Theorem 4atexlemex2 40947

Description: Lemma for 4atexlem7 40951. 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 40945	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
11	10	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ∈ 𝐴)
12	1, 2, 3, 4, 5, 6, 7, 8, 9	4atexlemnclw 40946	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝑊)
13	12	adantr 486	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ¬ 𝐶 ≤ 𝑊)
14	1, 2, 3, 4, 5, 6, 7, 8	4atexlemntlpq 40944	. . . . 5 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
15		id 23	. . . . . . . . . . 11 ⊢ (𝐶 = 𝑃 → 𝐶 = 𝑃)
16	9, 15	eqtr3id 2809	. . . . . . . . . 10 ⊢ (𝐶 = 𝑃 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = 𝑃)
17	16	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = 𝑃)
18	1	4atexlemkl 40933	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Lat)
19	1, 3, 5	4atexlemqtb 40937	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
20	1, 3, 5	4atexlempsb 40936	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
21		eqid 2760	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝐾)
22	21, 2, 4	latmle1 18555	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
23	18, 19, 20, 22	syl3anc 1398	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
24	1	4atexlemk 40923	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ HL)
25	1	4atexlemq 40927	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
26	1	4atexlemt 40929	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
27	3, 5	hlatjcom 40244	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
28	24, 25, 26, 27	syl3anc 1398	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
29	23, 28	breqtrd 5131	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑇 ∨ 𝑄))
30	29	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑇 ∨ 𝑄))
31	17, 30	eqbrtrrd 5129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → 𝑃 ≤ (𝑇 ∨ 𝑄))
32	1	4atexlemkc 40934	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ CvLat)
33	1	4atexlemp 40926	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
34	1	4atexlempnq 40931	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
35	2, 3, 5	cvlatexch2 40213	. . . . . . . . . 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 1410	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ≤ (𝑇 ∨ 𝑄) → 𝑇 ≤ (𝑃 ∨ 𝑄)))
37	36	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → (𝑃 ≤ (𝑇 ∨ 𝑄) → 𝑇 ≤ (𝑃 ∨ 𝑄)))
38	31, 37	mpd 16	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → 𝑇 ≤ (𝑃 ∨ 𝑄))
39	38	ex 418	. . . . . 6 ⊢ (𝜑 → (𝐶 = 𝑃 → 𝑇 ≤ (𝑃 ∨ 𝑄)))
40	39	necon3bd 2969	. . . . 5 ⊢ (𝜑 → (¬ 𝑇 ≤ (𝑃 ∨ 𝑄) → 𝐶 ≠ 𝑃))
41	14, 40	mpd 16	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝑃)
42	41	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ≠ 𝑃)
43		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ≠ 𝑆)
44	21, 2, 4	latmle2 18556	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
45	18, 19, 20, 44	syl3anc 1398	. . . . 5 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
46	9, 45	eqbrtrid 5140	. . . 4 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
47	46	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ≤ (𝑃 ∨ 𝑆))
48	1	4atexlems 40928	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
49	1, 2, 3, 5	4atexlempns 40938	. . . . 5 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
50	5, 2, 3	cvlsupr2 40219	. . . . 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 1410	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶) ↔ (𝐶 ≠ 𝑃 ∧ 𝐶 ≠ 𝑆 ∧ 𝐶 ≤ (𝑃 ∨ 𝑆))))
52	51	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ((𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶) ↔ (𝐶 ≠ 𝑃 ∧ 𝐶 ≠ 𝑆 ∧ 𝐶 ≤ (𝑃 ∨ 𝑆))))
53	42, 43, 47, 52	mpbir3and 1361	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))
54		breq1 5106	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧 ≤ 𝑊 ↔ 𝐶 ≤ 𝑊))
55	54	notbid 321	. . . 4 ⊢ (𝑧 = 𝐶 → (¬ 𝑧 ≤ 𝑊 ↔ ¬ 𝐶 ≤ 𝑊))
56		oveq2 7422	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑃 ∨ 𝑧) = (𝑃 ∨ 𝐶))
57		oveq2 7422	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝐶))
58	56, 57	eqeq12d 2776	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶)))
59	55, 58	anbi12d 644	. . 3 ⊢ (𝑧 = 𝐶 → ((¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (¬ 𝐶 ≤ 𝑊 ∧ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))))
60	59	rspcev 3576	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ (¬ 𝐶 ≤ 𝑊 ∧ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
61	11, 13, 53, 60	syl12anc 850	1 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086   class class class wbr 5103  ‘cfv 6533  (class class class)co 7414  Basecbs 17304  lecple 17352  joincjn 18402  meetcmee 18403  Latclat 18522  Atomscatm 40139  CvLatclc 40141  HLchlt 40226  LHypclh 40860

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-proset 18385  df-poset 18404  df-plt 18419  df-lub 18435  df-glb 18436  df-join 18437  df-meet 18438  df-p0 18514  df-p1 18515  df-lat 18523  df-clat 18590  df-oposet 40052  df-ol 40054  df-oml 40055  df-covers 40142  df-ats 40143  df-atl 40174  df-cvlat 40198  df-hlat 40227  df-llines 40374  df-lplanes 40375  df-lhyp 40864

This theorem is used by:  4atexlemex4  40949  4atexlemex6  40950