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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 7519 | . 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 1628 tpos ctpos 7516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-sep 4929 ax-nul 4937 ax-pr 5051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1631 df-ex 1850 df-nf 1855 df-sb 2043 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-rab 3055 df-v 3338 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-nul 4055 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-br 4801 df-opab 4861 df-mpt 4878 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-res 5274 df-tpos 7517 |
This theorem is referenced by: oppcval 16570 oppchomfval 16571 oppccofval 16573 oppchomfpropd 16583 oppcmon 16595 oppgval 17973 oppgplusfval 17974 oppglsm 18253 opprval 18820 opprmulfval 18821 mattposvs 20459 mattpos1 20460 mamutpos 20462 mattposm 20463 madulid 20649 |
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