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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 8252 | . 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 8249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-tpos 8250 |
This theorem is referenced by: oppcval 17758 oppchomfval 17759 oppchomfvalOLD 17760 oppccofval 17762 oppchomfpropd 17773 oppcmon 17786 oppgval 19378 oppgplusfval 19379 oppglsm 19675 opprval 20352 opprmulfval 20353 mattposvs 22477 mattpos1 22478 mamutpos 22480 mattposm 22481 madulid 22667 oppgoppcco 48900 |
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