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Theorem monmatcollpw 20863
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 20854 (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 783 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑁 ∈ Fin)
2 crngring 18825 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3 monmatcollpw.p . . . . . . 7 𝑃 = (Poly1𝑅)
43ply1ring 19891 . . . . . 6 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
52, 4syl 17 . . . . 5 (𝑅 ∈ CRing → 𝑃 ∈ Ring)
65ad2antlr 718 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑃 ∈ Ring)
72adantl 473 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
8 simp2 1167 . . . . . 6 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝐿 ∈ ℕ0)
9 monmatcollpw.x . . . . . . 7 𝑋 = (var1𝑅)
10 eqid 2765 . . . . . . 7 (mulGrp‘𝑃) = (mulGrp‘𝑃)
11 monmatcollpw.e . . . . . . 7 = (.g‘(mulGrp‘𝑃))
12 eqid 2765 . . . . . . 7 (Base‘𝑃) = (Base‘𝑃)
133, 9, 10, 11, 12ply1moncl 19914 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐿 ∈ ℕ0) → (𝐿 𝑋) ∈ (Base‘𝑃))
147, 8, 13syl2an 589 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝐿 𝑋) ∈ (Base‘𝑃))
152anim2i 610 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
16 simp1 1166 . . . . . . . 8 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝑀𝐾)
1715, 16anim12i 606 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀𝐾))
18 df-3an 1109 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀𝐾))
1917, 18sylibr 225 . . . . . 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 20810 . . . . . 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 507 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝐿 𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑀) ∈ (Base‘𝐶)))
27 eqid 2765 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
28 monmatcollpw.m . . . . 5 · = ( ·𝑠𝐶)
2912, 23, 27, 28matvscl 20513 . . . 4 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝐿 𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑀) ∈ (Base‘𝐶))) → ((𝐿 𝑋) · (𝑇𝑀)) ∈ (Base‘𝐶))
301, 6, 26, 29syl21anc 866 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝐿 𝑋) · (𝑇𝑀)) ∈ (Base‘𝐶))
31 simpr3 1252 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝐼 ∈ ℕ0)
3223, 27decpmatval 20849 . . 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 579 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (((𝐿 𝑋) · (𝑇𝑀)) decompPMat 𝐼) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗))‘𝐼)))
3463ad2ant1 1163 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑃 ∈ Ring)
35263ad2ant1 1163 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑀) ∈ (Base‘𝐶)))
36 3simpc 1182 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑁𝑗𝑁))
37 eqid 2765 . . . . . . . 8 (.r𝑃) = (.r𝑃)
3823, 27, 12, 28, 37matvscacell 20518 . . . . . . 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 1490 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗) = ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)))
4039fveq2d 6379 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘(𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗)) = (coe1‘((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗))))
4140fveq1d 6377 . . . 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 610 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀𝐾))
43 df-3an 1109 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀𝐾))
4442, 43sylibr 225 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾))
45443ad2ant1 1163 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾))
46 eqid 2765 . . . . . . . . . 10 (algSc‘𝑃) = (algSc‘𝑃)
4720, 21, 22, 3, 46mat2pmatvalel 20809 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇𝑀)𝑗) = ((algSc‘𝑃)‘(𝑖𝑀𝑗)))
4845, 36, 47syl2anc 579 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑇𝑀)𝑗) = ((algSc‘𝑃)‘(𝑖𝑀𝑗)))
4948oveq2d 6858 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)) = ((𝐿 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))))
503ply1assa 19842 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
5150ad2antlr 718 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑃 ∈ AssAlg)
52513ad2ant1 1163 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑃 ∈ AssAlg)
53 eqid 2765 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
54 eqid 2765 . . . . . . . . . 10 (Base‘𝐴) = (Base‘𝐴)
55 simp2 1167 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
56 simp3 1168 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
5722eleq2i 2836 . . . . . . . . . . . . . 14 (𝑀𝐾𝑀 ∈ (Base‘𝐴))
5857biimpi 207 . . . . . . . . . . . . 13 (𝑀𝐾𝑀 ∈ (Base‘𝐴))
59583ad2ant1 1163 . . . . . . . . . . . 12 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝑀 ∈ (Base‘𝐴))
6059adantl 473 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑀 ∈ (Base‘𝐴))
61603ad2ant1 1163 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑀 ∈ (Base‘𝐴))
6221, 53, 54, 55, 56, 61matecld 20508 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
633ply1sca 19896 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
6463adantl 473 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝑃))
6564eqcomd 2771 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝑃) = 𝑅)
6665fveq2d 6379 . . . . . . . . . . 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 1163 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
6962, 68eleqtrrd 2847 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Scalar‘𝑃)))
70143ad2ant1 1163 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝐿 𝑋) ∈ (Base‘𝑃))
71 eqid 2765 . . . . . . . . 9 (Scalar‘𝑃) = (Scalar‘𝑃)
72 eqid 2765 . . . . . . . . 9 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
73 eqid 2765 . . . . . . . . 9 ( ·𝑠𝑃) = ( ·𝑠𝑃)
7446, 71, 72, 12, 37, 73asclmul2 19614 . . . . . . . 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 1490 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))) = ((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))
7649, 75eqtrd 2799 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)) = ((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))
7776fveq2d 6379 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗))) = (coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋))))
7877fveq1d 6377 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)))‘𝐼) = ((coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))‘𝐼))
792ad2antlr 718 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑅 ∈ Ring)
80793ad2ant1 1163 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
81 simp1r2 1369 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝐿 ∈ ℕ0)
82 eqid 2765 . . . . . . 7 (0g𝑅) = (0g𝑅)
8382, 53, 3, 9, 73, 10, 11coe1tm 19916 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑖𝑀𝑗) ∈ (Base‘𝑅) ∧ 𝐿 ∈ ℕ0) → (coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
8480, 62, 81, 83syl3anc 1490 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
8584fveq1d 6377 . . . 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 2803 . . 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 6917 . 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 20501 . . . . . . . . . 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 2811 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 0 = (𝑧𝑁, 𝑤𝑁 ↦ (0g𝑅)))
94 eqidd 2766 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) ∧ (𝑧 = 𝑥𝑤 = 𝑦)) → (0g𝑅) = (0g𝑅))
95 simprl 787 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 𝑥𝑁)
96 simprr 789 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 𝑦𝑁)
97 fvexd 6390 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (0g𝑅) ∈ V)
9893, 94, 95, 96, 97ovmpt2d 6986 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥 0 𝑦) = (0g𝑅))
9998eqcomd 2771 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (0g𝑅) = (𝑥 0 𝑦))
10099ifeq2d 4262 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (𝑥 0 𝑦)))
101 eqidd 2766 . . . . . 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 6851 . . . . . . . . . 10 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
103102ifeq1d 4261 . . . . . . . . 9 ((𝑖 = 𝑥𝑗 = 𝑦) → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) = if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
104103mpteq2dv 4904 . . . . . . . 8 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅))))
105104fveq1d 6377 . . . . . . 7 ((𝑖 = 𝑥𝑗 = 𝑦) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼) = ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))‘𝐼))
106 eqidd 2766 . . . . . . . 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 2769 . . . . . . . . . 10 (𝑙 = 𝐼 → (𝑙 = 𝐿𝐼 = 𝐿))
108107ifbid 4265 . . . . . . . . 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 6874 . . . . . . . . . 10 (𝑥𝑀𝑦) ∈ V
112 fvex 6388 . . . . . . . . . 10 (0g𝑅) ∈ V
113111, 112ifex 4291 . . . . . . . . 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 6477 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
116105, 115sylan9eqr 2821 . . . . . 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 6986 . . . . 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 6938 . . . . . 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 2809 . . . 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 3118 . . 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 785 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑅 ∈ CRing)
123 eqidd 2766 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
124107ifbid 4265 . . . . . . . 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 1163 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝐼 ∈ ℕ0)
12753, 82ring0cl 18836 . . . . . . . . . . 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 1163 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (0g𝑅) ∈ (Base‘𝑅))
13162, 130ifcld 4288 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) ∈ (Base‘𝑅))
132123, 125, 126, 131fvmptd 6477 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))
133132, 131eqeltrd 2844 . . . . 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 20505 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼)) ∈ 𝐾)
13560, 57sylibr 225 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑀𝐾)
13621matring 20525 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
13722, 88ring0cl 18836 . . . . . . 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 4288 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → if(𝐼 = 𝐿, 𝑀, 0 ) ∈ 𝐾)
14121, 22eqmat 20506 . . . 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 579 . . 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 248 . 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 2803 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  ifcif 4243  cmpt 4888  cfv 6068  (class class class)co 6842  cmpt2 6844  Fincfn 8160  0cn0 11538  Basecbs 16130  .rcmulr 16215  Scalarcsca 16217   ·𝑠 cvsca 16218  0gc0g 16366  .gcmg 17807  mulGrpcmgp 18756  Ringcrg 18814  CRingccrg 18815  AssAlgcasa 19583  algSccascl 19585  var1cv1 19819  Poly1cpl1 19820  coe1cco1 19821   Mat cmat 20489   matToPolyMat cmat2pmat 20788   decompPMat cdecpmat 20846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-ot 4343  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-ofr 7096  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-sup 8555  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-sca 16230  df-vsca 16231  df-ip 16232  df-tset 16233  df-ple 16234  df-ds 16236  df-hom 16238  df-cco 16239  df-0g 16368  df-gsum 16369  df-prds 16374  df-pws 16376  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-mhm 17601  df-submnd 17602  df-grp 17692  df-minusg 17693  df-sbg 17694  df-mulg 17808  df-subg 17855  df-ghm 17922  df-cntz 18013  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-ring 18816  df-cring 18817  df-subrg 19047  df-lmod 19134  df-lss 19202  df-sra 19446  df-rgmod 19447  df-assa 19586  df-ascl 19588  df-psr 19630  df-mvr 19631  df-mpl 19632  df-opsr 19634  df-psr1 19823  df-vr1 19824  df-ply1 19825  df-coe1 19826  df-dsmm 20352  df-frlm 20367  df-mamu 20466  df-mat 20490  df-mat2pmat 20791  df-decpmat 20847
This theorem is referenced by:  monmat2matmon  20908
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