Theorem cdleme0ex2N 40269

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 1136	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp2l 1200	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
3		simp2rl 1243	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ 𝐴)
4		simp3 1138	. . 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 40268	. . 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 1377	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑄) ∧ 𝑢 ≤ 𝑊))
13		simp11l 1285	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝐾 ∈ HL)
14		hlcvl 39404	. . . . . . . 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 1241	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ 𝐴)
17	16	3ad2ant1 1133	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝑃 ∈ 𝐴)
18	3	3ad2ant1 1133	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝑄 ∈ 𝐴)
19		simp2 1137	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝑢 ∈ 𝐴)
20		simp13 1206	. . . . . . 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 39388	. . . . . . 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 1385	. . . . . 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 1088	. . . . . . 7 ⊢ ((𝑢 ≠ 𝑃 ∧ 𝑢 ≠ 𝑄 ∧ 𝑢 ≤ (𝑃 ∨ 𝑄)) ↔ ((𝑢 ≠ 𝑃 ∧ 𝑢 ≠ 𝑄) ∧ 𝑢 ≤ (𝑃 ∨ 𝑄)))
24		simp3 1138	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝑢 ≤ 𝑊)
25		simp2lr 1242	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ¬ 𝑃 ≤ 𝑊)
26	25	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → ¬ 𝑃 ≤ 𝑊)
27		nbrne2 5111	. . . . . . . . . 10 ⊢ ((𝑢 ≤ 𝑊 ∧ ¬ 𝑃 ≤ 𝑊) → 𝑢 ≠ 𝑃)
28	24, 26, 27	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → 𝑢 ≠ 𝑃)
29		simp2rr 1244	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) → ¬ 𝑄 ≤ 𝑊)
30	29	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑃 ≠ 𝑄) ∧ 𝑢 ∈ 𝐴 ∧ 𝑢 ≤ 𝑊) → ¬ 𝑄 ≤ 𝑊)
31		nbrne2 5111	. . . . . . . . . 10 ⊢ ((𝑢 ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊) → 𝑢 ≠ 𝑄)
32	24, 30, 31	syl2anc 584	. . . . . . . . 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 290	. . . . . 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 279	. . . . 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 1121	. . . 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 578	. . 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 3154	. 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 257	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∃wrex 3056   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346  lecple 17168  joincjn 18217  meetcmee 18218  Atomscatm 39308  CvLatclc 39310  HLchlt 39395  LHypclh 40029

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 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  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 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  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 39221  df-ol 39223  df-oml 39224  df-covers 39311  df-ats 39312  df-atl 39343  df-cvlat 39367  df-hlat 39396  df-lhyp 40033

This theorem is referenced by: (None)