Theorem tposeqd 8224

Description: Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.)

Hypothesis

Ref	Expression
tposeqd.1	⊢ (𝜑 → 𝐹 = 𝐺)

Assertion

Ref	Expression
tposeqd	⊢ (𝜑 → tpos 𝐹 = tpos 𝐺)

Proof of Theorem tposeqd

Step	Hyp	Ref	Expression
1		tposeqd.1	. 2 ⊢ (𝜑 → 𝐹 = 𝐺)
2		tposeq 8223	. 2 ⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)
3	1, 2	syl 18	1 ⊢ (𝜑 → tpos 𝐹 = tpos 𝐺)

Syntax hints:   → wi 4   = wceq 1567  tpos ctpos 8220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-mpt 5197  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-res 5674  df-tpos 8221

This theorem is referenced by:  oppcval  17768  oppchomfval  17769  oppccofval  17771  oppchomfpropd  17781  oppcmon  17794  oppgval  19416  oppgplusfval  19417  oppglsm  19711  opprval  20419  opprmulfval  20420  mattposvs  22580  mattpos1  22581  mamutpos  22583  mattposm  22584  madulid  22770  oppfvalg  49788  funcoppc4  49806  uptposlem  49859  oppgoppcco  50253