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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 8220 | . . . . 5 ⊢ (𝑦 ∈ V → (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴⟩𝐹𝑦)) | |
2 | 1 | elv 3481 | . . . 4 ⊢ (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴⟩𝐹𝑦) |
3 | 2 | iotabii 6529 | . . 3 ⊢ (℩𝑦⟨𝐴, 𝐵⟩tpos 𝐹𝑦) = (℩𝑦⟨𝐵, 𝐴⟩𝐹𝑦) |
4 | df-fv 6552 | . . 3 ⊢ (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑦⟨𝐴, 𝐵⟩tpos 𝐹𝑦) | |
5 | df-fv 6552 | . . 3 ⊢ (𝐹‘⟨𝐵, 𝐴⟩) = (℩𝑦⟨𝐵, 𝐴⟩𝐹𝑦) | |
6 | 3, 4, 5 | 3eqtr4i 2771 | . 2 ⊢ (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐵, 𝐴⟩) |
7 | df-ov 7412 | . 2 ⊢ (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘⟨𝐴, 𝐵⟩) | |
8 | df-ov 7412 | . 2 ⊢ (𝐵𝐹𝐴) = (𝐹‘⟨𝐵, 𝐴⟩) | |
9 | 6, 7, 8 | 3eqtr4i 2771 | 1 ⊢ (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 Vcvv 3475 ⟨cop 4635 class class class wbr 5149 ℩cio 6494 ‘cfv 6544 (class class class)co 7409 tpos ctpos 8210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-fv 6552 df-ov 7412 df-tpos 8211 |
This theorem is referenced by: tpossym 8243 oppchom 17660 oppcco 17662 oppcmon 17685 funcoppc 17825 fulloppc 17873 fthoppc 17874 fthepi 17879 yonedalem22 18231 oppgplus 19213 oppglsm 19510 opprmul 20153 mamutpos 21960 mdettpos 22113 madutpos 22144 mdetpmtr2 32804 |
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