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Theorem mat2pmatlin 20359
 Description: The transformation of matrices into polynomial matrices is "linear", analogous to lmhmlin 18856. Since 𝐴 and 𝐶 have different scalar rings, 𝑇 cannot be a left module homomorphism as defined in df-lmhm 18843, see lmhmsca 18851. (Contributed by AV, 13-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.)
Hypotheses
Ref Expression
mat2pmatbas.t 𝑇 = (𝑁 matToPolyMat 𝑅)
mat2pmatbas.a 𝐴 = (𝑁 Mat 𝑅)
mat2pmatbas.b 𝐵 = (Base‘𝐴)
mat2pmatbas.p 𝑃 = (Poly1𝑅)
mat2pmatbas.c 𝐶 = (𝑁 Mat 𝑃)
mat2pmatbas0.h 𝐻 = (Base‘𝐶)
mat2pmatlin.k 𝐾 = (Base‘𝑅)
mat2pmatlin.s 𝑆 = (algSc‘𝑃)
mat2pmatlin.m · = ( ·𝑠𝐴)
mat2pmatlin.n × = ( ·𝑠𝐶)
Assertion
Ref Expression
mat2pmatlin (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑇‘(𝑋 · 𝑌)) = ((𝑆𝑋) × (𝑇𝑌)))

Proof of Theorem mat2pmatlin
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ CRing)
2 mat2pmatbas.p . . . . . . . . . . 11 𝑃 = (Poly1𝑅)
32ply1assa 19390 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
4 mat2pmatlin.s . . . . . . . . . . 11 𝑆 = (algSc‘𝑃)
5 eqid 2610 . . . . . . . . . . 11 (Scalar‘𝑃) = (Scalar‘𝑃)
64, 5asclrhm 19163 . . . . . . . . . 10 (𝑃 ∈ AssAlg → 𝑆 ∈ ((Scalar‘𝑃) RingHom 𝑃))
71, 3, 63syl 18 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑆 ∈ ((Scalar‘𝑃) RingHom 𝑃))
82ply1sca 19444 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
98adantl 481 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝑃))
109oveq1d 6564 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 RingHom 𝑃) = ((Scalar‘𝑃) RingHom 𝑃))
117, 10eleqtrrd 2691 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑆 ∈ (𝑅 RingHom 𝑃))
1211adantr 480 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → 𝑆 ∈ (𝑅 RingHom 𝑃))
1312adantr 480 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑆 ∈ (𝑅 RingHom 𝑃))
14 mat2pmatlin.k . . . . . . . . . 10 𝐾 = (Base‘𝑅)
1514eleq2i 2680 . . . . . . . . 9 (𝑋𝐾𝑋 ∈ (Base‘𝑅))
1615biimpi 205 . . . . . . . 8 (𝑋𝐾𝑋 ∈ (Base‘𝑅))
1716adantr 480 . . . . . . 7 ((𝑋𝐾𝑌𝐵) → 𝑋 ∈ (Base‘𝑅))
1817ad2antlr 759 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑋 ∈ (Base‘𝑅))
19 mat2pmatbas.a . . . . . . 7 𝐴 = (𝑁 Mat 𝑅)
20 eqid 2610 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
21 mat2pmatbas.b . . . . . . 7 𝐵 = (Base‘𝐴)
22 simprl 790 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑖𝑁)
23 simpr 476 . . . . . . . 8 ((𝑖𝑁𝑗𝑁) → 𝑗𝑁)
2423adantl 481 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑗𝑁)
25 simplrr 797 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑌𝐵)
2619, 20, 21, 22, 24, 25matecld 20051 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖𝑌𝑗) ∈ (Base‘𝑅))
27 eqid 2610 . . . . . . 7 (.r𝑅) = (.r𝑅)
28 eqid 2610 . . . . . . 7 (.r𝑃) = (.r𝑃)
2920, 27, 28rhmmul 18550 . . . . . 6 ((𝑆 ∈ (𝑅 RingHom 𝑃) ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑖𝑌𝑗) ∈ (Base‘𝑅)) → (𝑆‘(𝑋(.r𝑅)(𝑖𝑌𝑗))) = ((𝑆𝑋)(.r𝑃)(𝑆‘(𝑖𝑌𝑗))))
3013, 18, 26, 29syl3anc 1318 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑆‘(𝑋(.r𝑅)(𝑖𝑌𝑗))) = ((𝑆𝑋)(.r𝑃)(𝑆‘(𝑖𝑌𝑗))))
31 crngring 18381 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3231ad2antlr 759 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → 𝑅 ∈ Ring)
3332adantr 480 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 ∈ Ring)
34 simpr 476 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑋𝐾𝑌𝐵))
3534adantr 480 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑋𝐾𝑌𝐵))
36 simpr 476 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖𝑁𝑗𝑁))
37 mat2pmatlin.m . . . . . . . 8 · = ( ·𝑠𝐴)
3819, 21, 14, 37, 27matvscacell 20061 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑋𝐾𝑌𝐵) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑋 · 𝑌)𝑗) = (𝑋(.r𝑅)(𝑖𝑌𝑗)))
3933, 35, 36, 38syl3anc 1318 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑋 · 𝑌)𝑗) = (𝑋(.r𝑅)(𝑖𝑌𝑗)))
4039fveq2d 6107 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)) = (𝑆‘(𝑋(.r𝑅)(𝑖𝑌𝑗))))
4131anim2i 591 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
42 simpr 476 . . . . . . . . 9 ((𝑋𝐾𝑌𝐵) → 𝑌𝐵)
4341, 42anim12i 588 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐵))
44 df-3an 1033 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐵))
4543, 44sylibr 223 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐵))
46 mat2pmatbas.t . . . . . . . 8 𝑇 = (𝑁 matToPolyMat 𝑅)
4746, 19, 21, 2, 4mat2pmatvalel 20349 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐵) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇𝑌)𝑗) = (𝑆‘(𝑖𝑌𝑗)))
4845, 47sylan 487 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇𝑌)𝑗) = (𝑆‘(𝑖𝑌𝑗)))
4948oveq2d 6565 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑆𝑋)(.r𝑃)(𝑖(𝑇𝑌)𝑗)) = ((𝑆𝑋)(.r𝑃)(𝑆‘(𝑖𝑌𝑗))))
5030, 40, 493eqtr4d 2654 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)) = ((𝑆𝑋)(.r𝑃)(𝑖(𝑇𝑌)𝑗)))
51 simpll 786 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → 𝑁 ∈ Fin)
5251adantr 480 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑁 ∈ Fin)
5314, 19, 21, 37matvscl 20056 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐾𝑌𝐵)) → (𝑋 · 𝑌) ∈ 𝐵)
5441, 53sylan 487 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑋 · 𝑌) ∈ 𝐵)
5554adantr 480 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑋 · 𝑌) ∈ 𝐵)
5646, 19, 21, 2, 4mat2pmatvalel 20349 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑋 · 𝑌) ∈ 𝐵) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)))
5752, 33, 55, 36, 56syl31anc 1321 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)))
582ply1ring 19439 . . . . . . . 8 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5931, 58syl 17 . . . . . . 7 (𝑅 ∈ CRing → 𝑃 ∈ Ring)
6059ad2antlr 759 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → 𝑃 ∈ Ring)
6160adantr 480 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑃 ∈ Ring)
6231adantl 481 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
63 simpl 472 . . . . . . . 8 ((𝑋𝐾𝑌𝐵) → 𝑋𝐾)
64 eqid 2610 . . . . . . . . 9 (Base‘𝑃) = (Base‘𝑃)
652, 4, 14, 64ply1sclcl 19477 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑋𝐾) → (𝑆𝑋) ∈ (Base‘𝑃))
6662, 63, 65syl2an 493 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑆𝑋) ∈ (Base‘𝑃))
67 mat2pmatbas.c . . . . . . . . 9 𝐶 = (𝑁 Mat 𝑃)
68 mat2pmatbas0.h . . . . . . . . 9 𝐻 = (Base‘𝐶)
6946, 19, 21, 2, 67, 68mat2pmatbas0 20351 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐵) → (𝑇𝑌) ∈ 𝐻)
7045, 69syl 17 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑇𝑌) ∈ 𝐻)
7166, 70jca 553 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → ((𝑆𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑌) ∈ 𝐻))
7271adantr 480 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑆𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑌) ∈ 𝐻))
73 mat2pmatlin.n . . . . . 6 × = ( ·𝑠𝐶)
7467, 68, 64, 73, 28matvscacell 20061 . . . . 5 ((𝑃 ∈ Ring ∧ ((𝑆𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑌) ∈ 𝐻) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗) = ((𝑆𝑋)(.r𝑃)(𝑖(𝑇𝑌)𝑗)))
7561, 72, 36, 74syl3anc 1318 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗) = ((𝑆𝑋)(.r𝑃)(𝑖(𝑇𝑌)𝑗)))
7650, 57, 753eqtr4d 2654 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗))
7776ralrimivva 2954 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → ∀𝑖𝑁𝑗𝑁 (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗))
7846, 19, 21, 2, 67, 68mat2pmatbas0 20351 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑋 · 𝑌) ∈ 𝐵) → (𝑇‘(𝑋 · 𝑌)) ∈ 𝐻)
7951, 32, 54, 78syl3anc 1318 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑇‘(𝑋 · 𝑌)) ∈ 𝐻)
8064, 67, 68, 73matvscl 20056 . . . 4 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝑆𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑌) ∈ 𝐻)) → ((𝑆𝑋) × (𝑇𝑌)) ∈ 𝐻)
8151, 60, 71, 80syl21anc 1317 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → ((𝑆𝑋) × (𝑇𝑌)) ∈ 𝐻)
8267, 68eqmat 20049 . . 3 (((𝑇‘(𝑋 · 𝑌)) ∈ 𝐻 ∧ ((𝑆𝑋) × (𝑇𝑌)) ∈ 𝐻) → ((𝑇‘(𝑋 · 𝑌)) = ((𝑆𝑋) × (𝑇𝑌)) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗)))
8379, 81, 82syl2anc 691 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → ((𝑇‘(𝑋 · 𝑌)) = ((𝑆𝑋) × (𝑇𝑌)) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗)))
8477, 83mpbird 246 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑇‘(𝑋 · 𝑌)) = ((𝑆𝑋) × (𝑇𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  Basecbs 15695  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  Ringcrg 18370  CRingccrg 18371   RingHom crh 18535  AssAlgcasa 19130  algSccascl 19132  Poly1cpl1 19368   Mat cmat 20032   matToPolyMat cmat2pmat 20328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-rnghom 18538  df-subrg 18601  df-lmod 18688  df-lss 18754  df-sra 18993  df-rgmod 18994  df-assa 19133  df-ascl 19135  df-psr 19177  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-ply1 19373  df-dsmm 19895  df-frlm 19910  df-mat 20033  df-mat2pmat 20331 This theorem is referenced by:  cpmidgsumm2pm  20493
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