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Mirrors > Home > ILE Home > Th. List > ovtposg | Unicode version |
Description: The transposition swaps the arguments in a two-argument function. When is a matrix, which is to say a function from ( 1 ... m ) ( 1 ... n ) to the reals or some ring, tpos is the transposition of , which is where the name comes from. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
ovtposg | tpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2663 | . . . . 5 | |
2 | brtposg 6119 | . . . . 5 tpos | |
3 | 1, 2 | mp3an3 1289 | . . . 4 tpos |
4 | 3 | iotabidv 5079 | . . 3 tpos |
5 | df-fv 5101 | . . 3 tpos tpos | |
6 | df-fv 5101 | . . 3 | |
7 | 4, 5, 6 | 3eqtr4g 2175 | . 2 tpos |
8 | df-ov 5745 | . 2 tpos tpos | |
9 | df-ov 5745 | . 2 | |
10 | 7, 8, 9 | 3eqtr4g 2175 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 cvv 2660 cop 3500 class class class wbr 3899 cio 5056 cfv 5093 (class class class)co 5742 tpos ctpos 6109 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 df-ov 5745 df-tpos 6110 |
This theorem is referenced by: tpossym 6141 |
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