Theorem atcvrj0 39430

Description: Two atoms covering the zero subspace are equal. (atcv1 32399 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 5146	. . . . . . . 8 ⊢ (𝑋 = 0 → (𝑋𝐶(𝑃 ∨ 𝑄) ↔ 0 𝐶(𝑃 ∨ 𝑄)))
2	1	biimpd 229	. . . . . . 7 ⊢ (𝑋 = 0 → (𝑋𝐶(𝑃 ∨ 𝑄) → 0 𝐶(𝑃 ∨ 𝑄)))
3	2	adantl 481	. . . . . 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 39428	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ( 0 𝐶(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
9	8	3adant3r1 1183	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ( 0 𝐶(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
10	9	adantr 480	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋 = 0 ) → ( 0 𝐶(𝑃 ∨ 𝑄) ↔ 𝑃 = 𝑄))
11	3, 10	sylibd 239	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋 = 0 ) → (𝑋𝐶(𝑃 ∨ 𝑄) → 𝑃 = 𝑄))
12	11	ex 412	. . . 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 1118	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐶(𝑃 ∨ 𝑄)) → (𝑋 = 0 → 𝑃 = 𝑄))
15		oveq1 7438	. . . . . . 7 ⊢ (𝑃 = 𝑄 → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑄))
16	15	breq2d 5155	. . . . . 6 ⊢ (𝑃 = 𝑄 → (𝑋𝐶(𝑃 ∨ 𝑄) ↔ 𝑋𝐶(𝑄 ∨ 𝑄)))
17	16	biimpac 478	. . . . 5 ⊢ ((𝑋𝐶(𝑃 ∨ 𝑄) ∧ 𝑃 = 𝑄) → 𝑋𝐶(𝑄 ∨ 𝑄))
18		simpr3 1197	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
19	4, 7	hlatjidm 39370	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑄) = 𝑄)
20	18, 19	syldan 591	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑄 ∨ 𝑄) = 𝑄)
21	20	breq2d 5155	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶(𝑄 ∨ 𝑄) ↔ 𝑋𝐶𝑄))
22		hlatl 39361	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
23	22	adantr 480	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ AtLat)
24		simpr1 1195	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
25		atcvrj0.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
26		eqid 2737	. . . . . . . . 9 ⊢ (le‘𝐾) = (le‘𝐾)
27	25, 26, 5, 6, 7	atcvreq0 39315	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑋𝐶𝑄 ↔ 𝑋 = 0 ))
28	23, 24, 18, 27	syl3anc 1373	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶𝑄 ↔ 𝑋 = 0 ))
29	28	biimpd 229	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶𝑄 → 𝑋 = 0 ))
30	21, 29	sylbid 240	. . . . 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 415	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋𝐶(𝑃 ∨ 𝑄) → (𝑃 = 𝑄 → 𝑋 = 0 )))
33	32	3impia 1118	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐶(𝑃 ∨ 𝑄)) → (𝑃 = 𝑄 → 𝑋 = 0 ))
34	14, 33	impbid 212	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋𝐶(𝑃 ∨ 𝑄)) → (𝑋 = 0 ↔ 𝑃 = 𝑄))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  0.cp0 18468   ⋖ ccvr 39263  Atomscatm 39264  AtLatcal 39265  HLchlt 39351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352

This theorem is referenced by:  cvrat2  39431  atcvrneN  39432  atcvrj2b  39434