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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 7943 | . . . . 5 ⊢ (𝑦 ∈ V → (〈𝐴, 𝐵〉tpos 𝐹𝑦 ↔ 〈𝐵, 𝐴〉𝐹𝑦)) | |
2 | 1 | elv 3406 | . . . 4 ⊢ (〈𝐴, 𝐵〉tpos 𝐹𝑦 ↔ 〈𝐵, 𝐴〉𝐹𝑦) |
3 | 2 | iotabii 6335 | . . 3 ⊢ (℩𝑦〈𝐴, 𝐵〉tpos 𝐹𝑦) = (℩𝑦〈𝐵, 𝐴〉𝐹𝑦) |
4 | df-fv 6358 | . . 3 ⊢ (tpos 𝐹‘〈𝐴, 𝐵〉) = (℩𝑦〈𝐴, 𝐵〉tpos 𝐹𝑦) | |
5 | df-fv 6358 | . . 3 ⊢ (𝐹‘〈𝐵, 𝐴〉) = (℩𝑦〈𝐵, 𝐴〉𝐹𝑦) | |
6 | 3, 4, 5 | 3eqtr4i 2772 | . 2 ⊢ (tpos 𝐹‘〈𝐴, 𝐵〉) = (𝐹‘〈𝐵, 𝐴〉) |
7 | df-ov 7186 | . 2 ⊢ (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘〈𝐴, 𝐵〉) | |
8 | df-ov 7186 | . 2 ⊢ (𝐵𝐹𝐴) = (𝐹‘〈𝐵, 𝐴〉) | |
9 | 6, 7, 8 | 3eqtr4i 2772 | 1 ⊢ (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1542 Vcvv 3400 〈cop 4532 class class class wbr 5040 ℩cio 6306 ‘cfv 6350 (class class class)co 7183 tpos ctpos 7933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6308 df-fun 6352 df-fn 6353 df-fv 6358 df-ov 7186 df-tpos 7934 |
This theorem is referenced by: tpossym 7966 oppchom 17102 oppcco 17104 oppcmon 17126 funcoppc 17263 fulloppc 17310 fthoppc 17311 fthepi 17316 yonedalem22 17657 oppgplus 18608 oppglsm 18898 opprmul 19511 mamutpos 21222 mdettpos 21375 madutpos 21406 mdetpmtr2 31359 |
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