Theorem atcvrj0 36724

Description: Two atoms covering the zero subspace are equal. (atcv1 30163 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 5033	. . . . . . . 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 485	. . . . . 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 36722	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ( 0 𝐶(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
9	8	3adant3r1 1179	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ( 0 𝐶(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
10	9	adantr 484	. . . . . 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 416	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 = 0 → (𝑋𝐶(𝑃 ∨ 𝑄) → 𝑃 = 𝑄)))
13	12	com23 86	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶(𝑃 ∨ 𝑄) → (𝑋 = 0 → 𝑃 = 𝑄)))
14	13	3impia 1114	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐶(𝑃 ∨ 𝑄)) → (𝑋 = 0 → 𝑃 = 𝑄))
15		oveq1 7142	. . . . . . 7 ⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
16	15	breq2d 5042	. . . . . 6 ⊢ (𝑃 = 𝑄 → (𝑋𝐶(𝑃 ∨ 𝑄) ↔ 𝑋𝐶(𝑄 ∨ 𝑄)))
17	16	biimpac 482	. . . . 5 ⊢ ((𝑋𝐶(𝑃 ∨ 𝑄) ∧ 𝑃 = 𝑄) → 𝑋𝐶(𝑄 ∨ 𝑄))
18		simpr3 1193	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
19	4, 7	hlatjidm 36665	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
20	18, 19	syldan 594	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄 ∨ 𝑄) = 𝑄)
21	20	breq2d 5042	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶(𝑄 ∨ 𝑄) ↔ 𝑋𝐶𝑄))
22		hlatl 36656	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
23	22	adantr 484	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ AtLat)
24		simpr1 1191	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
25		atcvrj0.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
26		eqid 2798	. . . . . . . . 9 ⊢ (le‘𝐾) = (le‘𝐾)
27	25, 26, 5, 6, 7	atcvreq0 36610	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑋𝐶𝑄 ↔ 𝑋 = 0 ))
28	23, 24, 18, 27	syl3anc 1368	. . . . . . 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 34	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋𝐶(𝑃 ∨ 𝑄) ∧ 𝑃 = 𝑄) → 𝑋 = 0 ))
32	31	expd 419	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶(𝑃 ∨ 𝑄) → (𝑃 = 𝑄 → 𝑋 = 0 )))
33	32	3impia 1114	. 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  joincjn 17546  0.cp0 17639   ⋖ ccvr 36558  Atomscatm 36559  AtLatcal 36560  HLchlt 36646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36472  df-ol 36474  df-oml 36475  df-covers 36562  df-ats 36563  df-atl 36594  df-cvlat 36618  df-hlat 36647

This theorem is referenced by:  cvrat2  36725  atcvrneN  36726  atcvrj2b  36728