Theorem 4atexlemex2 40696

Description: Lemma for 4atexlem7 40700. 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 40694	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
11	10	adantr 484	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ∈ 𝐴)
12	1, 2, 3, 4, 5, 6, 7, 8, 9	4atexlemnclw 40695	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝑊)
13	12	adantr 484	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ¬ 𝐶 ≤ 𝑊)
14	1, 2, 3, 4, 5, 6, 7, 8	4atexlemntlpq 40693	. . . . 5 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
15		id 22	. . . . . . . . . . 11 ⊢ (𝐶 = 𝑃 → 𝐶 = 𝑃)
16	9, 15	eqtr3id 2812	. . . . . . . . . 10 ⊢ (𝐶 = 𝑃 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = 𝑃)
17	16	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = 𝑃)
18	1	4atexlemkl 40682	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Lat)
19	1, 3, 5	4atexlemqtb 40686	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
20	1, 3, 5	4atexlempsb 40685	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
21		eqid 2763	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝐾)
22	21, 2, 4	latmle1 18497	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
23	18, 19, 20, 22	syl3anc 1391	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
24	1	4atexlemk 40672	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ HL)
25	1	4atexlemq 40676	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
26	1	4atexlemt 40678	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
27	3, 5	hlatjcom 39993	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
28	24, 25, 26, 27	syl3anc 1391	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
29	23, 28	breqtrd 5127	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑇 ∨ 𝑄))
30	29	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑇 ∨ 𝑄))
31	17, 30	eqbrtrrd 5125	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → 𝑃 ≤ (𝑇 ∨ 𝑄))
32	1	4atexlemkc 40683	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ CvLat)
33	1	4atexlemp 40675	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
34	1	4atexlempnq 40680	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
35	2, 3, 5	cvlatexch2 39962	. . . . . . . . . 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 1403	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ≤ (𝑇 ∨ 𝑄) → 𝑇 ≤ (𝑃 ∨ 𝑄)))
37	36	adantr 484	. . . . . . . 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 416	. . . . . 6 ⊢ (𝜑 → (𝐶 = 𝑃 → 𝑇 ≤ (𝑃 ∨ 𝑄)))
40	39	necon3bd 2972	. . . . 5 ⊢ (𝜑 → (¬ 𝑇 ≤ (𝑃 ∨ 𝑄) → 𝐶 ≠ 𝑃))
41	14, 40	mpd 15	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝑃)
42	41	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ≠ 𝑃)
43		simpr 488	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ≠ 𝑆)
44	21, 2, 4	latmle2 18498	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
45	18, 19, 20, 44	syl3anc 1391	. . . . 5 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
46	9, 45	eqbrtrid 5136	. . . 4 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
47	46	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ≤ (𝑃 ∨ 𝑆))
48	1	4atexlems 40677	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
49	1, 2, 3, 5	4atexlempns 40687	. . . . 5 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
50	5, 2, 3	cvlsupr2 39968	. . . . 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 1403	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶) ↔ (𝐶 ≠ 𝑃 ∧ 𝐶 ≠ 𝑆 ∧ 𝐶 ≤ (𝑃 ∨ 𝑆))))
52	51	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ((𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶) ↔ (𝐶 ≠ 𝑃 ∧ 𝐶 ≠ 𝑆 ∧ 𝐶 ≤ (𝑃 ∨ 𝑆))))
53	42, 43, 47, 52	mpbir3and 1357	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))
54		breq1 5104	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧 ≤ 𝑊 ↔ 𝐶 ≤ 𝑊))
55	54	notbid 320	. . . 4 ⊢ (𝑧 = 𝐶 → (¬ 𝑧 ≤ 𝑊 ↔ ¬ 𝐶 ≤ 𝑊))
56		oveq2 7405	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑃 ∨ 𝑧) = (𝑃 ∨ 𝐶))
57		oveq2 7405	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝐶))
58	56, 57	eqeq12d 2779	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶)))
59	55, 58	anbi12d 641	. . 3 ⊢ (𝑧 = 𝐶 → ((¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (¬ 𝐶 ≤ 𝑊 ∧ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))))
60	59	rspcev 3582	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ (¬ 𝐶 ≤ 𝑊 ∧ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
61	11, 13, 53, 60	syl12anc 847	1 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3087   class class class wbr 5101  ‘cfv 6522  (class class class)co 7397  Basecbs 17246  lecple 17294  joincjn 18344  meetcmee 18345  Latclat 18464  Atomscatm 39888  CvLatclc 39890  HLchlt 39975  LHypclh 40609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-proset 18327  df-poset 18346  df-plt 18361  df-lub 18377  df-glb 18378  df-join 18379  df-meet 18380  df-p0 18456  df-p1 18457  df-lat 18465  df-clat 18532  df-oposet 39801  df-ol 39803  df-oml 39804  df-covers 39891  df-ats 39892  df-atl 39923  df-cvlat 39947  df-hlat 39976  df-llines 40123  df-lplanes 40124  df-lhyp 40613

This theorem is referenced by:  4atexlemex4  40698  4atexlemex6  40699