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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 8253 | . 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 1540 tpos ctpos 8250 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-tpos 8251 |
| This theorem is referenced by: oppcval 17756 oppchomfval 17757 oppccofval 17759 oppchomfpropd 17769 oppcmon 17782 oppgval 19365 oppgplusfval 19366 oppglsm 19660 opprval 20335 opprmulfval 20336 mattposvs 22461 mattpos1 22462 mamutpos 22464 mattposm 22465 madulid 22651 oppgoppcco 49188 |
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