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Theorem monmatcollpw 22785
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 22776 (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 767 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑁 ∈ Fin)
2 crngring 20242 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3 monmatcollpw.p . . . . . . 7 𝑃 = (Poly1𝑅)
43ply1ring 22249 . . . . . 6 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
52, 4syl 17 . . . . 5 (𝑅 ∈ CRing → 𝑃 ∈ Ring)
65ad2antlr 727 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑃 ∈ Ring)
72adantl 481 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
8 simp2 1138 . . . . . 6 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝐿 ∈ ℕ0)
9 monmatcollpw.x . . . . . . 7 𝑋 = (var1𝑅)
10 eqid 2737 . . . . . . 7 (mulGrp‘𝑃) = (mulGrp‘𝑃)
11 monmatcollpw.e . . . . . . 7 = (.g‘(mulGrp‘𝑃))
12 eqid 2737 . . . . . . 7 (Base‘𝑃) = (Base‘𝑃)
133, 9, 10, 11, 12ply1moncl 22274 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐿 ∈ ℕ0) → (𝐿 𝑋) ∈ (Base‘𝑃))
147, 8, 13syl2an 596 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝐿 𝑋) ∈ (Base‘𝑃))
152anim2i 617 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
16 simp1 1137 . . . . . . . 8 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝑀𝐾)
1715, 16anim12i 613 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀𝐾))
18 df-3an 1089 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀𝐾))
1917, 18sylibr 234 . . . . . 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 22732 . . . . . 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 511 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝐿 𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑀) ∈ (Base‘𝐶)))
27 eqid 2737 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
28 monmatcollpw.m . . . . 5 · = ( ·𝑠𝐶)
2912, 23, 27, 28matvscl 22437 . . . 4 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝐿 𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑀) ∈ (Base‘𝐶))) → ((𝐿 𝑋) · (𝑇𝑀)) ∈ (Base‘𝐶))
301, 6, 26, 29syl21anc 838 . . 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 22771 . . 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 584 . 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 2737 . . . . . . . 8 (.r𝑃) = (.r𝑃)
3823, 27, 12, 28, 37matvscacell 22442 . . . . . . 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 1373 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗) = ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)))
4039fveq2d 6910 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘(𝑖((𝐿 𝑋) · (𝑇𝑀))𝑗)) = (coe1‘((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗))))
4140fveq1d 6908 . . . 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 617 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀𝐾))
43 df-3an 1089 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀𝐾))
4442, 43sylibr 234 . . . . . . . . . 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 2737 . . . . . . . . . 10 (algSc‘𝑃) = (algSc‘𝑃)
4720, 21, 22, 3, 46mat2pmatvalel 22731 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇𝑀)𝑗) = ((algSc‘𝑃)‘(𝑖𝑀𝑗)))
4845, 36, 47syl2anc 584 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖(𝑇𝑀)𝑗) = ((algSc‘𝑃)‘(𝑖𝑀𝑗)))
4948oveq2d 7447 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)) = ((𝐿 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))))
503ply1assa 22201 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
5150ad2antlr 727 . . . . . . . . 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 2737 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
54 eqid 2737 . . . . . . . . . 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 2833 . . . . . . . . . . . . . 14 (𝑀𝐾𝑀 ∈ (Base‘𝐴))
5857biimpi 216 . . . . . . . . . . . . 13 (𝑀𝐾𝑀 ∈ (Base‘𝐴))
59583ad2ant1 1134 . . . . . . . . . . . 12 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝑀 ∈ (Base‘𝐴))
6059adantl 481 . . . . . . . . . . 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 22432 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
633ply1sca 22254 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
6463adantl 481 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝑃))
6564eqcomd 2743 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝑃) = 𝑅)
6665fveq2d 6910 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
6766adantr 480 . . . . . . . . . 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 2844 . . . . . . . 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 2737 . . . . . . . . 9 (Scalar‘𝑃) = (Scalar‘𝑃)
72 eqid 2737 . . . . . . . . 9 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
73 eqid 2737 . . . . . . . . 9 ( ·𝑠𝑃) = ( ·𝑠𝑃)
7446, 71, 72, 12, 37, 73asclmul2 21907 . . . . . . . 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 1373 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))) = ((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))
7649, 75eqtrd 2777 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)) = ((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))
7776fveq2d 6910 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗))) = (coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋))))
7877fveq1d 6908 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)))‘𝐼) = ((coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))‘𝐼))
792ad2antlr 727 . . . . . . 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 2737 . . . . . . 7 (0g𝑅) = (0g𝑅)
8382, 53, 3, 9, 73, 10, 11coe1tm 22276 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑖𝑀𝑗) ∈ (Base‘𝑅) ∧ 𝐿 ∈ ℕ0) → (coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
8480, 62, 81, 83syl3anc 1373 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
8584fveq1d 6908 . . . 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 2781 . . 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 7510 . 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 480 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
9089adantr 480 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
9121, 82mat0op 22425 . . . . . . . . . 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 2789 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 0 = (𝑧𝑁, 𝑤𝑁 ↦ (0g𝑅)))
94 eqidd 2738 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) ∧ (𝑧 = 𝑥𝑤 = 𝑦)) → (0g𝑅) = (0g𝑅))
95 simprl 771 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 𝑥𝑁)
96 simprr 773 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 𝑦𝑁)
97 fvexd 6921 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (0g𝑅) ∈ V)
9893, 94, 95, 96, 97ovmpod 7585 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥 0 𝑦) = (0g𝑅))
9998eqcomd 2743 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (0g𝑅) = (𝑥 0 𝑦))
10099ifeq2d 4546 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (𝑥 0 𝑦)))
101 eqidd 2738 . . . . . 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 7440 . . . . . . . . . 10 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
103102ifeq1d 4545 . . . . . . . . 9 ((𝑖 = 𝑥𝑗 = 𝑦) → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) = if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
104103mpteq2dv 5244 . . . . . . . 8 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅))))
105104fveq1d 6908 . . . . . . 7 ((𝑖 = 𝑥𝑗 = 𝑦) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼) = ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))‘𝐼))
106 eqidd 2738 . . . . . . . 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 2741 . . . . . . . . . 10 (𝑙 = 𝐼 → (𝑙 = 𝐿𝐼 = 𝐿))
108107ifbid 4549 . . . . . . . . 9 (𝑙 = 𝐼 → if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
109108adantl 481 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) ∧ 𝑙 = 𝐼) → if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
11031adantr 480 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 𝐼 ∈ ℕ0)
111 ovex 7464 . . . . . . . . . 10 (𝑥𝑀𝑦) ∈ V
112 fvex 6919 . . . . . . . . . 10 (0g𝑅) ∈ V
113111, 112ifex 4576 . . . . . . . . 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 7023 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
116105, 115sylan9eqr 2799 . . . . . 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 7585 . . . . 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 7534 . . . . . 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 2787 . . . 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 3202 . . 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 769 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑅 ∈ CRing)
123 eqidd 2738 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
124107ifbid 4549 . . . . . . . 8 (𝑙 = 𝐼 → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))
125124adantl 481 . . . . . . 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 20264 . . . . . . . . . . 11 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
1287, 127syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (0g𝑅) ∈ (Base‘𝑅))
129128adantr 480 . . . . . . . . 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 4572 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) ∈ (Base‘𝑅))
132123, 125, 126, 131fvmptd 7023 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))
133132, 131eqeltrd 2841 . . . . 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 22429 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼)) ∈ 𝐾)
13560, 57sylibr 234 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑀𝐾)
13621matring 22449 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
13722, 88ring0cl 20264 . . . . . . 7 (𝐴 ∈ Ring → 0𝐾)
13815, 136, 1373syl 18 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 0𝐾)
139138adantr 480 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 0𝐾)
140135, 139ifcld 4572 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → if(𝐼 = 𝐿, 𝑀, 0 ) ∈ 𝐾)
14121, 22eqmat 22430 . . . 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 584 . . 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 2781 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  ifcif 4525  cmpt 5225  cfv 6561  (class class class)co 7431  cmpo 7433  Fincfn 8985  0cn0 12526  Basecbs 17247  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484  .gcmg 19085  mulGrpcmgp 20137  Ringcrg 20230  CRingccrg 20231  AssAlgcasa 21870  algSccascl 21872  var1cv1 22177  Poly1cpl1 22178  coe1cco1 22179   Mat cmat 22411   matToPolyMat cmat2pmat 22710   decompPMat cdecpmat 22768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-dsmm 21752  df-frlm 21767  df-assa 21873  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-mamu 22395  df-mat 22412  df-mat2pmat 22713  df-decpmat 22769
This theorem is referenced by:  monmat2matmon  22830
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