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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 2554 | . . . . 5 ⊢ y ∈ V | |
2 | brtposg 5810 | . . . . 5 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊 ∧ y ∈ V) → (〈A, B〉tpos 𝐹y ↔ 〈B, A〉𝐹y)) | |
3 | 1, 2 | mp3an3 1220 | . . . 4 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (〈A, B〉tpos 𝐹y ↔ 〈B, A〉𝐹y)) |
4 | 3 | iotabidv 4831 | . . 3 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (℩y〈A, B〉tpos 𝐹y) = (℩y〈B, A〉𝐹y)) |
5 | df-fv 4853 | . . 3 ⊢ (tpos 𝐹‘〈A, B〉) = (℩y〈A, B〉tpos 𝐹y) | |
6 | df-fv 4853 | . . 3 ⊢ (𝐹‘〈B, A〉) = (℩y〈B, A〉𝐹y) | |
7 | 4, 5, 6 | 3eqtr4g 2094 | . 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (tpos 𝐹‘〈A, B〉) = (𝐹‘〈B, A〉)) |
8 | df-ov 5458 | . 2 ⊢ (Atpos 𝐹B) = (tpos 𝐹‘〈A, B〉) | |
9 | df-ov 5458 | . 2 ⊢ (B𝐹A) = (𝐹‘〈B, A〉) | |
10 | 7, 8, 9 | 3eqtr4g 2094 | 1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (Atpos 𝐹B) = (B𝐹A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 Vcvv 2551 〈cop 3370 class class class wbr 3755 ℩cio 4808 ‘cfv 4845 (class class class)co 5455 tpos ctpos 5800 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-fv 4853 df-ov 5458 df-tpos 5801 |
This theorem is referenced by: tpossym 5832 |
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