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Theorem mat2pmatlin 22741
Description: The transformation of matrices into polynomial matrices is "linear", analogous to lmhmlin 21034. Since 𝐴 and 𝐶 have different scalar rings, 𝑇 cannot be a left module homomorphism as defined in df-lmhm 21021, see lmhmsca 21029. (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 484 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ CRing)
2 mat2pmatbas.p . . . . . . . . . . 11 𝑃 = (Poly1𝑅)
32ply1assa 22201 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
4 mat2pmatlin.s . . . . . . . . . . 11 𝑆 = (algSc‘𝑃)
5 eqid 2737 . . . . . . . . . . 11 (Scalar‘𝑃) = (Scalar‘𝑃)
64, 5asclrhm 21910 . . . . . . . . . 10 (𝑃 ∈ AssAlg → 𝑆 ∈ ((Scalar‘𝑃) RingHom 𝑃))
71, 3, 63syl 18 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑆 ∈ ((Scalar‘𝑃) RingHom 𝑃))
82ply1sca 22254 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
98adantl 481 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝑃))
109oveq1d 7446 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 RingHom 𝑃) = ((Scalar‘𝑃) RingHom 𝑃))
117, 10eleqtrrd 2844 . . . . . . . 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 2833 . . . . . . . . 9 (𝑋𝐾𝑋 ∈ (Base‘𝑅))
1615biimpi 216 . . . . . . . 8 (𝑋𝐾𝑋 ∈ (Base‘𝑅))
1716adantr 480 . . . . . . 7 ((𝑋𝐾𝑌𝐵) → 𝑋 ∈ (Base‘𝑅))
1817ad2antlr 727 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑋 ∈ (Base‘𝑅))
19 mat2pmatbas.a . . . . . . 7 𝐴 = (𝑁 Mat 𝑅)
20 eqid 2737 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
21 mat2pmatbas.b . . . . . . 7 𝐵 = (Base‘𝐴)
22 simprl 771 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑖𝑁)
23 simpr 484 . . . . . . . 8 ((𝑖𝑁𝑗𝑁) → 𝑗𝑁)
2423adantl 481 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑗𝑁)
25 simplrr 778 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑌𝐵)
2619, 20, 21, 22, 24, 25matecld 22432 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖𝑌𝑗) ∈ (Base‘𝑅))
27 eqid 2737 . . . . . . 7 (.r𝑅) = (.r𝑅)
28 eqid 2737 . . . . . . 7 (.r𝑃) = (.r𝑃)
2920, 27, 28rhmmul 20486 . . . . . 6 ((𝑆 ∈ (𝑅 RingHom 𝑃) ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑖𝑌𝑗) ∈ (Base‘𝑅)) → (𝑆‘(𝑋(.r𝑅)(𝑖𝑌𝑗))) = ((𝑆𝑋)(.r𝑃)(𝑆‘(𝑖𝑌𝑗))))
3013, 18, 26, 29syl3anc 1373 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑆‘(𝑋(.r𝑅)(𝑖𝑌𝑗))) = ((𝑆𝑋)(.r𝑃)(𝑆‘(𝑖𝑌𝑗))))
31 crngring 20242 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3231ad2antlr 727 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → 𝑅 ∈ Ring)
3332adantr 480 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑅 ∈ Ring)
34 simpr 484 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑋𝐾𝑌𝐵))
3534adantr 480 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑋𝐾𝑌𝐵))
36 simpr 484 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖𝑁𝑗𝑁))
37 mat2pmatlin.m . . . . . . . 8 · = ( ·𝑠𝐴)
3819, 21, 14, 37, 27matvscacell 22442 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑋𝐾𝑌𝐵) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑋 · 𝑌)𝑗) = (𝑋(.r𝑅)(𝑖𝑌𝑗)))
3933, 35, 36, 38syl3anc 1373 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑋 · 𝑌)𝑗) = (𝑋(.r𝑅)(𝑖𝑌𝑗)))
4039fveq2d 6910 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)) = (𝑆‘(𝑋(.r𝑅)(𝑖𝑌𝑗))))
4131anim2i 617 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
42 simpr 484 . . . . . . . . 9 ((𝑋𝐾𝑌𝐵) → 𝑌𝐵)
4341, 42anim12i 613 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐵))
44 df-3an 1089 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑌𝐵))
4543, 44sylibr 234 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐵))
46 mat2pmatbas.t . . . . . . . 8 𝑇 = (𝑁 matToPolyMat 𝑅)
4746, 19, 21, 2, 4mat2pmatvalel 22731 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐵) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇𝑌)𝑗) = (𝑆‘(𝑖𝑌𝑗)))
4845, 47sylan 580 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇𝑌)𝑗) = (𝑆‘(𝑖𝑌𝑗)))
4948oveq2d 7447 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑆𝑋)(.r𝑃)(𝑖(𝑇𝑌)𝑗)) = ((𝑆𝑋)(.r𝑃)(𝑆‘(𝑖𝑌𝑗))))
5030, 40, 493eqtr4d 2787 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)) = ((𝑆𝑋)(.r𝑃)(𝑖(𝑇𝑌)𝑗)))
51 simpll 767 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → 𝑁 ∈ Fin)
5251adantr 480 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → 𝑁 ∈ Fin)
5314, 19, 21, 37matvscl 22437 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐾𝑌𝐵)) → (𝑋 · 𝑌) ∈ 𝐵)
5441, 53sylan 580 . . . . . 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 22731 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑋 · 𝑌) ∈ 𝐵) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)))
5752, 33, 55, 36, 56syl31anc 1375 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)))
582ply1ring 22249 . . . . . . . 8 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5931, 58syl 17 . . . . . . 7 (𝑅 ∈ CRing → 𝑃 ∈ Ring)
6059ad2antlr 727 . . . . . 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 482 . . . . . . . 8 ((𝑋𝐾𝑌𝐵) → 𝑋𝐾)
64 eqid 2737 . . . . . . . . 9 (Base‘𝑃) = (Base‘𝑃)
652, 4, 14, 64ply1sclcl 22289 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑋𝐾) → (𝑆𝑋) ∈ (Base‘𝑃))
6662, 63, 65syl2an 596 . . . . . . 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 22733 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌𝐵) → (𝑇𝑌) ∈ 𝐻)
7045, 69syl 17 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑇𝑌) ∈ 𝐻)
7166, 70jca 511 . . . . . 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 22442 . . . . 5 ((𝑃 ∈ Ring ∧ ((𝑆𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑌) ∈ 𝐻) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗) = ((𝑆𝑋)(.r𝑃)(𝑖(𝑇𝑌)𝑗)))
7561, 72, 36, 74syl3anc 1373 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗) = ((𝑆𝑋)(.r𝑃)(𝑖(𝑇𝑌)𝑗)))
7650, 57, 753eqtr4d 2787 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗))
7776ralrimivva 3202 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → ∀𝑖𝑁𝑗𝑁 (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗))
7846, 19, 21, 2, 67, 68mat2pmatbas0 22733 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑋 · 𝑌) ∈ 𝐵) → (𝑇‘(𝑋 · 𝑌)) ∈ 𝐻)
7951, 32, 54, 78syl3anc 1373 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑇‘(𝑋 · 𝑌)) ∈ 𝐻)
8064, 67, 68, 73matvscl 22437 . . . 4 (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝑆𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑌) ∈ 𝐻)) → ((𝑆𝑋) × (𝑇𝑌)) ∈ 𝐻)
8151, 60, 71, 80syl21anc 838 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → ((𝑆𝑋) × (𝑇𝑌)) ∈ 𝐻)
8267, 68eqmat 22430 . . 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 584 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → ((𝑇‘(𝑋 · 𝑌)) = ((𝑆𝑋) × (𝑇𝑌)) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗)))
8477, 83mpbird 257 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑇‘(𝑋 · 𝑌)) = ((𝑆𝑋) × (𝑇𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cfv 6561  (class class class)co 7431  Fincfn 8985  Basecbs 17247  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  Ringcrg 20230  CRingccrg 20231   RingHom crh 20469  AssAlgcasa 21870  algSccascl 21872  Poly1cpl1 22178   Mat cmat 22411   matToPolyMat cmat2pmat 22710
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-rhm 20472  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-mpl 21931  df-opsr 21933  df-psr1 22181  df-ply1 22183  df-mat 22412  df-mat2pmat 22713
This theorem is referenced by:  cpmidgsumm2pm  22875
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