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 ) )

Metamath Proof Explorer

Theorem atcvrj0

Proof