Theorem qlaxr4 5527

Description: One of the conditions showing C_ℋ is an ortholattice. (This corresponds to axiom "ax-r4" in the Quantum Logic Explorer.)

Hypotheses

Ref	Expression
qlaxr4.1	⊢ A ∈ Cℋ
qlaxr4.2	⊢ B ∈ Cℋ
qlaxr4.3	⊢ A = B

Assertion

Ref	Expression
qlaxr4	⊢ (⊥ ‘A) = (⊥ ‘B)

Proof of Theorem qlaxr4

Step	Hyp	Ref	Expression
1		qlaxr4.3	. 2 ⊢ A = B
2	1	fveq2i 2835	1 ⊢ (⊥ ‘A) = (⊥ ‘B)

Syntax hints:   = wceq 1091   ∈ wcel 1092   ‘cfv 2422   C_ℋ cch 4968  ⊥cort 4969

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438

metamath.org