This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem atcvrj0

Description: Two atoms covering the zero subspace are equal. ( atcv1 analog.) (Contributed by NM, 29-Nov-2011)

Ref Expression
Hypotheses atcvrj0.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
atcvrj0.j ⊢ ∨ = ( join ‘ 𝐾 )
atcvrj0.z ⊢ 0 = ( 0. ‘ 𝐾 )
atcvrj0.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
atcvrj0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion atcvrj0 ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → ( 𝑋 = 0 ↔ 𝑃 = 𝑄 ) )

Proof

Step Hyp Ref Expression
1 atcvrj0.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 atcvrj0.j ⊢ ∨ = ( join ‘ 𝐾 )
3 atcvrj0.z ⊢ 0 = ( 0. ‘ 𝐾 )
4 atcvrj0.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
5 atcvrj0.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 breq1 ⊢ ( 𝑋 = 0 → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ↔ 0 𝐶 ( 𝑃 ∨ 𝑄 ) ) )
7 6 biimpd ⊢ ( 𝑋 = 0 → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → 0 𝐶 ( 𝑃 ∨ 𝑄 ) ) )
8 7 adantl ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑋 = 0 ) → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → 0 𝐶 ( 𝑃 ∨ 𝑄 ) ) )
9 2 3 4 5 atcvr0eq ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 0 𝐶 ( 𝑃 ∨ 𝑄 ) ↔ 𝑃 = 𝑄 ) )
10 9 3adant3r1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 0 𝐶 ( 𝑃 ∨ 𝑄 ) ↔ 𝑃 = 𝑄 ) )
11 10 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑋 = 0 ) → ( 0 𝐶 ( 𝑃 ∨ 𝑄 ) ↔ 𝑃 = 𝑄 ) )
12 8 11 sylibd ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) ∧ 𝑋 = 0 ) → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → 𝑃 = 𝑄 ) )
13 12 ex ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 = 0 → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → 𝑃 = 𝑄 ) ) )
14 13 com23 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → ( 𝑋 = 0 → 𝑃 = 𝑄 ) ) )
15 14 3impia ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → ( 𝑋 = 0 → 𝑃 = 𝑄 ) )
16 oveq1 ⊢ ( 𝑃 = 𝑄 → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑄 ) )
17 16 breq2d ⊢ ( 𝑃 = 𝑄 → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ↔ 𝑋 𝐶 ( 𝑄 ∨ 𝑄 ) ) )
18 17 biimpac ⊢ ( ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 = 𝑄 ) → 𝑋 𝐶 ( 𝑄 ∨ 𝑄 ) )
19 simpr3 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐴 )
20 2 5 hlatjidm ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑄 ) = 𝑄 )
21 19 20 syldan ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑄 ∨ 𝑄 ) = 𝑄 )
22 21 breq2d ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 ( 𝑄 ∨ 𝑄 ) ↔ 𝑋 𝐶 𝑄 ) )
23 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
24 23 adantr ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝐾 ∈ AtLat )
25 simpr1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → 𝑋 ∈ 𝐵 )
26 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
27 1 26 3 4 5 atcvreq0 ⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑋 𝐶 𝑄 ↔ 𝑋 = 0 ) )
28 24 25 19 27 syl3anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 𝑄 ↔ 𝑋 = 0 ) )
29 28 biimpd ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 𝑄 → 𝑋 = 0 ) )
30 22 29 sylbid ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 ( 𝑄 ∨ 𝑄 ) → 𝑋 = 0 ) )
31 18 30 syl5 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 = 𝑄 ) → 𝑋 = 0 ) )
32 31 expd ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) → ( 𝑃 = 𝑄 → 𝑋 = 0 ) ) )
33 32 3impia ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 = 𝑄 → 𝑋 = 0 ) )
34 15 33 impbid ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ 𝑋 𝐶 ( 𝑃 ∨ 𝑄 ) ) → ( 𝑋 = 0 ↔ 𝑃 = 𝑄 ) )