Theorem cdleme27N 41229

Description: Part of proof of Lemma E in [Crawley] p. 113. Eliminate the 𝑠 ≠ 𝑡 antecedent in cdleme27a 41227. (Contributed by NM, 3-Feb-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme26.b	⊢ 𝐵 = (Base‘𝐾)
cdleme26.l	⊢ ≤ = (le‘𝐾)
cdleme26.j	⊢ ∨ = (join‘𝐾)
cdleme26.m	⊢ ∧ = (meet‘𝐾)
cdleme26.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme26.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme27.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme27.f	⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme27.z	⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
cdleme27.n	⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊)))
cdleme27.d	⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))
cdleme27.c	⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹)
cdleme27.g	⊢ 𝐺 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme27.o	⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊)))
cdleme27.e	⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))
cdleme27.y	⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺)

Assertion

Ref	Expression
cdleme27N	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐶 ≤ (𝑌 ∨ 𝑉))

Distinct variable groups:   𝑡,𝑠,𝑢,𝑧,𝐴   𝐵,𝑠,𝑡,𝑢,𝑧   𝑢,𝐹   𝑢,𝐺   𝐻,𝑠,𝑡,𝑧   ∨ ,𝑠,𝑡,𝑢,𝑧   𝐾,𝑠,𝑡,𝑧   ≤ ,𝑠,𝑡,𝑢,𝑧   ∧ ,𝑠,𝑡,𝑢,𝑧   𝑡,𝑁,𝑢   𝑂,𝑠,𝑢   𝑃,𝑠,𝑡,𝑢,𝑧   𝑄,𝑠,𝑡,𝑢,𝑧   𝑈,𝑠,𝑡,𝑢,𝑧   𝑧,𝑉   𝑊,𝑠,𝑡,𝑢,𝑧

Allowed substitution hints:   𝐶(𝑧, 𝑢, 𝑡, 𝑠)   𝐷(𝑧, 𝑢, 𝑡, 𝑠)   𝐸(𝑧, 𝑢, 𝑡, 𝑠)   𝐹(𝑧, 𝑡, 𝑠)   𝐺(𝑧, 𝑡, 𝑠)   𝐻(𝑢)   𝐾(𝑢)   𝑁(𝑧, 𝑠)   𝑂(𝑧, 𝑡)   𝑉(𝑢, 𝑡, 𝑠)   𝑌(𝑧, 𝑢, 𝑡, 𝑠)   𝑍(𝑧, 𝑢, 𝑡, 𝑠)

Proof of Theorem cdleme27N

Step	Hyp	Ref	Expression
1		cdleme26.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		cdleme26.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		cdleme26.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
4		cdleme26.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
5		cdleme26.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdleme26.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdleme27.u	. . . . 5 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8		cdleme27.f	. . . . 5 ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
9		cdleme27.z	. . . . 5 ⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
10		cdleme27.n	. . . . 5 ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊)))
11		cdleme27.d	. . . . 5 ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))
12		cdleme27.c	. . . . 5 ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹)
13		cdleme27.g	. . . . 5 ⊢ 𝐺 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
14		cdleme27.o	. . . . 5 ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊)))
15		cdleme27.e	. . . . 5 ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))
16		cdleme27.y	. . . . 5 ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺)
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	cdleme27b 41228	. . . 4 ⊢ (𝑠 = 𝑡 → 𝐶 = 𝑌)
18	17	adantl 487	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 = 𝑡) → 𝐶 = 𝑌)
19		simp11l 1303	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐾 ∈ HL)
20	19	hllatd 40224	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐾 ∈ Lat)
21		simp11r 1304	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝑊 ∈ 𝐻)
22		simp21 1225	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
23		simp22 1226	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
24		simp23 1227	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))
25		simp12 1223	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝑃 ≠ 𝑄)
26	1, 2, 3, 4, 5, 6, 7, 13, 9, 14, 15, 16	cdleme27cl 41226	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊) ∧ 𝑃 ≠ 𝑄)) → 𝑌 ∈ 𝐵)
27	19, 21, 22, 23, 24, 25, 26	syl222anc 1413	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝑌 ∈ 𝐵)
28		simp3rl 1265	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝑉 ∈ 𝐴)
29	1, 5	atbase 40149	. . . . . 6 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ 𝐵)
30	28, 29	syl 18	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝑉 ∈ 𝐵)
31	1, 2, 3	latlej1 18540	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵) → 𝑌 ≤ (𝑌 ∨ 𝑉))
32	20, 27, 30, 31	syl3anc 1398	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝑌 ≤ (𝑌 ∨ 𝑉))
33	32	adantr 486	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 = 𝑡) → 𝑌 ≤ (𝑌 ∨ 𝑉))
34	18, 33	eqbrtrd 5131	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 = 𝑡) → 𝐶 ≤ (𝑌 ∨ 𝑉))
35		simpl11 1267	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 ≠ 𝑡) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
36		simpl12 1268	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 ≠ 𝑡) → 𝑃 ≠ 𝑄)
37		simpl13 1269	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 ≠ 𝑡) → (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊))
38		simpl21 1270	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 ≠ 𝑡) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
39		simpl22 1271	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 ≠ 𝑡) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
40		simpl23 1272	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 ≠ 𝑡) → (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))
41		simpr 490	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 ≠ 𝑡) → 𝑠 ≠ 𝑡)
42		simpl3l 1247	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 ≠ 𝑡) → 𝑠 ≤ (𝑡 ∨ 𝑉))
43	41, 42	jca 521	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 ≠ 𝑡) → (𝑠 ≠ 𝑡 ∧ 𝑠 ≤ (𝑡 ∨ 𝑉)))
44		simpl3r 1248	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 ≠ 𝑡) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
45	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	cdleme27a 41227	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ ((𝑠 ≠ 𝑡 ∧ 𝑠 ≤ (𝑡 ∨ 𝑉)) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐶 ≤ (𝑌 ∨ 𝑉))
46	35, 36, 37, 38, 39, 40, 43, 44, 45	syl332anc 1428	. 2 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) ∧ 𝑠 ≠ 𝑡) → 𝐶 ≤ (𝑌 ∨ 𝑉))
47	34, 46	pm2.61dane 3044	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ≠ 𝑄 ∧ (𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊)) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊)) ∧ (𝑠 ≤ (𝑡 ∨ 𝑉) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))) → 𝐶 ≤ (𝑌 ∨ 𝑉))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ifcif 4485   class class class wbr 5107  ‘cfv 6537  ℩crio 7372  (class class class)co 7416  Basecbs 17305  lecple 17353  joincjn 18403  meetcmee 18404  Latclat 18523  Atomscatm 40123  HLchlt 40210  LHypclh 40844

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-riotaBAD 39813

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-undef 8274  df-proset 18386  df-poset 18405  df-plt 18420  df-lub 18436  df-glb 18437  df-join 18438  df-meet 18439  df-p0 18515  df-p1 18516  df-lat 18524  df-clat 18591  df-oposet 40036  df-ol 40038  df-oml 40039  df-covers 40126  df-ats 40127  df-atl 40158  df-cvlat 40182  df-hlat 40211  df-llines 40358  df-lplanes 40359  df-lvols 40360  df-lines 40361  df-psubsp 40363  df-pmap 40364  df-padd 40656  df-lhyp 40848

This theorem is used by: (None)