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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 8219 | . . . . 5 ⊢ (𝑦 ∈ V → (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴⟩𝐹𝑦)) | |
2 | 1 | elv 3480 | . . . 4 ⊢ (⟨𝐴, 𝐵⟩tpos 𝐹𝑦 ↔ ⟨𝐵, 𝐴⟩𝐹𝑦) |
3 | 2 | iotabii 6528 | . . 3 ⊢ (℩𝑦⟨𝐴, 𝐵⟩tpos 𝐹𝑦) = (℩𝑦⟨𝐵, 𝐴⟩𝐹𝑦) |
4 | df-fv 6551 | . . 3 ⊢ (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (℩𝑦⟨𝐴, 𝐵⟩tpos 𝐹𝑦) | |
5 | df-fv 6551 | . . 3 ⊢ (𝐹‘⟨𝐵, 𝐴⟩) = (℩𝑦⟨𝐵, 𝐴⟩𝐹𝑦) | |
6 | 3, 4, 5 | 3eqtr4i 2770 | . 2 ⊢ (tpos 𝐹‘⟨𝐴, 𝐵⟩) = (𝐹‘⟨𝐵, 𝐴⟩) |
7 | df-ov 7411 | . 2 ⊢ (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘⟨𝐴, 𝐵⟩) | |
8 | df-ov 7411 | . 2 ⊢ (𝐵𝐹𝐴) = (𝐹‘⟨𝐵, 𝐴⟩) | |
9 | 6, 7, 8 | 3eqtr4i 2770 | 1 ⊢ (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 Vcvv 3474 ⟨cop 4634 class class class wbr 5148 ℩cio 6493 ‘cfv 6543 (class class class)co 7408 tpos ctpos 8209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 df-ov 7411 df-tpos 8210 |
This theorem is referenced by: tpossym 8242 oppchom 17659 oppcco 17661 oppcmon 17684 funcoppc 17824 fulloppc 17872 fthoppc 17873 fthepi 17878 yonedalem22 18230 oppgplus 19212 oppglsm 19509 opprmul 20152 mamutpos 21959 mdettpos 22112 madutpos 22143 mdetpmtr2 32799 |
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