Theorem 4atexlemex2 40517

Description: Lemma for 4atexlem7 40521. 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 40515	. . 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 40516	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝑊)
13	12	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ¬ 𝐶 ≤ 𝑊)
14	1, 2, 3, 4, 5, 6, 7, 8	4atexlemntlpq 40514	. . . . 5 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
15		id 22	. . . . . . . . . . 11 ⊢ (𝐶 = 𝑃 → 𝐶 = 𝑃)
16	9, 15	eqtr3id 2785	. . . . . . . . . 10 ⊢ (𝐶 = 𝑃 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = 𝑃)
17	16	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = 𝑃)
18	1	4atexlemkl 40503	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Lat)
19	1, 3, 5	4atexlemqtb 40507	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
20	1, 3, 5	4atexlempsb 40506	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
21		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘𝐾) = (Base‘𝐾)
22	21, 2, 4	latmle1 18430	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
23	18, 19, 20, 22	syl3anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑄 ∨ 𝑇))
24	1	4atexlemk 40493	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ HL)
25	1	4atexlemq 40497	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
26	1	4atexlemt 40499	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
27	3, 5	hlatjcom 39814	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
28	24, 25, 26, 27	syl3anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑄))
29	23, 28	breqtrd 5111	. . . . . . . . . 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 5109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶 = 𝑃) → 𝑃 ≤ (𝑇 ∨ 𝑄))
32	1	4atexlemkc 40504	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ CvLat)
33	1	4atexlemp 40496	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
34	1	4atexlempnq 40501	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
35	2, 3, 5	cvlatexch2 39783	. . . . . . . . . 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 1386	. . . . . . . . 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 2946	. . . . 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 18431	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
45	18, 19, 20, 44	syl3anc 1374	. . . . 5 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) ≤ (𝑃 ∨ 𝑆))
46	9, 45	eqbrtrid 5120	. . . 4 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
47	46	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → 𝐶 ≤ (𝑃 ∨ 𝑆))
48	1	4atexlems 40498	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
49	1, 2, 3, 5	4atexlempns 40508	. . . . 5 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
50	5, 2, 3	cvlsupr2 39789	. . . . 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 1386	. . . 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 1344	. 2 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))
54		breq1 5088	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧 ≤ 𝑊 ↔ 𝐶 ≤ 𝑊))
55	54	notbid 318	. . . 4 ⊢ (𝑧 = 𝐶 → (¬ 𝑧 ≤ 𝑊 ↔ ¬ 𝐶 ≤ 𝑊))
56		oveq2 7375	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑃 ∨ 𝑧) = (𝑃 ∨ 𝐶))
57		oveq2 7375	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑆 ∨ 𝑧) = (𝑆 ∨ 𝐶))
58	56, 57	eqeq12d 2752	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧) ↔ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶)))
59	55, 58	anbi12d 633	. . 3 ⊢ (𝑧 = 𝐶 → ((¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)) ↔ (¬ 𝐶 ≤ 𝑊 ∧ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))))
60	59	rspcev 3564	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ (¬ 𝐶 ≤ 𝑊 ∧ (𝑃 ∨ 𝐶) = (𝑆 ∨ 𝐶))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
61	11, 13, 53, 60	syl12anc 837	1 ⊢ ((𝜑 ∧ 𝐶 ≠ 𝑆) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  lecple 17227  joincjn 18277  meetcmee 18278  Latclat 18397  Atomscatm 39709  CvLatclc 39711  HLchlt 39796  LHypclh 40430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-proset 18260  df-poset 18279  df-plt 18294  df-lub 18310  df-glb 18311  df-join 18312  df-meet 18313  df-p0 18389  df-p1 18390  df-lat 18398  df-clat 18465  df-oposet 39622  df-ol 39624  df-oml 39625  df-covers 39712  df-ats 39713  df-atl 39744  df-cvlat 39768  df-hlat 39797  df-llines 39944  df-lplanes 39945  df-lhyp 40434

This theorem is referenced by:  4atexlemex4  40519  4atexlemex6  40520