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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 | vex 3176 | . . . . 5 ⊢ 𝑦 ∈ V | |
2 | brtpos 7248 | . . . . 5 ⊢ (𝑦 ∈ V → (〈𝐴, 𝐵〉tpos 𝐹𝑦 ↔ 〈𝐵, 𝐴〉𝐹𝑦)) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (〈𝐴, 𝐵〉tpos 𝐹𝑦 ↔ 〈𝐵, 𝐴〉𝐹𝑦) |
4 | 3 | iotabii 5790 | . . 3 ⊢ (℩𝑦〈𝐴, 𝐵〉tpos 𝐹𝑦) = (℩𝑦〈𝐵, 𝐴〉𝐹𝑦) |
5 | df-fv 5812 | . . 3 ⊢ (tpos 𝐹‘〈𝐴, 𝐵〉) = (℩𝑦〈𝐴, 𝐵〉tpos 𝐹𝑦) | |
6 | df-fv 5812 | . . 3 ⊢ (𝐹‘〈𝐵, 𝐴〉) = (℩𝑦〈𝐵, 𝐴〉𝐹𝑦) | |
7 | 4, 5, 6 | 3eqtr4i 2642 | . 2 ⊢ (tpos 𝐹‘〈𝐴, 𝐵〉) = (𝐹‘〈𝐵, 𝐴〉) |
8 | df-ov 6552 | . 2 ⊢ (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘〈𝐴, 𝐵〉) | |
9 | df-ov 6552 | . 2 ⊢ (𝐵𝐹𝐴) = (𝐹‘〈𝐵, 𝐴〉) | |
10 | 7, 8, 9 | 3eqtr4i 2642 | 1 ⊢ (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 class class class wbr 4583 ℩cio 5766 ‘cfv 5804 (class class class)co 6549 tpos ctpos 7238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-ov 6552 df-tpos 7239 |
This theorem is referenced by: tpossym 7271 oppchom 16198 oppcco 16200 oppcmon 16221 funcoppc 16358 fulloppc 16405 fthoppc 16406 fthepi 16411 yonedalem22 16741 oppgplus 17602 oppglsm 17880 opprmul 18449 mamutpos 20083 mdettpos 20236 madutpos 20267 mdetpmtr2 29218 |
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