MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  monmatcollpw Structured version   Visualization version   GIF version

Theorem monmatcollpw 22800
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 22791 (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 20262 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3 monmatcollpw.p . . . . . . 7 𝑃 = (Poly1𝑅)
43ply1ring 22264 . . . . . 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 1136 . . . . . 6 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝐿 ∈ ℕ0)
9 monmatcollpw.x . . . . . . 7 𝑋 = (var1𝑅)
10 eqid 2734 . . . . . . 7 (mulGrp‘𝑃) = (mulGrp‘𝑃)
11 monmatcollpw.e . . . . . . 7 = (.g‘(mulGrp‘𝑃))
12 eqid 2734 . . . . . . 7 (Base‘𝑃) = (Base‘𝑃)
133, 9, 10, 11, 12ply1moncl 22289 . . . . . 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 1135 . . . . . . . 8 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝑀𝐾)
1715, 16anim12i 613 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀𝐾))
18 df-3an 1088 . . . . . . 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 22747 . . . . . 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 2734 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
28 monmatcollpw.m . . . . 5 · = ( ·𝑠𝐶)
2912, 23, 27, 28matvscl 22452 . . . 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 1195 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝐼 ∈ ℕ0)
3223, 27decpmatval 22786 . . 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 1132 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑃 ∈ Ring)
35263ad2ant1 1132 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑀) ∈ (Base‘𝐶)))
36 3simpc 1149 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑁𝑗𝑁))
37 eqid 2734 . . . . . . . 8 (.r𝑃) = (.r𝑃)
3823, 27, 12, 28, 37matvscacell 22457 . . . . . . 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 1370 . . . . . 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 1088 . . . . . . . . . . 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 1132 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐾))
46 eqid 2734 . . . . . . . . . 10 (algSc‘𝑃) = (algSc‘𝑃)
4720, 21, 22, 3, 46mat2pmatvalel 22746 . . . . . . . . 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 7446 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)(𝑖(𝑇𝑀)𝑗)) = ((𝐿 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))))
503ply1assa 22216 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
5150ad2antlr 727 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑃 ∈ AssAlg)
52513ad2ant1 1132 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑃 ∈ AssAlg)
53 eqid 2734 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
54 eqid 2734 . . . . . . . . . 10 (Base‘𝐴) = (Base‘𝐴)
55 simp2 1136 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
56 simp3 1137 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
5722eleq2i 2830 . . . . . . . . . . . . . 14 (𝑀𝐾𝑀 ∈ (Base‘𝐴))
5857biimpi 216 . . . . . . . . . . . . 13 (𝑀𝐾𝑀 ∈ (Base‘𝐴))
59583ad2ant1 1132 . . . . . . . . . . . 12 ((𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0) → 𝑀 ∈ (Base‘𝐴))
6059adantl 481 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → 𝑀 ∈ (Base‘𝐴))
61603ad2ant1 1132 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑀 ∈ (Base‘𝐴))
6221, 53, 54, 55, 56, 61matecld 22447 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
633ply1sca 22269 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
6463adantl 481 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝑃))
6564eqcomd 2740 . . . . . . . . . . . 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 1132 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
6962, 68eleqtrrd 2841 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Scalar‘𝑃)))
70143ad2ant1 1132 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝐿 𝑋) ∈ (Base‘𝑃))
71 eqid 2734 . . . . . . . . 9 (Scalar‘𝑃) = (Scalar‘𝑃)
72 eqid 2734 . . . . . . . . 9 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
73 eqid 2734 . . . . . . . . 9 ( ·𝑠𝑃) = ( ·𝑠𝑃)
7446, 71, 72, 12, 37, 73asclmul2 21924 . . . . . . . 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 1370 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝐿 𝑋)(.r𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))) = ((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋)))
7649, 75eqtrd 2774 . . . . . 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 1132 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
81 simp1r2 1269 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝐿 ∈ ℕ0)
82 eqid 2734 . . . . . . 7 (0g𝑅) = (0g𝑅)
8382, 53, 3, 9, 73, 10, 11coe1tm 22291 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑖𝑀𝑗) ∈ (Base‘𝑅) ∧ 𝐿 ∈ ℕ0) → (coe1‘((𝑖𝑀𝑗)( ·𝑠𝑃)(𝐿 𝑋))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
8480, 62, 81, 83syl3anc 1370 . . . . 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 2778 . . 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 7509 . 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 22440 . . . . . . . . . 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 2786 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → 0 = (𝑧𝑁, 𝑤𝑁 ↦ (0g𝑅)))
94 eqidd 2735 . . . . . . . 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 7584 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (𝑥 0 𝑦) = (0g𝑅))
9998eqcomd 2740 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → (0g𝑅) = (𝑥 0 𝑦))
10099ifeq2d 4550 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (𝑥 0 𝑦)))
101 eqidd 2735 . . . . . 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 7439 . . . . . . . . . 10 ((𝑖 = 𝑥𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
103102ifeq1d 4549 . . . . . . . . 9 ((𝑖 = 𝑥𝑗 = 𝑦) → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) = if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
104103mpteq2dv 5249 . . . . . . . 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 2735 . . . . . . . 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 2738 . . . . . . . . . 10 (𝑙 = 𝐼 → (𝑙 = 𝐿𝐼 = 𝐿))
108107ifbid 4553 . . . . . . . . 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 7463 . . . . . . . . . 10 (𝑥𝑀𝑦) ∈ V
112 fvex 6919 . . . . . . . . . 10 (0g𝑅) ∈ V
113111, 112ifex 4580 . . . . . . . . 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 7022 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ (𝑥𝑁𝑦𝑁)) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g𝑅)))
116105, 115sylan9eqr 2796 . . . . . 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 7584 . . . . 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 7533 . . . . . 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 2784 . . . 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 3199 . . 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 2735 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅))))
124107ifbid 4553 . . . . . . . 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 1132 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → 𝐼 ∈ ℕ0)
12753, 82ring0cl 20280 . . . . . . . . . . 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 1132 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → (0g𝑅) ∈ (Base‘𝑅))
13162, 130ifcld 4576 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)) ∈ (Base‘𝑅))
132123, 125, 126, 131fvmptd 7022 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) ∧ 𝑖𝑁𝑗𝑁) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g𝑅)))
133132, 131eqeltrd 2838 . . . . 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 22444 . . . 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 22464 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
13722, 88ring0cl 20280 . . . . . . 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 4576 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → if(𝐼 = 𝐿, 𝑀, 0 ) ∈ 𝐾)
14121, 22eqmat 22445 . . . 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 2778 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 1086   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  ifcif 4530  cmpt 5230  cfv 6562  (class class class)co 7430  cmpo 7432  Fincfn 8983  0cn0 12523  Basecbs 17244  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17485  .gcmg 19097  mulGrpcmgp 20151  Ringcrg 20250  CRingccrg 20251  AssAlgcasa 21887  algSccascl 21889  var1cv1 22192  Poly1cpl1 22193  coe1cco1 22194   Mat cmat 22426   matToPolyMat cmat2pmat 22725   decompPMat cdecpmat 22783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-ot 4639  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  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 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-cring 20253  df-subrng 20562  df-subrg 20586  df-lmod 20876  df-lss 20947  df-sra 21189  df-rgmod 21190  df-dsmm 21769  df-frlm 21784  df-assa 21890  df-ascl 21892  df-psr 21946  df-mvr 21947  df-mpl 21948  df-opsr 21950  df-psr1 22196  df-vr1 22197  df-ply1 22198  df-coe1 22199  df-mamu 22410  df-mat 22427  df-mat2pmat 22728  df-decpmat 22784
This theorem is referenced by:  monmat2matmon  22845
  Copyright terms: Public domain W3C validator