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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 8158 | . 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 1541 tpos ctpos 8155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-mpt 5189 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-res 5645 df-tpos 8156 |
This theorem is referenced by: oppcval 17592 oppchomfval 17593 oppchomfvalOLD 17594 oppccofval 17596 oppchomfpropd 17607 oppcmon 17620 oppgval 19123 oppgplusfval 19124 oppglsm 19422 opprval 20048 opprmulfval 20049 mattposvs 21802 mattpos1 21803 mamutpos 21805 mattposm 21806 madulid 21992 |
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