 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼)) = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼)))
102 oveq12 7405 . . . . . . . . . 10 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
103102ifeq1d 4543 . . . . . . . . 9 ((𝑖 = 𝑥𝑗 = 𝑦) → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) = if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
104103mpteq2dv 5246 . . . . . . . 8 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅))))
105104fveq1d 6883 . . . . . . 7 ((𝑖 = 𝑥𝑗 = 𝑦) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼) = ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))‘𝐼))
106 eqidd 2734 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅))))
107 eqeq1 2737 . . . . . . . . . 10 (𝑙 = 𝐼 → (𝑙 = 𝐿𝐼 = 𝐿))
108107ifbid 4547 . . . . . . . . 9 (𝑙 = 𝐼 → if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
109108adantl 483 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑙 = 𝐼) → if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
11031adantr 482 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 𝐼 ∈ ℕ0)
111 ovex 7429 . . . . . . . . . 10 (𝑥𝑀𝑦) ∈ V
112 fvex 6894 . . . . . . . . . 10 (0g𝑅) ∈ V
113111, 112ifex 4574 . . . . . . . . 9 if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)) ∈ V
114113a1i 11 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)) ∈ V)
115106, 109, 110, 114fvmptd 6994 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
116105, 115sylan9eqr 2795 . . . . . 6 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) ∧ (𝑖 = 𝑥𝑗 = 𝑦)) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
117101, 116, 95, 96, 114ovmpod 7547 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼))𝑦) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
118 ifov 7496 . . . . . 6 (𝑥if(𝐼 = 𝐿, 𝑀, 0 )𝑦) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (𝑥 0 𝑦))
119118a1i 11 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥if(𝐼 = 𝐿, 𝑀, 0 )𝑦) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (𝑥 0 𝑦)))
120100, 117, 1193eqtr4d 2783 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼))𝑦) = (𝑥if(𝐼 = 𝐿, 𝑀, 0 )𝑦))
121120ralrimivva 3201 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼))𝑦) = (𝑥if(𝐼 = 𝐿, 𝑀, 0 )𝑦))
122 simplr 768 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑅 ∈ CRing)
123 eqidd 2734 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
124107ifbid 4547 . . . . . . . 8 (𝑙 = 𝐼 → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))
125124adantl 483 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) ∧ 𝑙 = 𝐼) → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))
126313ad2ant1 1134 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝐼 ∈ ℕ0)
12753, 82ring0cl 20065 . . . . . . . . . . 11 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
1287, 127syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (0g𝑅) ∈ (Base‘𝑅))
129128adantr 482 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (0g𝑅) ∈ (Base‘𝑅))
1301293ad2ant1 1134 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (0g𝑅) ∈ (Base‘𝑅))
13162, 130ifcld 4570 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) ∈ (Base‘𝑅))
132123, 125, 126, 131fvmptd 6994 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))
133132, 131eqeltrd 2834 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼) ∈ (Base‘𝑅))
13421, 53, 22, 1, 122, 133matbas2d 21894 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼)) ∈ 𝐾)
13560, 57sylibr 233 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑀𝐾)
13621matring 21914 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
13722, 88ring0cl 20065 . . . . . . 7 (𝐴 ∈ Ring → 0𝐾)
13815, 136, 1373syl 18 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 0𝐾)
139138adantr 482 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 0𝐾)
140135, 139ifcld 4570 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → if(𝐼 = 𝐿, 𝑀, 0 ) ∈ 𝐾)
14121, 22eqmat 21895 . . . 4 (((𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼)) ∈ 𝐾 ∧ if(𝐼 = 𝐿, 𝑀, 0 ) ∈ 𝐾) → ((𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼)) = if(𝐼 = 𝐿, 𝑀, 0 ) ↔ ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼))𝑦) = (𝑥if(𝐼 = 𝐿, 𝑀, 0 )𝑦)))
142134, 140, 141syl2anc 585 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼)) = if(𝐼 = 𝐿, 𝑀, 0 ) ↔ ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼))𝑦) = (𝑥if(𝐼 = 𝐿, 𝑀, 0 )𝑦)))
143121, 142mpbird 257 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼)) = if(𝐼 = 𝐿, 𝑀, 0 ))
14433, 87, 1433eqtrd 2777 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (((𝐿 𝑋) · (𝑇𝑀)) decompPMat 𝐼) = if(𝐼 = 𝐿, 𝑀, 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  ifcif 4524  cmpt 5227  cfv 6535  (class class class)co 7396  cmpo 7398  Fincfn 8927  0cn0 12459  Basecbs 17131  .rcmulr 17185  Scalarcsca 17187   ·𝑠 cvsca 17188  0gc0g 17372  .gcmg 18935  mulGrpcmgp 19970  Ringcrg 20038  CRingccrg 20039  AssAlgcasa 21378  algSccascl 21380  var1cv1 21669  Poly1cpl1 21670  coe1cco1 21671   Mat cmat 21876   matToPolyMat cmat2pmat 22175   decompPMat cdecpmat 22233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-cnex 11153  ax-resscn 11154  ax-1cn 11155  ax-icn 11156  ax-addcl 11157  ax-addrcl 11158  ax-mulcl 11159  ax-mulrcl 11160  ax-mulcom 11161  ax-addass 11162  ax-mulass 11163  ax-distr 11164  ax-i2m1 11165  ax-1ne0 11166  ax-1rid 11167  ax-rnegex 11168  ax-rrecex 11169  ax-cnre 11170  ax-pre-lttri 11171  ax-pre-lttrn 11172  ax-pre-ltadd 11173  ax-pre-mulgt0 11174
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-ot 4633  df-uni 4905  df-int 4947  df-iun 4995  df-iin 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7657  df-ofr 7658  df-om 7843  df-1st 7962  df-2nd 7963  df-supp 8134  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8691  df-map 8810  df-pm 8811  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9350  df-sup 9424  df-oi 9492  df-card 9921  df-pnf 11237  df-mnf 11238  df-xr 11239  df-ltxr 11240  df-le 11241  df-sub 11433  df-neg 11434  df-nn 12200  df-2 12262  df-3 12263  df-4 12264  df-5 12265  df-6 12266  df-7 12267  df-8 12268  df-9 12269  df-n0 12460  df-z 12546  df-dec 12665  df-uz 12810  df-fz 13472  df-fzo 13615  df-seq 13954  df-hash 14278  df-struct 17067  df-sets 17084  df-slot 17102  df-ndx 17114  df-base 17132  df-ress 17161  df-plusg 17197  df-mulr 17198  df-sca 17200  df-vsca 17201  df-ip 17202  df-tset 17203  df-ple 17204  df-ds 17206  df-hom 17208  df-cco 17209  df-0g 17374  df-gsum 17375  df-prds 17380  df-pws 17382  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18548  df-sgrp 18597  df-mnd 18613  df-mhm 18658  df-submnd 18659  df-grp 18809  df-minusg 18810  df-sbg 18811  df-mulg 18936  df-subg 18988  df-ghm 19075  df-cntz 19166  df-cmn 19634  df-abl 19635  df-mgp 19971  df-ur 19988  df-ring 20040  df-cring 20041  df-subrg 20338  df-lmod 20450  df-lss 20520  df-sra 20762  df-rgmod 20763  df-dsmm 21260  df-frlm 21275  df-assa 21381  df-ascl 21383  df-psr 21433  df-mvr 21434  df-mpl 21435  df-opsr 21437  df-psr1 21673  df-vr1 21674  df-ply1 21675  df-coe1 21676  df-mamu 21855  df-mat 21877  df-mat2pmat 22178  df-decpmat 22234
This theorem is referenced by:  monmat2matmon  22295
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