Theorem atcvrj0 36015

Description: Two atoms covering the zero subspace are equal. (atcv1 29938 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 4932	. . . . . . . 8 ⊢ (𝑋 = 0 → (𝑋𝐶(𝑃 ∨ 𝑄) ↔ 0 𝐶(𝑃 ∨ 𝑄)))
2	1	biimpd 221	. . . . . . 7 ⊢ (𝑋 = 0 → (𝑋𝐶(𝑃 ∨ 𝑄) → 0 𝐶(𝑃 ∨ 𝑄)))
3	2	adantl 474	. . . . . 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 36013	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ( 0 𝐶(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
9	8	3adant3r1 1162	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ( 0 𝐶(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
10	9	adantr 473	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋 = 0 ) → ( 0 𝐶(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
11	3, 10	sylibd 231	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋 = 0 ) → (𝑋𝐶(𝑃 ∨ 𝑄) → 𝑃 = 𝑄))
12	11	ex 405	. . . 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 1097	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐶(𝑃 ∨ 𝑄)) → (𝑋 = 0 → 𝑃 = 𝑄))
15		oveq1 6983	. . . . . . 7 ⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
16	15	breq2d 4941	. . . . . 6 ⊢ (𝑃 = 𝑄 → (𝑋𝐶(𝑃 ∨ 𝑄) ↔ 𝑋𝐶(𝑄 ∨ 𝑄)))
17	16	biimpac 471	. . . . 5 ⊢ ((𝑋𝐶(𝑃 ∨ 𝑄) ∧ 𝑃 = 𝑄) → 𝑋𝐶(𝑄 ∨ 𝑄))
18		simpr3 1176	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
19	4, 7	hlatjidm 35956	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
20	18, 19	syldan 582	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄 ∨ 𝑄) = 𝑄)
21	20	breq2d 4941	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶(𝑄 ∨ 𝑄) ↔ 𝑋𝐶𝑄))
22		hlatl 35947	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
23	22	adantr 473	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ AtLat)
24		simpr1 1174	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
25		atcvrj0.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
26		eqid 2778	. . . . . . . . 9 ⊢ (le‘𝐾) = (le‘𝐾)
27	25, 26, 5, 6, 7	atcvreq0 35901	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑋𝐶𝑄 ↔ 𝑋 = 0 ))
28	23, 24, 18, 27	syl3anc 1351	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶𝑄 ↔ 𝑋 = 0 ))
29	28	biimpd 221	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶𝑄 → 𝑋 = 0 ))
30	21, 29	sylbid 232	. . . . 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 408	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶(𝑃 ∨ 𝑄) → (𝑃 = 𝑄 → 𝑋 = 0 )))
33	32	3impia 1097	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐶(𝑃 ∨ 𝑄)) → (𝑃 = 𝑄 → 𝑋 = 0 ))
34	14, 33	impbid 204	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐶(𝑃 ∨ 𝑄)) → (𝑋 = 0 ↔ 𝑃 = 𝑄))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050   class class class wbr 4929  ‘cfv 6188  (class class class)co 6976  Basecbs 16339  lecple 16428  joincjn 17412  0.cp0 17505   ⋖ ccvr 35849  Atomscatm 35850  AtLatcal 35851  HLchlt 35937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-proset 17396  df-poset 17414  df-plt 17426  df-lub 17442  df-glb 17443  df-join 17444  df-meet 17445  df-p0 17507  df-lat 17514  df-clat 17576  df-oposet 35763  df-ol 35765  df-oml 35766  df-covers 35853  df-ats 35854  df-atl 35885  df-cvlat 35909  df-hlat 35938

This theorem is referenced by:  cvrat2  36016  atcvrneN  36017  atcvrj2b  36019