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Mirrors > Home > ILE Home > Th. List > ovtposg | 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 the reals 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 |
---|---|
ovtposg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2692 | . . . . 5 ⊢ 𝑦 ∈ V | |
2 | brtposg 6159 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → (〈𝐴, 𝐵〉tpos 𝐹𝑦 ↔ 〈𝐵, 𝐴〉𝐹𝑦)) | |
3 | 1, 2 | mp3an3 1305 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉tpos 𝐹𝑦 ↔ 〈𝐵, 𝐴〉𝐹𝑦)) |
4 | 3 | iotabidv 5117 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (℩𝑦〈𝐴, 𝐵〉tpos 𝐹𝑦) = (℩𝑦〈𝐵, 𝐴〉𝐹𝑦)) |
5 | df-fv 5139 | . . 3 ⊢ (tpos 𝐹‘〈𝐴, 𝐵〉) = (℩𝑦〈𝐴, 𝐵〉tpos 𝐹𝑦) | |
6 | df-fv 5139 | . . 3 ⊢ (𝐹‘〈𝐵, 𝐴〉) = (℩𝑦〈𝐵, 𝐴〉𝐹𝑦) | |
7 | 4, 5, 6 | 3eqtr4g 2198 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (tpos 𝐹‘〈𝐴, 𝐵〉) = (𝐹‘〈𝐵, 𝐴〉)) |
8 | df-ov 5785 | . 2 ⊢ (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘〈𝐴, 𝐵〉) | |
9 | df-ov 5785 | . 2 ⊢ (𝐵𝐹𝐴) = (𝐹‘〈𝐵, 𝐴〉) | |
10 | 7, 8, 9 | 3eqtr4g 2198 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 Vcvv 2689 〈cop 3535 class class class wbr 3937 ℩cio 5094 ‘cfv 5131 (class class class)co 5782 tpos ctpos 6149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-fv 5139 df-ov 5785 df-tpos 6150 |
This theorem is referenced by: tpossym 6181 |
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