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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 7464 | . . . 4 ⊢ (𝑅 freeLMod (𝑁 × 𝑁)) ∈ V | |
2 | matmulr.t | . . . . 5 ⊢ · = (𝑅 maMul 〈𝑁, 𝑁, 𝑁〉) | |
3 | 2 | ovexi 7465 | . . . 4 ⊢ · ∈ V |
4 | 1, 3 | pm3.2i 470 | . . 3 ⊢ ((𝑅 freeLMod (𝑁 × 𝑁)) ∈ V ∧ · ∈ V) |
5 | mulridx 17340 | . . . 4 ⊢ .r = Slot (.r‘ndx) | |
6 | 5 | setsid 17242 | . . 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 2735 | . . . 4 ⊢ (𝑅 freeLMod (𝑁 × 𝑁)) = (𝑅 freeLMod (𝑁 × 𝑁)) | |
10 | 8, 9, 2 | matval 22431 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐴 = ((𝑅 freeLMod (𝑁 × 𝑁)) sSet 〈(.r‘ndx), · 〉)) |
11 | 10 | fveq2d 6911 | . 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (.r‘𝐴) = (.r‘((𝑅 freeLMod (𝑁 × 𝑁)) sSet 〈(.r‘ndx), · 〉))) |
12 | 7, 11 | eqtr4d 2778 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → · = (.r‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 〈cotp 4639 × cxp 5687 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 sSet csts 17197 ndxcnx 17227 .rcmulr 17299 freeLMod cfrlm 21784 maMul cmmul 22410 Mat cmat 22427 |
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-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
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-ral 3060 df-rex 3069 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-op 4638 df-ot 4640 df-uni 4913 df-iun 4998 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-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-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-mulr 17312 df-mat 22428 |
This theorem is referenced by: matring 22465 matassa 22466 matmulcell 22467 mpomatmul 22468 mat1 22469 mattposm 22481 mat1dimmul 22498 dmatmul 22519 mdetmul 22645 madurid 22666 slesolinv 22702 slesolinvbi 22703 cramerimplem3 22707 mat2pmatmul 22753 decpmatmullem 22793 decpmatmul 22794 matunitlindflem2 37604 |
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