Theorem tposeqd 8221

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 8220	. 2 ⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺)
3	1, 2	syl 18	1 ⊢ (𝜑 → tpos 𝐹 = tpos 𝐺)

Syntax hints:   → wi 4   = wceq 1570  tpos ctpos 8217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-mpt 5193  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-tpos 8218

This theorem is referenced by:  oppcval  17764  oppchomfval  17765  oppccofval  17767  oppchomfpropd  17777  oppcmon  17790  oppgval  19412  oppgplusfval  19413  oppglsm  19707  opprval  20416  opprmulfval  20417  mattposvs  22612  mattpos1  22613  mamutpos  22615  mattposm  22616  madulid  22802  oppfvalg  49904  funcoppc4  49922  uptposlem  49975  oppgoppcco  50369