Theorem opexmid

Description: Law of excluded middle for orthoposets. ( chjo analog.) (Contributed by NM, 13-Sep-2011)

	Ref	Expression
Hypotheses	opexmid.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	opexmid.o	⊢ ⊥ = ( oc ‘ 𝐾 )
	opexmid.j	⊢ ∨ = ( join ‘ 𝐾 )
	opexmid.u	⊢ 1 = ( 1. ‘ 𝐾 )
Assertion	opexmid	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∨ ( ⊥ ‘ 𝑋 ) ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		opexmid.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		opexmid.o	⊢ ⊥ = ( oc ‘ 𝐾 )
3		opexmid.j	⊢ ∨ = ( join ‘ 𝐾 )
4		opexmid.u	⊢ 1 = ( 1. ‘ 𝐾 )
5		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
6		eqid	⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
7		eqid	⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
8	1 5 2 3 6 7 4	oposlem	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ( ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ ( 𝑋 ( le ‘ 𝐾 ) 𝑋 → ( ⊥ ‘ 𝑋 ) ( le ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) ) ∧ ( 𝑋 ∨ ( ⊥ ‘ 𝑋 ) ) = 1 ∧ ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) = ( 0. ‘ 𝐾 ) ) )
9	8	3anidm23	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( ( ( ⊥ ‘ 𝑋 ) ∈ 𝐵 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ∧ ( 𝑋 ( le ‘ 𝐾 ) 𝑋 → ( ⊥ ‘ 𝑋 ) ( le ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) ) ∧ ( 𝑋 ∨ ( ⊥ ‘ 𝑋 ) ) = 1 ∧ ( 𝑋 ( meet ‘ 𝐾 ) ( ⊥ ‘ 𝑋 ) ) = ( 0. ‘ 𝐾 ) ) )
10	9	simp2d	⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∨ ( ⊥ ‘ 𝑋 ) ) = 1 )