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Mirrors > Home > MPE Home > Th. List > matmulr | Structured version Visualization version GIF version |
Description: Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
Ref | Expression |
---|---|
matmulr.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
matmulr.t | ⊢ · = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) |
Ref | Expression |
---|---|
matmulr | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → · = (.r‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7181 | . . . 4 ⊢ (𝑅 freeLMod (𝑁 × 𝑁)) ∈ V | |
2 | matmulr.t | . . . . 5 ⊢ · = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) | |
3 | 2 | ovexi 7182 | . . . 4 ⊢ · ∈ V |
4 | 1, 3 | pm3.2i 473 | . . 3 ⊢ ((𝑅 freeLMod (𝑁 × 𝑁)) ∈ V ∧ · ∈ V) |
5 | mulrid 16608 | . . . 4 ⊢ .r = Slot (.r‘ndx) | |
6 | 5 | setsid 16530 | . . 3 ⊢ (((𝑅 freeLMod (𝑁 × 𝑁)) ∈ V ∧ · ∈ V) → · = (.r‘((𝑅 freeLMod (𝑁 × 𝑁)) sSet 〈(.r‘ndx), · 〉))) |
7 | 4, 6 | mp1i 13 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → · = (.r‘((𝑅 freeLMod (𝑁 × 𝑁)) sSet 〈(.r‘ndx), · 〉))) |
8 | matmulr.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
9 | eqid 2819 | . . . 4 ⊢ (𝑅 freeLMod (𝑁 × 𝑁)) = (𝑅 freeLMod (𝑁 × 𝑁)) | |
10 | 8, 9, 2 | matval 21012 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = ((𝑅 freeLMod (𝑁 × 𝑁)) sSet 〈(.r‘ndx), · 〉)) |
11 | 10 | fveq2d 6667 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (.r‘𝐴) = (.r‘((𝑅 freeLMod (𝑁 × 𝑁)) sSet 〈(.r‘ndx), · 〉))) |
12 | 7, 11 | eqtr4d 2857 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → · = (.r‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 Vcvv 3493 〈cop 4565 〈cotp 4567 × cxp 5546 ‘cfv 6348 (class class class)co 7148 Fincfn 8501 ndxcnx 16472 sSet csts 16473 .rcmulr 16558 freeLMod cfrlm 20882 maMul cmmul 20986 Mat cmat 21008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-1cn 10587 ax-addcl 10589 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-ot 4568 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-nn 11631 df-2 11692 df-3 11693 df-ndx 16478 df-slot 16479 df-sets 16482 df-mulr 16571 df-mat 21009 |
This theorem is referenced by: matring 21044 matassa 21045 matmulcell 21046 mpomatmul 21047 mat1 21048 mattposm 21060 mat1dimmul 21077 dmatmul 21098 mdetmul 21224 madurid 21245 slesolinv 21281 slesolinvbi 21282 cramerimplem3 21286 mat2pmatmul 21331 decpmatmullem 21371 decpmatmul 21372 matunitlindflem2 34881 |
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