Theorem pmap11

Description: The projective map of a Hilbert lattice is one-to-one. Part of Theorem 15.5 of MaedaMaeda p. 62. (Contributed by NM, 22-Oct-2011)

	Ref	Expression
Hypotheses	pmap11.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	pmap11.m	⊢ 𝑀 = ( pmap ‘ 𝐾 )
Assertion	pmap11	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		pmap11.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		pmap11.m	⊢ 𝑀 = ( pmap ‘ 𝐾 )
3		eqss	⊢ ( ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑌 ) ↔ ( ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ∧ ( 𝑀 ‘ 𝑌 ) ⊆ ( 𝑀 ‘ 𝑋 ) ) )
4		hllat	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
5		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
6	1 5	latasymb	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ↔ 𝑋 = 𝑌 ) )
7	4 6	syl3an1	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ↔ 𝑋 = 𝑌 ) )
8	1 5 2	pmaple	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ↔ ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ) )
9	1 5 2	pmaple	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 ( le ‘ 𝐾 ) 𝑋 ↔ ( 𝑀 ‘ 𝑌 ) ⊆ ( 𝑀 ‘ 𝑋 ) ) )
10	9	3com23	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑌 ( le ‘ 𝐾 ) 𝑋 ↔ ( 𝑀 ‘ 𝑌 ) ⊆ ( 𝑀 ‘ 𝑋 ) ) )
11	8 10	anbi12d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑌 ( le ‘ 𝐾 ) 𝑋 ) ↔ ( ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ∧ ( 𝑀 ‘ 𝑌 ) ⊆ ( 𝑀 ‘ 𝑋 ) ) ) )
12	7 11	bitr3d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 = 𝑌 ↔ ( ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ∧ ( 𝑀 ‘ 𝑌 ) ⊆ ( 𝑀 ‘ 𝑋 ) ) ) )
13	3 12	bitr4id	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )