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Mirrors > Home > MPE Home > Th. List > tposeqd | Structured version Visualization version GIF version |
Description: Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
Ref | Expression |
---|---|
tposeqd.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
Ref | Expression |
---|---|
tposeqd | ⊢ (𝜑 → tpos 𝐹 = tpos 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tposeqd.1 | . 2 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | tposeq 8269 | . 2 ⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → tpos 𝐹 = tpos 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 tpos ctpos 8266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-mpt 5250 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-res 5712 df-tpos 8267 |
This theorem is referenced by: oppcval 17771 oppchomfval 17772 oppchomfvalOLD 17773 oppccofval 17775 oppchomfpropd 17786 oppcmon 17799 oppgval 19387 oppgplusfval 19388 oppglsm 19684 opprval 20361 opprmulfval 20362 mattposvs 22482 mattpos1 22483 mamutpos 22485 mattposm 22486 madulid 22672 |
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