Theorem dihmeetlem1N 42014

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 1214	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝐾 ∈ HL)
2	1	hllatd 40088	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝐾 ∈ Lat)
3		simp2l 1216	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → 𝑋 ∈ 𝐵)
4		simp3l 1218	. . . . 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 18523	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑋)
9	2, 3, 4, 8	syl3anc 1396	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ≤ 𝑋)
10		simp1 1152	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11	5, 7	latmcl 18499	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
12	2, 3, 4, 11	syl3anc 1396	. . . . 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 41988	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊆ (𝐼‘𝑋) ↔ (𝑋 ∧ 𝑌) ≤ 𝑋))
16	10, 12, 3, 15	syl3anc 1396	. . . 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 18524	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌)
19	2, 3, 4, 18	syl3anc 1396	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ≤ 𝑌)
20	5, 6, 13, 14	dihord 41988	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊆ (𝐼‘𝑌) ↔ (𝑋 ∧ 𝑌) ≤ 𝑌))
21	10, 12, 4, 20	syl3anc 1396	. . . 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 4201	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) ⊆ ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
24	13, 14	dihvalrel 42003	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑋))
25		relin1 5803	. . . . 5 ⊢ (Rel (𝐼‘𝑋) → Rel ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
26	24, 25	syl 18	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
27	26	3ad2ant1 1149	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → Rel ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
28		elin 3929	. . . 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 40748	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
32	31	3adant3 1148	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
33		simpl1 1208	. . . . . . . 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 1209	. . . . . . . 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 782	. . . . . . . . 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 792	. . . . . . . . 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 520	. . . . . . . 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 793	. . . . . . . 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 3466	. . . . . . . . 9 ⊢ 𝑓 ∈ V
45		vex 3466	. . . . . . . . 9 ⊢ 𝑠 ∈ V
46	5, 6, 29, 7, 30, 13, 39, 40, 41, 42, 14, 43, 44, 45	dihopelvalc 41973	. . . . . . . 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 1399	. . . . . . 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 489	. . . . . . 7 ⊢ (((𝑓 ∈ 𝑇 ∧ 𝑠 ∈ 𝐸) ∧ (𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋) → (𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋)
49	47, 48	biimtrdi 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 1210	. . . . . . . 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 41962	. . . . . . . 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 595	. . . . . . 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 790	. . . . . . . . . 10 ⊢ (((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 )) → 𝑓 ∈ 𝑇)
56	55	3ad2ant3 1151	. . . . . . . . 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 1264	. . . . . . . . . . . . . . . . . 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 6887	. . . . . . . . . . . . . . . . 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 1220	. . . . . . . . . . . . . . . . . . 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 40743	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
61	59, 60	syl 18	. . . . . . . . . . . . . . . . . . 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 1216	. . . . . . . . . . . . . . . . . . 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 1259	. . . . . . . . . . . . . . . . . . 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 41303	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
65	59, 61, 62, 63, 64	syl112anc 1399	. . . . . . . . . . . . . . . . . 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 41511	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ 𝑇 → ( 0 ‘𝐺) = ( I ↾ 𝐵))
67	65, 66	syl 18	. . . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . . 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 5865	. . . . . . . . . . . . . . 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 6619	. . . . . . . . . . . . . . 15 ⊢ ◡( I ↾ 𝐵) = ( I ↾ 𝐵)
71	69, 70	eqtrdi 2821	. . . . . . . . . . . . . 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 5852	. . . . . . . . . . . . 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 40848	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → 𝑓:𝐵–1-1-onto→𝐵)
74	59, 56, 73	syl2anc 595	. . . . . . . . . . . . . 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 6824	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1-onto→𝐵 → 𝑓:𝐵⟶𝐵)
76		fcoi1 6756	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵⟶𝐵 → (𝑓 ∘ ( I ↾ 𝐵)) = 𝑓)
77	74, 75, 76	3syl 19	. . . . . . . . . . . . 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 2805	. . . . . . . . . . . 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 6889	. . . . . . . . . . 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 1218	. . . . . . . . . . 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 5140	. . . . . . . . . 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 791	. . . . . . . . . . 11 ⊢ (((𝑅‘(𝑓 ∘ ◡(𝑠‘𝐺))) ≤ 𝑋 ∧ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ 𝑌) ∧ 𝑠 = 0 )) → (𝑅‘𝑓) ≤ 𝑌)
83	82	3ad2ant3 1151	. . . . . . . . . 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 1301	. . . . . . . . . . . 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 40088	. . . . . . . . . . 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 40888	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝑅‘𝑓) ∈ 𝐵)
87	59, 56, 86	syl2anc 595	. . . . . . . . . . 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 1303	. . . . . . . . . . 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 1305	. . . . . . . . . . 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 18525	. . . . . . . . . . 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 1397	. . . . . . . . . 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 724	. . . . . . . . 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 520	. . . . . . . 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 1396	. . . . . . . . 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 1302	. . . . . . . . . . 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 40722	. . . . . . . . . . 11 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
97	95, 96	syl 18	. . . . . . . . . 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 1396	. . . . . . . . . 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 1306	. . . . . . . . . 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 18505	. . . . . . . . 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 41962	. . . . . . . . 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 849	. . . . . . . 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 725	. . . . . . 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 1137	. . . . . 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 619	. . . . 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 3179	. . . 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	biimtrid 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 5778	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ∩ (𝐼‘𝑌)) ⊆ (𝐼‘(𝑋 ∧ 𝑌)))
109	23, 108	eqssd 3962	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 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150  ∃wrex 3096   ∩ cin 3912   ⊆ wss 3913  ⟨cop 4600   class class class wbr 5114   ↦ cmpt 5197   I cid 5559  ^◡ccnv 5664   ↾ cres 5667   ∘ ccom 5669  Rel wrel 5670  ⟶wf 6536  –1-1-onto→wf1o 6539  ‘cfv 6540  ℩crio 7370  (class class class)co 7414  Basecbs 17272  lecple 17320  occoc 17321  joincjn 18370  meetcmee 18371  Latclat 18490  Atomscatm 39987  HLchlt 40074  LHypclh 40708  LTrncltrn 40825  trLctrl 40882  TEndoctendo 41476  DIsoHcdih 41952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736  ax-cnex 11159  ax-resscn 11160  ax-1cn 11161  ax-icn 11162  ax-addcl 11163  ax-addrcl 11164  ax-mulcl 11165  ax-mulrcl 11166  ax-mulcom 11167  ax-addass 11168  ax-mulass 11169  ax-distr 11170  ax-i2m1 11171  ax-1ne0 11172  ax-1rid 11173  ax-rnegex 11174  ax-rrecex 11175  ax-cnre 11176  ax-pre-lttri 11177  ax-pre-lttrn 11178  ax-pre-ltadd 11179  ax-pre-mulgt0 11180  ax-riotaBAD 39677

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-nel 3072  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5560  df-eprel 5565  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7866  df-1st 7989  df-2nd 7990  df-tpos 8225  df-undef 8272  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-er 8697  df-map 8829  df-en 8947  df-dom 8948  df-sdom 8949  df-fin 8950  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11446  df-neg 11447  df-nn 12237  df-2 12306  df-3 12307  df-4 12308  df-5 12309  df-6 12310  df-n0 12508  df-z 12595  df-uz 12866  df-fz 13539  df-struct 17210  df-sets 17227  df-slot 17245  df-ndx 17257  df-base 17273  df-ress 17294  df-plusg 17326  df-mulr 17327  df-sca 17329  df-vsca 17330  df-0g 17497  df-proset 18353  df-poset 18372  df-plt 18387  df-lub 18403  df-glb 18404  df-join 18405  df-meet 18406  df-p0 18482  df-p1 18483  df-lat 18491  df-clat 18558  df-mgm 18701  df-sgrp 18780  df-mnd 18796  df-submnd 18845  df-grp 19006  df-minusg 19007  df-sbg 19008  df-subg 19192  df-cntz 19390  df-lsm 19709  df-cmn 19855  df-abl 19856  df-mgp 20220  df-rng 20234  df-ur 20267  df-ring 20320  df-oppr 20422  df-dvdsr 20442  df-unit 20443  df-invr 20473  df-dvr 20486  df-drng 20818  df-lmod 20966  df-lss 21036  df-lsp 21076  df-lvec 21207  df-oposet 39900  df-ol 39902  df-oml 39903  df-covers 39990  df-ats 39991  df-atl 40022  df-cvlat 40046  df-hlat 40075  df-llines 40222  df-lplanes 40223  df-lvols 40224  df-lines 40225  df-psubsp 40227  df-pmap 40228  df-padd 40520  df-lhyp 40712  df-laut 40713  df-ldil 40828  df-ltrn 40829  df-trl 40883  df-tendo 41479  df-edring 41481  df-disoa 41753  df-dvech 41803  df-dib 41863  df-dic 41897  df-dih 41953

This theorem is referenced by:  dihmeetbN  42027