Theorem cdleme8 40252

Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114. 𝐶 represents s₁. In their notation, we prove p ∨ s₁ = p ∨ s. (Contributed by NM, 9-Jun-2012.)

Hypotheses

Ref	Expression
cdleme8.l	⊢ ≤ = (le‘𝐾)
cdleme8.j	⊢ ∨ = (join‘𝐾)
cdleme8.m	⊢ ∧ = (meet‘𝐾)
cdleme8.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme8.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme8.4	⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊)

Assertion

Ref	Expression
cdleme8	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝐶) = (𝑃 ∨ 𝑆))

Proof of Theorem cdleme8

Step	Hyp	Ref	Expression
1		cdleme8.4	. . 3 ⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
2	1	oveq2i 7442	. 2 ⊢ (𝑃 ∨ 𝐶) = (𝑃 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊))
3		simp1l 1198	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → 𝐾 ∈ HL)
4		simp2l 1200	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → 𝑃 ∈ 𝐴)
5	3	hllatd 39365	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → 𝐾 ∈ Lat)
6		eqid 2737	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		cdleme8.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
8	6, 7	atbase 39290	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
9	4, 8	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → 𝑃 ∈ (Base‘𝐾))
10	6, 7	atbase 39290	. . . . . 6 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
11	10	3ad2ant3 1136	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → 𝑆 ∈ (Base‘𝐾))
12		cdleme8.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
13	6, 12	latjcl 18484	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
14	5, 9, 11, 13	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
15		simp1r 1199	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → 𝑊 ∈ 𝐻)
16		cdleme8.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
17	6, 16	lhpbase 40000	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
18	15, 17	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → 𝑊 ∈ (Base‘𝐾))
19		cdleme8.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
20	6, 19, 12	latlej1 18493	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → 𝑃 ≤ (𝑃 ∨ 𝑆))
21	5, 9, 11, 20	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑆))
22		cdleme8.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
23	6, 19, 12, 22, 7	atmod3i1 39866	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 ≤ (𝑃 ∨ 𝑆)) → (𝑃 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) = ((𝑃 ∨ 𝑆) ∧ (𝑃 ∨ 𝑊)))
24	3, 4, 14, 18, 21, 23	syl131anc 1385	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) = ((𝑃 ∨ 𝑆) ∧ (𝑃 ∨ 𝑊)))
25		eqid 2737	. . . . . 6 ⊢ (1.‘𝐾) = (1.‘𝐾)
26	19, 12, 25, 7, 16	lhpjat2 40023	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
27	26	3adant3 1133	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
28	27	oveq2d 7447	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → ((𝑃 ∨ 𝑆) ∧ (𝑃 ∨ 𝑊)) = ((𝑃 ∨ 𝑆) ∧ (1.‘𝐾)))
29		hlol 39362	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
30	3, 29	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → 𝐾 ∈ OL)
31	6, 22, 25	olm11 39228	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ (1.‘𝐾)) = (𝑃 ∨ 𝑆))
32	30, 14, 31	syl2anc 584	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → ((𝑃 ∨ 𝑆) ∧ (1.‘𝐾)) = (𝑃 ∨ 𝑆))
33	24, 28, 32	3eqtrd 2781	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)) = (𝑃 ∨ 𝑆))
34	2, 33	eqtrid 2789	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝐶) = (𝑃 ∨ 𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ 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  meetcmee 18358  1.cp1 18469  Latclat 18476  OLcol 39175  Atomscatm 39264  HLchlt 39351  LHypclh 39986

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-iin 4994  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-mpo 7436  df-1st 8014  df-2nd 8015  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-p1 18471  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  df-psubsp 39505  df-pmap 39506  df-padd 39798  df-lhyp 39990

This theorem is referenced by:  cdleme8tN  40257  cdleme15a  40276  cdleme17b  40289  cdlemg3a  40599