Theorem atcvrj0 40235

Description: Two atoms covering the zero subspace are equal. (atcv1 32779 analog.) (Contributed by NM, 29-Nov-2011.)

Hypotheses

Ref	Expression
atcvrj0.b	⊢ 𝐵 = (Base‘𝐾)
atcvrj0.j	⊢ ∨ = (join‘𝐾)
atcvrj0.z	⊢ 0 = (0.‘𝐾)
atcvrj0.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
atcvrj0.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
atcvrj0	⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐶(𝑃 ∨ 𝑄)) → (𝑋 = 0 ↔ 𝑃 = 𝑄))

Proof of Theorem atcvrj0

Step	Hyp	Ref	Expression
1		breq1 5114	. . . . . . . 8 ⊢ (𝑋 = 0 → (𝑋𝐶(𝑃 ∨ 𝑄) ↔ 0 𝐶(𝑃 ∨ 𝑄)))
2	1	biimpd 232	. . . . . . 7 ⊢ (𝑋 = 0 → (𝑋𝐶(𝑃 ∨ 𝑄) → 0 𝐶(𝑃 ∨ 𝑄)))
3	2	adantl 487	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋 = 0 ) → (𝑋𝐶(𝑃 ∨ 𝑄) → 0 𝐶(𝑃 ∨ 𝑄)))
4		atcvrj0.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
5		atcvrj0.z	. . . . . . . . 9 ⊢ 0 = (0.‘𝐾)
6		atcvrj0.c	. . . . . . . . 9 ⊢ 𝐶 = ( ⋖ ‘𝐾)
7		atcvrj0.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
8	4, 5, 6, 7	atcvr0eq 40233	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ( 0 𝐶(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
9	8	3adant3r1 1201	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ( 0 𝐶(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
10	9	adantr 486	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋 = 0 ) → ( 0 𝐶(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
11	3, 10	sylibd 242	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋 = 0 ) → (𝑋𝐶(𝑃 ∨ 𝑄) → 𝑃 = 𝑄))
12	11	ex 418	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 = 0 → (𝑋𝐶(𝑃 ∨ 𝑄) → 𝑃 = 𝑄)))
13	12	com23 87	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶(𝑃 ∨ 𝑄) → (𝑋 = 0 → 𝑃 = 𝑄)))
14	13	3impia 1135	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐶(𝑃 ∨ 𝑄)) → (𝑋 = 0 → 𝑃 = 𝑄))
15		oveq1 7423	. . . . . . 7 ⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
16	15	breq2d 5123	. . . . . 6 ⊢ (𝑃 = 𝑄 → (𝑋𝐶(𝑃 ∨ 𝑄) ↔ 𝑋𝐶(𝑄 ∨ 𝑄)))
17	16	biimpac 484	. . . . 5 ⊢ ((𝑋𝐶(𝑃 ∨ 𝑄) ∧ 𝑃 = 𝑄) → 𝑋𝐶(𝑄 ∨ 𝑄))
18		simpr3 1215	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
19	4, 7	hlatjidm 40176	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
20	18, 19	syldan 603	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄 ∨ 𝑄) = 𝑄)
21	20	breq2d 5123	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶(𝑄 ∨ 𝑄) ↔ 𝑋𝐶𝑄))
22		hlatl 40167	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
23	22	adantr 486	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ AtLat)
24		simpr1 1213	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
25		atcvrj0.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
26		eqid 2765	. . . . . . . . 9 ⊢ (le‘𝐾) = (le‘𝐾)
27	25, 26, 5, 6, 7	atcvreq0 40121	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑋𝐶𝑄 ↔ 𝑋 = 0 ))
28	23, 24, 18, 27	syl3anc 1398	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶𝑄 ↔ 𝑋 = 0 ))
29	28	biimpd 232	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶𝑄 → 𝑋 = 0 ))
30	21, 29	sylbid 243	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶(𝑄 ∨ 𝑄) → 𝑋 = 0 ))
31	17, 30	syl5 35	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋𝐶(𝑃 ∨ 𝑄) ∧ 𝑃 = 𝑄) → 𝑋 = 0 ))
32	31	expd 421	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶(𝑃 ∨ 𝑄) → (𝑃 = 𝑄 → 𝑋 = 0 )))
33	32	3impia 1135	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐶(𝑃 ∨ 𝑄)) → (𝑃 = 𝑄 → 𝑋 = 0 ))
34	14, 33	impbid 215	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐶(𝑃 ∨ 𝑄)) → (𝑋 = 0 ↔ 𝑃 = 𝑄))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   class class class wbr 5111  ‘cfv 6540  (class class class)co 7416  Basecbs 17287  lecple 17335  joincjn 18385  0.cp0 18495   ⋖ ccvr 40069  Atomscatm 40070  AtLatcal 40071  HLchlt 40157

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-proset 18368  df-poset 18387  df-plt 18402  df-lub 18418  df-glb 18419  df-join 18420  df-meet 18421  df-p0 18497  df-lat 18506  df-clat 18573  df-oposet 39983  df-ol 39985  df-oml 39986  df-covers 40073  df-ats 40074  df-atl 40105  df-cvlat 40129  df-hlat 40158

This theorem is used by:  cvrat2  40236  atcvrneN  40237  atcvrj2b  40239