Theorem dihmeetlem1N 39041

Description: Isomorphism H of a conjunction. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dihglblem5a.b	⊢ 𝐵 = (Base‘𝐾)
dihglblem5a.m	⊢ ∧ = (meet‘𝐾)
dihglblem5a.h	⊢ 𝐻 = (LHyp‘𝐾)
dihglblem5a.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihglblem5a.l	⊢ ≤ = (le‘𝐾)
dihglblem5a.j	⊢ ∨ = (join‘𝐾)
dihglblem5a.a	⊢ 𝐴 = (Atoms‘𝐾)
dihglblem5a.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
dihglblem5a.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihglblem5a.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dihglblem5a.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihglblem5a.g	⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑞)
dihglblem5a.o	⊢ 0 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))

Assertion

Ref	Expression
dihmeetlem1N	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))

Distinct variable groups:   ∧ ,𝑞   ℎ,𝑞, ≤   𝐴,ℎ,𝑞   𝐵,ℎ,𝑞   ℎ,𝐻,𝑞   𝐼,𝑞   ℎ,𝐾,𝑞   𝑃,ℎ   𝑇,ℎ   ℎ,𝑊,𝑞   𝑋,𝑞   𝑌,𝑞

Allowed substitution hints:   𝑃(𝑞)   𝑅(ℎ,𝑞)   𝑇(𝑞)   𝐸(ℎ,𝑞)   𝐺(ℎ,𝑞)   𝐼(ℎ)   ∨ (ℎ,𝑞)   ∧ (ℎ)   𝑋(ℎ)   𝑌(ℎ)   0 (ℎ,𝑞)

Proof of Theorem dihmeetlem1N

Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1l 1199	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝐾 ∈ HL)
2	1	hllatd 37115	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝐾 ∈ Lat)
3		simp2l 1201	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑋 ∈ 𝐵)
4		simp3l 1203	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ 𝐵)
5		dihglblem5a.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6		dihglblem5a.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
7		dihglblem5a.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
8	5, 6, 7	latmle1 17970	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑋)
9	2, 3, 4, 8	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ≤ 𝑋)
10		simp1 1138	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11	5, 7	latmcl 17946	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
12	2, 3, 4, 11	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ∈ 𝐵)
13		dihglblem5a.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
14		dihglblem5a.i	. . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
15	5, 6, 13, 14	dihord 39015	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊆ (𝐼‘𝑋) ↔ (𝑋 ∧ 𝑌) ≤ 𝑋))
16	10, 12, 3, 15	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊆ (𝐼‘𝑋) ↔ (𝑋 ∧ 𝑌) ≤ 𝑋))
17	9, 16	mpbird 260	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) ⊆ (𝐼‘𝑋))
18	5, 6, 7	latmle2 17971	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌)
19	2, 3, 4, 18	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ≤ 𝑌)
20	5, 6, 13, 14	dihord 39015	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊆ (𝐼‘𝑌) ↔ (𝑋 ∧ 𝑌) ≤ 𝑌))
21	10, 12, 4, 20	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊆ (𝐼‘𝑌) ↔ (𝑋 ∧ 𝑌) ≤ 𝑌))
22	19, 21	mpbird 260	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) ⊆ (𝐼‘𝑌))
23	17, 22	ssind 4147	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) ⊆ ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
24	13, 14	dihvalrel 39030	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑋))
25		relin1 5682	. . . . 5 ⊢ (Rel (𝐼‘𝑋) → Rel ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
26	24, 25	syl 17	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
27	26	3ad2ant1 1135	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → Rel ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
28		elin 3882	. . . 4 ⊢ (⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑋) ∩ (𝐼‘𝑌)) ↔ (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑌)))
29		dihglblem5a.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
30		dihglblem5a.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
31	5, 6, 29, 7, 30, 13	lhpmcvr2 37775	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
32	31	3adant3 1134	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
33		simpl1 1193	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
34		simpl2 1194	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊))
35		simprl 771	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → 𝑞 ∈ 𝐴)
36		simprrl 781	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → ¬ 𝑞 ≤ 𝑊)
37	35, 36	jca 515	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊))
38		simprrr 782	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
39		dihglblem5a.p	. . . . . . . . 9 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
40		dihglblem5a.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
41		dihglblem5a.r	. . . . . . . . 9 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
42		dihglblem5a.e	. . . . . . . . 9 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
43		dihglblem5a.g	. . . . . . . . 9 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑞)
44		vex 3412	. . . . . . . . 9 ⊢ 𝑓 ∈ V
45		vex 3412	. . . . . . . . 9 ⊢ 𝑠 ∈ V
46	5, 6, 29, 7, 30, 13, 39, 40, 41, 42, 14, 43, 44, 45	dihopelvalc 39000	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑋) ↔ ((𝑓 ∈ 𝑇 ∧ 𝑠 ∈ 𝐸) ∧ (𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋)))
47	33, 34, 37, 38, 46	syl112anc 1376	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑋) ↔ ((𝑓 ∈ 𝑇 ∧ 𝑠 ∈ 𝐸) ∧ (𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋)))
48		simpr 488	. . . . . . 7 ⊢ (((𝑓 ∈ 𝑇 ∧ 𝑠 ∈ 𝐸) ∧ (𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋) → (𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋)
49	47, 48	syl6bi 256	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑋) → (𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋))
50		simpl3 1195	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊))
51		dihglblem5a.o	. . . . . . . . 9 ⊢ 0 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
52	5, 6, 13, 40, 41, 51, 14	dihopelvalbN 38989	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑌) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 )))
53	33, 50, 52	syl2anc 587	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑌) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 )))
54	53	biimpd 232	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑌) → ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 )))
55		simprll 779	. . . . . . . . . 10 ⊢ (((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 )) → 𝑓 ∈ 𝑇)
56	55	3ad2ant3 1137	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → 𝑓 ∈ 𝑇)
57		simp3rr 1249	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → 𝑠 = 0 )
58	57	fveq1d 6719	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑠‘𝐺) = ( 0 ‘𝐺))
59		simp11 1205	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
60	6, 30, 13, 39	lhpocnel2 37770	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
61	59, 60	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
62		simp2l 1201	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → 𝑞 ∈ 𝐴)
63		simp2rl 1244	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → ¬ 𝑞 ≤ 𝑊)
64	6, 30, 13, 40, 43	ltrniotacl 38330	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
65	59, 61, 62, 63, 64	syl112anc 1376	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → 𝐺 ∈ 𝑇)
66	51, 5	tendo02 38538	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ 𝑇 → ( 0 ‘𝐺) = ( I ↾ 𝐵))
67	65, 66	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → ( 0 ‘𝐺) = ( I ↾ 𝐵))
68	58, 67	eqtrd 2777	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑠‘𝐺) = ( I ↾ 𝐵))
69	68	cnveqd 5744	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → ◡(𝑠‘𝐺) = ◡( I ↾ 𝐵))
70		cnvresid 6459	. . . . . . . . . . . . . . 15 ⊢ ◡( I ↾ 𝐵) = ( I ↾ 𝐵)
71	69, 70	eqtrdi 2794	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → ◡(𝑠‘𝐺) = ( I ↾ 𝐵))
72	71	coeq2d 5731	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑓 ∘ ◡(𝑠‘𝐺)) = (𝑓 ∘ ( I ↾ 𝐵)))
73	5, 13, 40	ltrn1o 37875	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → 𝑓:𝐵–1-1-onto→𝐵)
74	59, 56, 73	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → 𝑓:𝐵–1-1-onto→𝐵)
75		f1of 6661	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1-onto→𝐵 → 𝑓:𝐵⟶𝐵)
76		fcoi1 6593	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵⟶𝐵 → (𝑓 ∘ ( I ↾ 𝐵)) = 𝑓)
77	74, 75, 76	3syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑓 ∘ ( I ↾ 𝐵)) = 𝑓)
78	72, 77	eqtrd 2777	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑓 ∘ ◡(𝑠‘𝐺)) = 𝑓)
79	78	fveq2d 6721	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) = (𝑅‘𝑓))
80		simp3l 1203	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋)
81	79, 80	eqbrtrrd 5077	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑅‘𝑓) ≤ 𝑋)
82		simprlr 780	. . . . . . . . . . 11 ⊢ (((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 )) → (𝑅‘𝑓) ≤ 𝑌)
83	82	3ad2ant3 1137	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑅‘𝑓) ≤ 𝑌)
84		simp11l 1286	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → 𝐾 ∈ HL)
85	84	hllatd 37115	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → 𝐾 ∈ Lat)
86	5, 13, 40, 41	trlcl 37915	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝑅‘𝑓) ∈ 𝐵)
87	59, 56, 86	syl2anc 587	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑅‘𝑓) ∈ 𝐵)
88		simp12l 1288	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → 𝑋 ∈ 𝐵)
89		simp13l 1290	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → 𝑌 ∈ 𝐵)
90	5, 6, 7	latlem12 17972	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ ((𝑅‘𝑓) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (((𝑅‘𝑓) ≤ 𝑋 ∧ (𝑅‘𝑓) ≤ 𝑌) ↔ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑌)))
91	85, 87, 88, 89, 90	syl13anc 1374	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (((𝑅‘𝑓) ≤ 𝑋 ∧ (𝑅‘𝑓) ≤ 𝑌) ↔ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑌)))
92	81, 83, 91	mpbi2and 712	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑌))
93	56, 92	jca 515	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑌)))
94	85, 88, 89, 11	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑋 ∧ 𝑌) ∈ 𝐵)
95		simp11r 1287	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → 𝑊 ∈ 𝐻)
96	5, 13	lhpbase 37749	. . . . . . . . . . 11 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
97	95, 96	syl 17	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → 𝑊 ∈ 𝐵)
98	85, 88, 89, 18	syl3anc 1373	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑋 ∧ 𝑌) ≤ 𝑌)
99		simp13r 1291	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → 𝑌 ≤ 𝑊)
100	5, 6, 85, 94, 89, 97, 98, 99	lattrd 17952	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (𝑋 ∧ 𝑌) ≤ 𝑊)
101	5, 6, 13, 40, 41, 51, 14	dihopelvalbN 38989	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑌)) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑌)) ∧ 𝑠 = 0 )))
102	59, 94, 100, 101	syl12anc 837	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑌)) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑌)) ∧ 𝑠 = 0 )))
103	93, 57, 102	mpbir2and 713	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 ))) → ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑌)))
104	103	3expia 1123	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 )) → ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑌))))
105	49, 54, 104	syl2and 611	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → ((⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑌)) → ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑌))))
106	32, 105	rexlimddv 3210	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑌)) → ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑌))))
107	28, 106	syl5bi 245	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (⟨𝑓, 𝑠⟩ ∈ ((𝐼‘𝑋) ∩ (𝐼‘𝑌)) → ⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝑋 ∧ 𝑌))))
108	27, 107	relssdv 5658	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ∩ (𝐼‘𝑌)) ⊆ (𝐼‘(𝑋 ∧ 𝑌)))
109	23, 108	eqssd 3918	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∃wrex 3062   ∩ cin 3865   ⊆ wss 3866  ⟨cop 4547   class class class wbr 5053   ↦ cmpt 5135   I cid 5454  ^◡ccnv 5550   ↾ cres 5553   ∘ ccom 5555  Rel wrel 5556  ⟶wf 6376  –1-1-onto→wf1o 6379  ‘cfv 6380  ℩crio 7169  (class class class)co 7213  Basecbs 16760  lecple 16809  occoc 16810  joincjn 17818  meetcmee 17819  Latclat 17937  Atomscatm 37014  HLchlt 37101  LHypclh 37735  LTrncltrn 37852  trLctrl 37909  TEndoctendo 38503  DIsoHcdih 38979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-riotaBAD 36704

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-tpos 7968  df-undef 8015  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-n0 12091  df-z 12177  df-uz 12439  df-fz 13096  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-sca 16818  df-vsca 16819  df-0g 16946  df-proset 17802  df-poset 17820  df-plt 17836  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-p0 17931  df-p1 17932  df-lat 17938  df-clat 18005  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-submnd 18219  df-grp 18368  df-minusg 18369  df-sbg 18370  df-subg 18540  df-cntz 18711  df-lsm 19025  df-cmn 19172  df-abl 19173  df-mgp 19505  df-ur 19517  df-ring 19564  df-oppr 19641  df-dvdsr 19659  df-unit 19660  df-invr 19690  df-dvr 19701  df-drng 19769  df-lmod 19901  df-lss 19969  df-lsp 20009  df-lvec 20140  df-oposet 36927  df-ol 36929  df-oml 36930  df-covers 37017  df-ats 37018  df-atl 37049  df-cvlat 37073  df-hlat 37102  df-llines 37249  df-lplanes 37250  df-lvols 37251  df-lines 37252  df-psubsp 37254  df-pmap 37255  df-padd 37547  df-lhyp 37739  df-laut 37740  df-ldil 37855  df-ltrn 37856  df-trl 37910  df-tendo 38506  df-edring 38508  df-disoa 38780  df-dvech 38830  df-dib 38890  df-dic 38924  df-dih 38980

This theorem is referenced by:  dihmeetbN  39054