Theorem 4atexlemex2 40180

Description: Lemma for 4atexlem7 40184. 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 40178	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
11	10	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ∈ 𝐴)
12	1, 2, 3, 4, 5, 6, 7, 8, 9	4atexlemnclw 40179	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝑊)
13	12	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ¬ 𝐶 ≤ 𝑊)
14	1, 2, 3, 4, 5, 6, 7, 8	4atexlemntlpq 40177	. . . . 5 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
15		id 22	. . . . . . . . . . 11 ⊢ (𝐶 = 𝑃 → 𝐶 = 𝑃)
16	9, 15	eqtr3id 2780	. . . . . . . . . 10 ⊢ (𝐶 = 𝑃 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = 𝑃)
17	16	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = 𝑃)
18	1	4atexlemkl 40166	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Lat)
19	1, 3, 5	4atexlemqtb 40170	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
20	1, 3, 5	4atexlempsb 40169	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
21		eqid 2731	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝐾)
22	21, 2, 4	latmle1 18370	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
23	18, 19, 20, 22	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
24	1	4atexlemk 40156	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ HL)
25	1	4atexlemq 40160	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
26	1	4atexlemt 40162	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
27	3, 5	hlatjcom 39477	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
28	24, 25, 26, 27	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
29	23, 28	breqtrd 5115	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑇 ∨ 𝑄))
30	29	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑇 ∨ 𝑄))
31	17, 30	eqbrtrrd 5113	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → 𝑃 ≤ (𝑇 ∨ 𝑄))
32	1	4atexlemkc 40167	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ CvLat)
33	1	4atexlemp 40159	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
34	1	4atexlempnq 40164	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
35	2, 3, 5	cvlatexch2 39446	. . . . . . . . . 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 1385	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ≤ (𝑇 ∨ 𝑄) → 𝑇 ≤ (𝑃 ∨ 𝑄)))
37	36	adantr 480	. . . . . . . 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 412	. . . . . 6 ⊢ (𝜑 → (𝐶 = 𝑃 → 𝑇 ≤ (𝑃 ∨ 𝑄)))
40	39	necon3bd 2942	. . . . 5 ⊢ (𝜑 → (¬ 𝑇 ≤ (𝑃 ∨ 𝑄) → 𝐶 ≠ 𝑃))
41	14, 40	mpd 15	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝑃)
42	41	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ≠ 𝑃)
43		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ≠ 𝑆)
44	21, 2, 4	latmle2 18371	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
45	18, 19, 20, 44	syl3anc 1373	. . . . 5 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
46	9, 45	eqbrtrid 5124	. . . 4 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
47	46	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ≤ (𝑃 ∨ 𝑆))
48	1	4atexlems 40161	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
49	1, 2, 3, 5	4atexlempns 40171	. . . . 5 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
50	5, 2, 3	cvlsupr2 39452	. . . . 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 1385	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶) ↔ (𝐶 ≠ 𝑃 ∧ 𝐶 ≠ 𝑆 ∧ 𝐶 ≤ (𝑃 ∨ 𝑆))))
52	51	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ((𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶) ↔ (𝐶 ≠ 𝑃 ∧ 𝐶 ≠ 𝑆 ∧ 𝐶 ≤ (𝑃 ∨ 𝑆))))
53	42, 43, 47, 52	mpbir3and 1343	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))
54		breq1 5092	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧 ≤ 𝑊 ↔ 𝐶 ≤ 𝑊))
55	54	notbid 318	. . . 4 ⊢ (𝑧 = 𝐶 → (¬ 𝑧 ≤ 𝑊 ↔ ¬ 𝐶 ≤ 𝑊))
56		oveq2 7354	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑃 ∨ 𝑧) = (𝑃 ∨ 𝐶))
57		oveq2 7354	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝐶))
58	56, 57	eqeq12d 2747	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶)))
59	55, 58	anbi12d 632	. . 3 ⊢ (𝑧 = 𝐶 → ((¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (¬ 𝐶 ≤ 𝑊 ∧ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))))
60	59	rspcev 3572	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ (¬ 𝐶 ≤ 𝑊 ∧ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
61	11, 13, 53, 60	syl12anc 836	1 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056   class class class wbr 5089  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  lecple 17168  joincjn 18217  meetcmee 18218  Latclat 18337  Atomscatm 39372  CvLatclc 39374  HLchlt 39459  LHypclh 40093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-proset 18200  df-poset 18219  df-plt 18234  df-lub 18250  df-glb 18251  df-join 18252  df-meet 18253  df-p0 18329  df-p1 18330  df-lat 18338  df-clat 18405  df-oposet 39285  df-ol 39287  df-oml 39288  df-covers 39375  df-ats 39376  df-atl 39407  df-cvlat 39431  df-hlat 39460  df-llines 39607  df-lplanes 39608  df-lhyp 40097

This theorem is referenced by:  4atexlemex4  40182  4atexlemex6  40183