Theorem cmt3N

Description: Commutation with orthocomplement. Remark in Kalmbach p. 23. ( cmcm4i analog.) (Contributed by NM, 8-Nov-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cmt2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cmt2.o	⊢ ⊥ = ( oc ‘ 𝐾 )
	cmt2.c	⊢ 𝐶 = ( cm ‘ 𝐾 )
Assertion	cmt3N	⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ ( ⊥ ‘ 𝑋 ) 𝐶 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		cmt2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cmt2.o	⊢ ⊥ = ( oc ‘ 𝐾 )
3		cmt2.c	⊢ 𝐶 = ( cm ‘ 𝐾 )
4	1 2 3	cmt2N	⊢ ( ( 𝐾 ∈ OML ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 𝐶 𝑋 ↔ 𝑌 𝐶 ( ⊥ ‘ 𝑋 ) ) )
5	4	3com23	⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 𝐶 𝑋 ↔ 𝑌 𝐶 ( ⊥ ‘ 𝑋 ) ) )
6	1 3	cmtcomN	⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ 𝑌 𝐶 𝑋 ) )
7		omlop	⊢ ( 𝐾 ∈ OML → 𝐾 ∈ OP )
8	7	3ad2ant1	⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ OP )
9		simp2	⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
10	1 2	opoccl	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )
11	8 9 10	syl2anc	⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐵 )
12	1 3	cmtcomN	⊢ ( ( 𝐾 ∈ OML ∧ ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑋 ) 𝐶 𝑌 ↔ 𝑌 𝐶 ( ⊥ ‘ 𝑋 ) ) )
13	11 12	syld3an2	⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ 𝑋 ) 𝐶 𝑌 ↔ 𝑌 𝐶 ( ⊥ ‘ 𝑋 ) ) )
14	5 6 13	3bitr4d	⊢ ( ( 𝐾 ∈ OML ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ ( ⊥ ‘ 𝑋 ) 𝐶 𝑌 ) )