Theorem cdleme0ex2N 38165

Description: Part of proof of Lemma E in [Crawley] p. 113. Note that (𝑃 ∨ 𝑢) = (𝑄 ∨ 𝑢) is a shorter way to express 𝑢 ≠ 𝑃 ∧ 𝑢 ≠ 𝑄 ∧ 𝑢 ≤ (𝑃 ∨ 𝑄). (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme0.l	⊢ ≤ = (le‘𝐾)
cdleme0.j	⊢ ∨ = (join‘𝐾)
cdleme0.m	⊢ ∧ = (meet‘𝐾)
cdleme0.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme0.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme0.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)

Assertion

Ref	Expression
cdleme0ex2N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ∃𝑢 ∈ 𝐴 ((𝑃 ∨ 𝑢) = (𝑄 ∨ 𝑢) ∧ 𝑢 ≤ 𝑊))

Distinct variable groups:   𝑢,𝐴   𝑢, ∨   𝑢, ≤   𝑢,𝑃   𝑢,𝑄   𝑢,𝑈   𝑢,𝑊   𝑢,𝐻   𝑢,𝐾

Allowed substitution hint:   ∧ (𝑢)

Proof of Theorem cdleme0ex2N

Step	Hyp	Ref	Expression
1		simp1 1134	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp2l 1197	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
3		simp2rl 1240	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ 𝐴)
4		simp3 1136	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄)
5		cdleme0.l	. . . 4 ⊢ ≤ = (le‘𝐾)
6		cdleme0.j	. . . 4 ⊢ ∨ = (join‘𝐾)
7		cdleme0.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
8		cdleme0.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
9		cdleme0.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
10		cdleme0.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
11	5, 6, 7, 8, 9, 10	cdleme0ex1N 38164	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑢 ≤ 𝑊))
12	1, 2, 3, 4, 11	syl121anc 1373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑢 ≤ 𝑊))
13		simp11l 1282	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝐾 ∈ HL)
14		hlcvl 37300	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)
15	13, 14	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝐾 ∈ CvLat)
16		simp2ll 1238	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ 𝐴)
17	16	3ad2ant1 1131	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝑃 ∈ 𝐴)
18	3	3ad2ant1 1131	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝑄 ∈ 𝐴)
19		simp2 1135	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝑢 ∈ 𝐴)
20		simp13 1203	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝑃 ≠ 𝑄)
21	8, 5, 6	cvlsupr2 37284	. . . . . . 7 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑢) = (𝑄 ∨ 𝑢) ↔ (𝑢 ≠ 𝑃 ∧ 𝑢 ≠ 𝑄 ∧ 𝑢 ≤ (𝑃 ∨ 𝑄))))
22	15, 17, 18, 19, 20, 21	syl131anc 1381	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → ((𝑃 ∨ 𝑢) = (𝑄 ∨ 𝑢) ↔ (𝑢 ≠ 𝑃 ∧ 𝑢 ≠ 𝑄 ∧ 𝑢 ≤ (𝑃 ∨ 𝑄))))
23		df-3an 1087	. . . . . . 7 ⊢ ((𝑢 ≠ 𝑃 ∧ 𝑢 ≠ 𝑄 ∧ 𝑢 ≤ (𝑃 ∨ 𝑄)) ↔ ((𝑢 ≠ 𝑃 ∧ 𝑢 ≠ 𝑄) ∧ 𝑢 ≤ (𝑃 ∨ 𝑄)))
24		simp3 1136	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝑢 ≤ 𝑊)
25		simp2lr 1239	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ¬ 𝑃 ≤ 𝑊)
26	25	3ad2ant1 1131	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → ¬ 𝑃 ≤ 𝑊)
27		nbrne2 5090	. . . . . . . . . 10 ⊢ ((𝑢 ≤ 𝑊 ∧ ¬ 𝑃 ≤ 𝑊) → 𝑢 ≠ 𝑃)
28	24, 26, 27	syl2anc 583	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝑢 ≠ 𝑃)
29		simp2rr 1241	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ¬ 𝑄 ≤ 𝑊)
30	29	3ad2ant1 1131	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → ¬ 𝑄 ≤ 𝑊)
31		nbrne2 5090	. . . . . . . . . 10 ⊢ ((𝑢 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊) → 𝑢 ≠ 𝑄)
32	24, 30, 31	syl2anc 583	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝑢 ≠ 𝑄)
33	28, 32	jca 511	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → (𝑢 ≠ 𝑃 ∧ 𝑢 ≠ 𝑄))
34	33	biantrurd 532	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → (𝑢 ≤ (𝑃 ∨ 𝑄) ↔ ((𝑢 ≠ 𝑃 ∧ 𝑢 ≠ 𝑄) ∧ 𝑢 ≤ (𝑃 ∨ 𝑄))))
35	23, 34	bitr4id 289	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → ((𝑢 ≠ 𝑃 ∧ 𝑢 ≠ 𝑄 ∧ 𝑢 ≤ (𝑃 ∨ 𝑄)) ↔ 𝑢 ≤ (𝑃 ∨ 𝑄)))
36	22, 35	bitrd 278	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → ((𝑃 ∨ 𝑢) = (𝑄 ∨ 𝑢) ↔ 𝑢 ≤ (𝑃 ∨ 𝑄)))
37	36	3expia 1119	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴) → (𝑢 ≤ 𝑊 → ((𝑃 ∨ 𝑢) = (𝑄 ∨ 𝑢) ↔ 𝑢 ≤ (𝑃 ∨ 𝑄))))
38	37	pm5.32rd 577	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴) → (((𝑃 ∨ 𝑢) = (𝑄 ∨ 𝑢) ∧ 𝑢 ≤ 𝑊) ↔ (𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑢 ≤ 𝑊)))
39	38	rexbidva 3224	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (∃𝑢 ∈ 𝐴 ((𝑃 ∨ 𝑢) = (𝑄 ∨ 𝑢) ∧ 𝑢 ≤ 𝑊) ↔ ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑢 ≤ 𝑊)))
40	12, 39	mpbird 256	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ∃𝑢 ∈ 𝐴 ((𝑃 ∨ 𝑢) = (𝑄 ∨ 𝑢) ∧ 𝑢 ≤ 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  lecple 16895  joincjn 17944  meetcmee 17945  Atomscatm 37204  CvLatclc 37206  HLchlt 37291  LHypclh 37925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-p1 18059  df-lat 18065  df-clat 18132  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-lhyp 37929

This theorem is referenced by: (None)