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| Mirrors > Home > MPE Home > Th. List > ovtpos | Structured version Visualization version GIF version | ||
| Description: The transposition swaps the arguments in a two-argument function. When 𝐹 is a matrix, which is to say a function from (1...𝑚) × (1...𝑛) to ℝ or some ring, tpos 𝐹 is the transposition of 𝐹, which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.) |
| Ref | Expression |
|---|---|
| ovtpos | ⊢ (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | brtpos 8260 | . . . . 5 ⊢ (𝑦 ∈ V → (〈𝐴, 𝐵〉tpos 𝐹𝑦 ↔ 〈𝐵, 𝐴〉𝐹𝑦)) | |
| 2 | 1 | elv 3485 | . . . 4 ⊢ (〈𝐴, 𝐵〉tpos 𝐹𝑦 ↔ 〈𝐵, 𝐴〉𝐹𝑦) |
| 3 | 2 | iotabii 6546 | . . 3 ⊢ (℩𝑦〈𝐴, 𝐵〉tpos 𝐹𝑦) = (℩𝑦〈𝐵, 𝐴〉𝐹𝑦) |
| 4 | df-fv 6569 | . . 3 ⊢ (tpos 𝐹‘〈𝐴, 𝐵〉) = (℩𝑦〈𝐴, 𝐵〉tpos 𝐹𝑦) | |
| 5 | df-fv 6569 | . . 3 ⊢ (𝐹‘〈𝐵, 𝐴〉) = (℩𝑦〈𝐵, 𝐴〉𝐹𝑦) | |
| 6 | 3, 4, 5 | 3eqtr4i 2775 | . 2 ⊢ (tpos 𝐹‘〈𝐴, 𝐵〉) = (𝐹‘〈𝐵, 𝐴〉) |
| 7 | df-ov 7434 | . 2 ⊢ (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘〈𝐴, 𝐵〉) | |
| 8 | df-ov 7434 | . 2 ⊢ (𝐵𝐹𝐴) = (𝐹‘〈𝐵, 𝐴〉) | |
| 9 | 6, 7, 8 | 3eqtr4i 2775 | 1 ⊢ (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 Vcvv 3480 〈cop 4632 class class class wbr 5143 ℩cio 6512 ‘cfv 6561 (class class class)co 7431 tpos ctpos 8250 |
| 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-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 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-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-ov 7434 df-tpos 8251 |
| This theorem is referenced by: tpossym 8283 oppchom 17758 oppcco 17760 oppcmon 17782 funcoppc 17920 fulloppc 17969 fthoppc 17970 fthepi 17975 yonedalem22 18323 oppgplus 19367 oppglsm 19660 opprmul 20337 mamutpos 22464 mdettpos 22617 madutpos 22648 mdetpmtr2 33823 tposid 48785 tposidres 48786 tposideq 48788 oppcup 48948 |
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