MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mat2pmatlin Structured version   Visualization version   GIF version

Theorem mat2pmatlin 22757
Description: The transformation of matrices into polynomial matrices is "linear", analogous to lmhmlin 21052. Since 𝐴 and 𝐶 have different scalar rings, 𝑇 cannot be a left module homomorphism as defined in df-lmhm 21039, see lmhmsca 21047. (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 22217 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
4 mat2pmatlin.s . . . . . . . . . . 11 𝑆 = (algSc‘𝑃)
5 eqid 2735 . . . . . . . . . . 11 (Scalar‘𝑃) = (Scalar‘𝑃)
64, 5asclrhm 21928 . . . . . . . . . 10 (𝑃 ∈ AssAlg → 𝑆 ∈ ((Scalar‘𝑃) RingHom 𝑃))
71, 3, 63syl 18 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑆 ∈ ((Scalar‘𝑃) RingHom 𝑃))
82ply1sca 22270 . . . . . . . . . . 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 2842 . . . . . . . 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 2831 . . . . . . . . 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 2735 . . . . . . 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 22448 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖𝑌𝑗) ∈ (Base‘𝑅))
27 eqid 2735 . . . . . . 7 (.r𝑅) = (.r𝑅)
28 eqid 2735 . . . . . . 7 (.r𝑃) = (.r𝑃)
2920, 27, 28rhmmul 20503 . . . . . 6 ((𝑆 ∈ (𝑅 RingHom 𝑃) ∧ 𝑋 ∈ (Base‘𝑅) ∧ (𝑖𝑌𝑗) ∈ (Base‘𝑅)) → (𝑆‘(𝑋(.r𝑅)(𝑖𝑌𝑗))) = ((𝑆𝑋)(.r𝑃)(𝑆‘(𝑖𝑌𝑗))))
3013, 18, 26, 29syl3anc 1370 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑆‘(𝑋(.r𝑅)(𝑖𝑌𝑗))) = ((𝑆𝑋)(.r𝑃)(𝑆‘(𝑖𝑌𝑗))))
31 crngring 20263 . . . . . . . . 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 22458 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑋𝐾𝑌𝐵) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑋 · 𝑌)𝑗) = (𝑋(.r𝑅)(𝑖𝑌𝑗)))
3933, 35, 36, 38syl3anc 1370 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑋 · 𝑌)𝑗) = (𝑋(.r𝑅)(𝑖𝑌𝑗)))
4039fveq2d 6911 . . . . 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 1088 . . . . . . . 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 22747 . . . . . . 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 2785 . . . 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 22453 . . . . . . 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 22747 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑋 · 𝑌) ∈ 𝐵) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)))
5752, 33, 55, 36, 56syl31anc 1372 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑆‘(𝑖(𝑋 · 𝑌)𝑗)))
582ply1ring 22265 . . . . . . . 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 2735 . . . . . . . . 9 (Base‘𝑃) = (Base‘𝑃)
652, 4, 14, 64ply1sclcl 22305 . . . . . . . 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 22749 . . . . . . . 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 22458 . . . . 5 ((𝑃 ∈ Ring ∧ ((𝑆𝑋) ∈ (Base‘𝑃) ∧ (𝑇𝑌) ∈ 𝐻) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗) = ((𝑆𝑋)(.r𝑃)(𝑖(𝑇𝑌)𝑗)))
7561, 72, 36, 74syl3anc 1370 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗) = ((𝑆𝑋)(.r𝑃)(𝑖(𝑇𝑌)𝑗)))
7650, 57, 753eqtr4d 2785 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗))
7776ralrimivva 3200 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → ∀𝑖𝑁𝑗𝑁 (𝑖(𝑇‘(𝑋 · 𝑌))𝑗) = (𝑖((𝑆𝑋) × (𝑇𝑌))𝑗))
7846, 19, 21, 2, 67, 68mat2pmatbas0 22749 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑋 · 𝑌) ∈ 𝐵) → (𝑇‘(𝑋 · 𝑌)) ∈ 𝐻)
7951, 32, 54, 78syl3anc 1370 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑇‘(𝑋 · 𝑌)) ∈ 𝐻)
8064, 67, 68, 73matvscl 22453 . . . 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 22446 . . 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 1086   = wceq 1537  wcel 2106  wral 3059  cfv 6563  (class class class)co 7431  Fincfn 8984  Basecbs 17245  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  Ringcrg 20251  CRingccrg 20252   RingHom crh 20486  AssAlgcasa 21888  algSccascl 21890  Poly1cpl1 22194   Mat cmat 22427   matToPolyMat cmat2pmat 22726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-rhm 20489  df-subrng 20563  df-subrg 20587  df-lmod 20877  df-lss 20948  df-sra 21190  df-rgmod 21191  df-dsmm 21770  df-frlm 21785  df-assa 21891  df-ascl 21893  df-psr 21947  df-mpl 21949  df-opsr 21951  df-psr1 22197  df-ply1 22199  df-mat 22428  df-mat2pmat 22729
This theorem is referenced by:  cpmidgsumm2pm  22891
  Copyright terms: Public domain W3C validator