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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 2684 | . . . . 5 ⊢ 𝑦 ∈ V | |
2 | brtposg 6144 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑦 ∈ V) → (〈𝐴, 𝐵〉tpos 𝐹𝑦 ↔ 〈𝐵, 𝐴〉𝐹𝑦)) | |
3 | 1, 2 | mp3an3 1304 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉tpos 𝐹𝑦 ↔ 〈𝐵, 𝐴〉𝐹𝑦)) |
4 | 3 | iotabidv 5104 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (℩𝑦〈𝐴, 𝐵〉tpos 𝐹𝑦) = (℩𝑦〈𝐵, 𝐴〉𝐹𝑦)) |
5 | df-fv 5126 | . . 3 ⊢ (tpos 𝐹‘〈𝐴, 𝐵〉) = (℩𝑦〈𝐴, 𝐵〉tpos 𝐹𝑦) | |
6 | df-fv 5126 | . . 3 ⊢ (𝐹‘〈𝐵, 𝐴〉) = (℩𝑦〈𝐵, 𝐴〉𝐹𝑦) | |
7 | 4, 5, 6 | 3eqtr4g 2195 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (tpos 𝐹‘〈𝐴, 𝐵〉) = (𝐹‘〈𝐵, 𝐴〉)) |
8 | df-ov 5770 | . 2 ⊢ (𝐴tpos 𝐹𝐵) = (tpos 𝐹‘〈𝐴, 𝐵〉) | |
9 | df-ov 5770 | . 2 ⊢ (𝐵𝐹𝐴) = (𝐹‘〈𝐵, 𝐴〉) | |
10 | 7, 8, 9 | 3eqtr4g 2195 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴tpos 𝐹𝐵) = (𝐵𝐹𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2681 〈cop 3525 class class class wbr 3924 ℩cio 5081 ‘cfv 5118 (class class class)co 5767 tpos ctpos 6134 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 df-ov 5770 df-tpos 6135 |
This theorem is referenced by: tpossym 6166 |
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