Theorem 4atexlemex2 35357

Description: Lemma for 4atexlem7 35361. Show that when 𝐶 ≠ 𝑆, 𝐶 satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012.)

Hypotheses

Ref	Expression
4thatlem.ph	⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))))
4thatlem0.l	⊢ ≤ = (le‘𝐾)
4thatlem0.j	⊢ ∨ = (join‘𝐾)
4thatlem0.m	⊢ ∧ = (meet‘𝐾)
4thatlem0.a	⊢ 𝐴 = (Atoms‘𝐾)
4thatlem0.h	⊢ 𝐻 = (LHyp‘𝐾)
4thatlem0.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
4thatlem0.v	⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
4thatlem0.c	⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))

Assertion

Ref	Expression
4atexlemex2	⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))

Distinct variable groups:   𝑧,𝐴   𝑧,𝐶   𝑧, ∨   𝑧, ≤   𝑧,𝑃   𝑧,𝑆   𝑧,𝑊

Allowed substitution hints:   𝜑(𝑧)   𝑄(𝑧)   𝑅(𝑧)   𝑇(𝑧)   𝑈(𝑧)   𝐻(𝑧)   𝐾(𝑧)   ∧ (𝑧)   𝑉(𝑧)

Proof of Theorem 4atexlemex2

Step	Hyp	Ref	Expression
1		4thatlem.ph	. . . 4 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))))
2		4thatlem0.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		4thatlem0.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		4thatlem0.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
5		4thatlem0.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6		4thatlem0.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
7		4thatlem0.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8		4thatlem0.v	. . . 4 ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
9		4thatlem0.c	. . . 4 ⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	4atexlemc 35355	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
11	10	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ∈ 𝐴)
12	1, 2, 3, 4, 5, 6, 7, 8, 9	4atexlemnclw 35356	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝑊)
13	12	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ¬ 𝐶 ≤ 𝑊)
14	1, 2, 3, 4, 5, 6, 7, 8	4atexlemntlpq 35354	. . . . 5 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
15		id 22	. . . . . . . . . . 11 ⊢ (𝐶 = 𝑃 → 𝐶 = 𝑃)
16	9, 15	syl5eqr 2670	. . . . . . . . . 10 ⊢ (𝐶 = 𝑃 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = 𝑃)
17	16	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = 𝑃)
18	1	4atexlemkl 35343	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Lat)
19	1, 3, 5	4atexlemqtb 35347	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
20	1, 3, 5	4atexlempsb 35346	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
21		eqid 2622	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝐾)
22	21, 2, 4	latmle1 17076	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
23	18, 19, 20, 22	syl3anc 1326	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
24	1	4atexlemk 35333	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ HL)
25	1	4atexlemq 35337	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
26	1	4atexlemt 35339	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
27	3, 5	hlatjcom 34654	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
28	24, 25, 26, 27	syl3anc 1326	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
29	23, 28	breqtrd 4679	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑇 ∨ 𝑄))
30	29	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑇 ∨ 𝑄))
31	17, 30	eqbrtrrd 4677	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → 𝑃 ≤ (𝑇 ∨ 𝑄))
32	1	4atexlemkc 35344	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ CvLat)
33	1	4atexlemp 35336	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
34	1	4atexlempnq 35341	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
35	2, 3, 5	cvlatexch2 34624	. . . . . . . . . 10 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ≤ (𝑇 ∨ 𝑄) → 𝑇 ≤ (𝑃 ∨ 𝑄)))
36	32, 33, 26, 25, 34, 35	syl131anc 1339	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ≤ (𝑇 ∨ 𝑄) → 𝑇 ≤ (𝑃 ∨ 𝑄)))
37	36	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → (𝑃 ≤ (𝑇 ∨ 𝑄) → 𝑇 ≤ (𝑃 ∨ 𝑄)))
38	31, 37	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → 𝑇 ≤ (𝑃 ∨ 𝑄))
39	38	ex 450	. . . . . 6 ⊢ (𝜑 → (𝐶 = 𝑃 → 𝑇 ≤ (𝑃 ∨ 𝑄)))
40	39	necon3bd 2808	. . . . 5 ⊢ (𝜑 → (¬ 𝑇 ≤ (𝑃 ∨ 𝑄) → 𝐶 ≠ 𝑃))
41	14, 40	mpd 15	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝑃)
42	41	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ≠ 𝑃)
43		simpr 477	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ≠ 𝑆)
44	21, 2, 4	latmle2 17077	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
45	18, 19, 20, 44	syl3anc 1326	. . . . 5 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
46	9, 45	syl5eqbr 4688	. . . 4 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
47	46	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ≤ (𝑃 ∨ 𝑆))
48	1	4atexlems 35338	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
49	1, 2, 3, 5	4atexlempns 35348	. . . . 5 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
50	5, 2, 3	cvlsupr2 34630	. . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ 𝑃 ≠ 𝑆) → ((𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶) ↔ (𝐶 ≠ 𝑃 ∧ 𝐶 ≠ 𝑆 ∧ 𝐶 ≤ (𝑃 ∨ 𝑆))))
51	32, 33, 48, 10, 49, 50	syl131anc 1339	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶) ↔ (𝐶 ≠ 𝑃 ∧ 𝐶 ≠ 𝑆 ∧ 𝐶 ≤ (𝑃 ∨ 𝑆))))
52	51	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ((𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶) ↔ (𝐶 ≠ 𝑃 ∧ 𝐶 ≠ 𝑆 ∧ 𝐶 ≤ (𝑃 ∨ 𝑆))))
53	42, 43, 47, 52	mpbir3and 1245	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))
54		breq1 4656	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧 ≤ 𝑊 ↔ 𝐶 ≤ 𝑊))
55	54	notbid 308	. . . 4 ⊢ (𝑧 = 𝐶 → (¬ 𝑧 ≤ 𝑊 ↔ ¬ 𝐶 ≤ 𝑊))
56		oveq2 6658	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑃 ∨ 𝑧) = (𝑃 ∨ 𝐶))
57		oveq2 6658	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝐶))
58	56, 57	eqeq12d 2637	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶)))
59	55, 58	anbi12d 747	. . 3 ⊢ (𝑧 = 𝐶 → ((¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (¬ 𝐶 ≤ 𝑊 ∧ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))))
60	59	rspcev 3309	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ (¬ 𝐶 ≤ 𝑊 ∧ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
61	11, 13, 53, 60	syl12anc 1324	1 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  lecple 15948  joincjn 16944  meetcmee 16945  Latclat 17045  Atomscatm 34550  CvLatclc 34552  HLchlt 34637  LHypclh 35270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784  df-lplanes 34785  df-lhyp 35274

This theorem is referenced by:  4atexlemex4  35359  4atexlemex6  35360