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Theorem monmatcollpw 20786
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 20777 (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 807 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑁 ∈ Fin)
2 crngring 18758 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3 monmatcollpw.p . . . . . . 7 𝑃 = (Poly1𝑅)
43ply1ring 19820 . . . . . 6 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
52, 4syl 17 . . . . 5 (𝑅 ∈ CRing → 𝑃 ∈ Ring)
65ad2antlr 765 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑃 ∈ Ring)
72adantl 473 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
8 simp2 1132 . . . . . 6 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝐿 ∈ ℕ0)
9 monmatcollpw.x . . . . . . 7 𝑋 = (var1𝑅)
10 eqid 2760 . . . . . . 7 (mulGrp‘𝑃) = (mulGrp‘𝑃)
11 monmatcollpw.e . . . . . . 7 = (.g‘(mulGrp‘𝑃))
12 eqid 2760 . . . . . . 7 (Base‘𝑃) = (Base‘𝑃)
133, 9, 10, 11, 12ply1moncl 19843 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐿 ∈ ℕ0) → (𝐿 𝑋) ∈ (Base‘𝑃))
147, 8, 13syl2an 495 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝐿 𝑋) ∈ (Base‘𝑃))
152anim2i 594 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
16 simp1 1131 . . . . . . . 8 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝑀𝐾)
1715, 16anim12i 591 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀𝐾))
18 df-3an 1074 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀𝐾))
1917, 18sylibr 224 . . . . . 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 20733 . . . . . 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 555 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝐿 𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑀) ∈ (Base‘𝐶)))
27 eqid 2760 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
28 monmatcollpw.m . . . . 5 · = ( ·𝑠𝐶)
2912, 23, 27, 28matvscl 20439 . . . 4 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝐿 𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑀) ∈ (Base‘𝐶))) → ((𝐿 𝑋) · (𝑇𝑀)) ∈ (Base‘𝐶))
301, 6, 26, 29syl21anc 1476 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝐿 𝑋) · (𝑇𝑀)) ∈ (Base‘𝐶))
31 simpr3 1238 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝐼 ∈ ℕ0)
3223, 27decpmatval 20772 . . 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 696 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (((𝐿 𝑋) · (𝑇𝑀)) decompPMat 𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗))‘𝐼)))
3463ad2ant1 1128 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑃 ∈ Ring)
35263ad2ant1 1128 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑀) ∈ (Base‘𝐶)))
36 3simpc 1147 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑁𝑗𝑁))
37 eqid 2760 . . . . . . . 8 (.r𝑃) = (.r𝑃)
3823, 27, 12, 28, 37matvscacell 20444 . . . . . . 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 1477 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗) = ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)))
4039fveq2d 6356 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘(𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗)) = (coe1‘((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗))))
4140fveq1d 6354 . . . 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 594 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀𝐾))
43 df-3an 1074 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀𝐾))
4442, 43sylibr 224 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾))
45443ad2ant1 1128 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾))
46 eqid 2760 . . . . . . . . . 10 (algSc‘𝑃) = (algSc‘𝑃)
4720, 21, 22, 3, 46mat2pmatvalel 20732 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇𝑀)𝑗) = ((algSc‘𝑃)‘(𝑖𝑀𝑗)))
4845, 36, 47syl2anc 696 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑇𝑀)𝑗) = ((algSc‘𝑃)‘(𝑖𝑀𝑗)))
4948oveq2d 6829 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)) = ((𝐿 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))))
503ply1assa 19771 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
5150ad2antlr 765 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑃 ∈ AssAlg)
52513ad2ant1 1128 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑃 ∈ AssAlg)
53 eqid 2760 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
54 eqid 2760 . . . . . . . . . 10 (Base‘𝐴) = (Base‘𝐴)
55 simp2 1132 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
56 simp3 1133 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
5722eleq2i 2831 . . . . . . . . . . . . . 14 (𝑀𝐾𝑀 ∈ (Base‘𝐴))
5857biimpi 206 . . . . . . . . . . . . 13 (𝑀𝐾𝑀 ∈ (Base‘𝐴))
59583ad2ant1 1128 . . . . . . . . . . . 12 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝑀 ∈ (Base‘𝐴))
6059adantl 473 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑀 ∈ (Base‘𝐴))
61603ad2ant1 1128 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑀 ∈ (Base‘𝐴))
6221, 53, 54, 55, 56, 61matecld 20434 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
633ply1sca 19825 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
6463adantl 473 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝑃))
6564eqcomd 2766 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝑃) = 𝑅)
6665fveq2d 6356 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
6766adantr 472 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
68673ad2ant1 1128 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
6962, 68eleqtrrd 2842 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Scalar‘𝑃)))
70143ad2ant1 1128 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝐿 𝑋) ∈ (Base‘𝑃))
71 eqid 2760 . . . . . . . . 9 (Scalar‘𝑃) = (Scalar‘𝑃)
72 eqid 2760 . . . . . . . . 9 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
73 eqid 2760 . . . . . . . . 9 ( ·𝑠𝑃) = ( ·𝑠𝑃)
7446, 71, 72, 12, 37, 73asclmul2 19542 . . . . . . . 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 1477 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))) = ((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))
7649, 75eqtrd 2794 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)) = ((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))
7776fveq2d 6356 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗))) = (coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋))))
7877fveq1d 6354 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)))‘𝐼) = ((coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))‘𝐼))
792ad2antlr 765 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑅 ∈ Ring)
80793ad2ant1 1128 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
81 simp1r2 1355 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝐿 ∈ ℕ0)
82 eqid 2760 . . . . . . 7 (0g𝑅) = (0g𝑅)
8382, 53, 3, 9, 73, 10, 11coe1tm 19845 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑖𝑀𝑗) ∈ (Base‘𝑅) ∧ 𝐿 ∈ ℕ0) → (coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
8480, 62, 81, 83syl3anc 1477 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
8584fveq1d 6354 . . . 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 2798 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗))‘𝐼) = ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼))
8786mpt2eq3dva 6884 . 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 472 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
9089adantr 472 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
9121, 82mat0op 20427 . . . . . . . . . 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, 92syl5eq 2806 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 0 = (𝑧𝑁, 𝑤𝑁 ↦ (0g𝑅)))
94 eqidd 2761 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) ∧ (𝑧 = 𝑥𝑤 = 𝑦)) → (0g𝑅) = (0g𝑅))
95 simprl 811 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 𝑥𝑁)
96 simprr 813 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 𝑦𝑁)
97 fvexd 6364 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (0g𝑅) ∈ V)
9893, 94, 95, 96, 97ovmpt2d 6953 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥 0 𝑦) = (0g𝑅))
9998eqcomd 2766 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (0g𝑅) = (𝑥 0 𝑦))
10099ifeq2d 4249 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (𝑥 0 𝑦)))
101 eqidd 2761 . . . . . 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 6822 . . . . . . . . . 10 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
103102ifeq1d 4248 . . . . . . . . 9 ((𝑖 = 𝑥𝑗 = 𝑦) → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) = if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
104103mpteq2dv 4897 . . . . . . . 8 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅))))
105104fveq1d 6354 . . . . . . 7 ((𝑖 = 𝑥𝑗 = 𝑦) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼) = ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))‘𝐼))
106 eqidd 2761 . . . . . . . 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 2764 . . . . . . . . . 10 (𝑙 = 𝐼 → (𝑙 = 𝐿𝐼 = 𝐿))
108107ifbid 4252 . . . . . . . . 9 (𝑙 = 𝐼 → if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
109108adantl 473 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑙 = 𝐼) → if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
11031adantr 472 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 𝐼 ∈ ℕ0)
111 ovex 6841 . . . . . . . . . 10 (𝑥𝑀𝑦) ∈ V
112 fvex 6362 . . . . . . . . . 10 (0g𝑅) ∈ V
113111, 112ifex 4300 . . . . . . . . 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 6450 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
116105, 115sylan9eqr 2816 . . . . . 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, 114ovmpt2d 6953 . . . . 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 6905 . . . . . 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 2804 . . . 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 3109 . . 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 809 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑅 ∈ CRing)
123 eqidd 2761 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
124107ifbid 4252 . . . . . . . 8 (𝑙 = 𝐼 → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))
125124adantl 473 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) ∧ 𝑙 = 𝐼) → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))
126313ad2ant1 1128 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝐼 ∈ ℕ0)
12753, 82ring0cl 18769 . . . . . . . . . . 11 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
1287, 127syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (0g𝑅) ∈ (Base‘𝑅))
129128adantr 472 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (0g𝑅) ∈ (Base‘𝑅))
1301293ad2ant1 1128 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (0g𝑅) ∈ (Base‘𝑅))
13162, 130ifcld 4275 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) ∈ (Base‘𝑅))
132123, 125, 126, 131fvmptd 6450 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))
133132, 131eqeltrd 2839 . . . . 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 20431 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼)) ∈ 𝐾)
13560, 57sylibr 224 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑀𝐾)
13621matring 20451 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
13722, 88ring0cl 18769 . . . . . . 7 (𝐴 ∈ Ring → 0𝐾)
13815, 136, 1373syl 18 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 0𝐾)
139138adantr 472 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 0𝐾)
140135, 139ifcld 4275 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → if(𝐼 = 𝐿, 𝑀, 0 ) ∈ 𝐾)
14121, 22eqmat 20432 . . . 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 696 . . 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 247 . 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 2798 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  ifcif 4230  cmpt 4881  cfv 6049  (class class class)co 6813  cmpt2 6815  Fincfn 8121  0cn0 11484  Basecbs 16059  .rcmulr 16144  Scalarcsca 16146   ·𝑠 cvsca 16147  0gc0g 16302  .gcmg 17741  mulGrpcmgp 18689  Ringcrg 18747  CRingccrg 18748  AssAlgcasa 19511  algSccascl 19513  var1cv1 19748  Poly1cpl1 19749  coe1cco1 19750   Mat cmat 20415   matToPolyMat cmat2pmat 20711   decompPMat cdecpmat 20769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-ot 4330  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-ofr 7063  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-sup 8513  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-fzo 12660  df-seq 12996  df-hash 13312  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-hom 16168  df-cco 16169  df-0g 16304  df-gsum 16305  df-prds 16310  df-pws 16312  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-mulg 17742  df-subg 17792  df-ghm 17859  df-cntz 17950  df-cmn 18395  df-abl 18396  df-mgp 18690  df-ur 18702  df-ring 18749  df-cring 18750  df-subrg 18980  df-lmod 19067  df-lss 19135  df-sra 19374  df-rgmod 19375  df-assa 19514  df-ascl 19516  df-psr 19558  df-mvr 19559  df-mpl 19560  df-opsr 19562  df-psr1 19752  df-vr1 19753  df-ply1 19754  df-coe1 19755  df-dsmm 20278  df-frlm 20293  df-mamu 20392  df-mat 20416  df-mat2pmat 20714  df-decpmat 20770
This theorem is referenced by:  monmat2matmon  20831
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