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Metamath Proof Explorer

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 ) )