diaintclN - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > diaintclN		Structured version Visualization version GIF version

Theorem diaintclN 41059

Description: The intersection of partial isomorphism A closed subspaces is a closed subspace. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
diaintcl.h	⊢ 𝐻 = (LHyp‘𝐾)
diaintcl.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
diaintclN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑆 ∈ ran 𝐼)

Proof of Theorem diaintclN Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		diaintcl.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
2		diaintcl.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
3	1, 2	diaf11N 41050	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼)
4	3	adantr 480	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼)
5		f1ofn 6804	. . . . . 6 ⊢ (𝐼:dom 𝐼–1-1-onto→ran 𝐼 → 𝐼 Fn dom 𝐼)
6	4, 5	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝐼 Fn dom 𝐼)
7		cnvimass 6056	. . . . 5 ⊢ (^◡𝐼 “ 𝑆) ⊆ dom 𝐼
8		fnssres 6644	. . . . 5 ⊢ ((𝐼 Fn dom 𝐼 ∧ (^◡𝐼 “ 𝑆) ⊆ dom 𝐼) → (𝐼 ↾ (^◡𝐼 “ 𝑆)) Fn (^◡𝐼 “ 𝑆))
9	6, 7, 8	sylancl 586	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼 ↾ (^◡𝐼 “ 𝑆)) Fn (^◡𝐼 “ 𝑆))
10		fniinfv 6942	. . . 4 ⊢ ((𝐼 ↾ (^◡𝐼 “ 𝑆)) Fn (^◡𝐼 “ 𝑆) → ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) = ∩ ran (𝐼 ↾ (^◡𝐼 “ 𝑆)))
11	9, 10	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) = ∩ ran (𝐼 ↾ (^◡𝐼 “ 𝑆)))
12		df-ima 5654	. . . . 5 ⊢ (𝐼 “ (^◡𝐼 “ 𝑆)) = ran (𝐼 ↾ (^◡𝐼 “ 𝑆))
13		f1ofo 6810	. . . . . . . 8 ⊢ (𝐼:dom 𝐼–1-1-onto→ran 𝐼 → 𝐼:dom 𝐼–onto→ran 𝐼)
14	3, 13	syl 17	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–onto→ran 𝐼)
15	14	adantr 480	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝐼:dom 𝐼–onto→ran 𝐼)
16		simprl 770	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ ran 𝐼)
17		foimacnv 6820	. . . . . 6 ⊢ ((𝐼:dom 𝐼–onto→ran 𝐼 ∧ 𝑆 ⊆ ran 𝐼) → (𝐼 “ (^◡𝐼 “ 𝑆)) = 𝑆)
18	15, 16, 17	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼 “ (^◡𝐼 “ 𝑆)) = 𝑆)
19	12, 18	eqtr3id 2779	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ran (𝐼 ↾ (^◡𝐼 “ 𝑆)) = 𝑆)
20	19	inteqd 4918	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ ran (𝐼 ↾ (^◡𝐼 “ 𝑆)) = ∩ 𝑆)
21	11, 20	eqtrd 2765	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) = ∩ 𝑆)
22		simpl 482	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
23	7	a1i 11	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ⊆ dom 𝐼)
24		simprr 772	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ≠ ∅)
25		n0 4319	. . . . . . 7 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑆)
26	24, 25	sylib 218	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∃𝑦 𝑦 ∈ 𝑆)
27	16	sselda 3949	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ran 𝐼)
28	3	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼)
29	28, 5	syl 17	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → 𝐼 Fn dom 𝐼)
30		fvelrnb 6924	. . . . . . . . 9 ⊢ (𝐼 Fn dom 𝐼 → (𝑦 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝑦))
31	29, 30	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → (𝑦 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝑦))
32	27, 31	mpbid 232	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → ∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝑦)
33		f1ofun 6805	. . . . . . . . . . . . . . . 16 ⊢ (𝐼:dom 𝐼–1-1-onto→ran 𝐼 → Fun 𝐼)
34	3, 33	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Fun 𝐼)
35	34	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → Fun 𝐼)
36		fvimacnv 7028	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐼 ∧ 𝑥 ∈ dom 𝐼) → ((𝐼‘𝑥) ∈ 𝑆 ↔ 𝑥 ∈ (^◡𝐼 “ 𝑆)))
37	35, 36	sylan 580	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ dom 𝐼) → ((𝐼‘𝑥) ∈ 𝑆 ↔ 𝑥 ∈ (^◡𝐼 “ 𝑆)))
38		ne0i 4307	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (^◡𝐼 “ 𝑆) → (^◡𝐼 “ 𝑆) ≠ ∅)
39	37, 38	biimtrdi 253	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ dom 𝐼) → ((𝐼‘𝑥) ∈ 𝑆 → (^◡𝐼 “ 𝑆) ≠ ∅))
40	39	ex 412	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝑥 ∈ dom 𝐼 → ((𝐼‘𝑥) ∈ 𝑆 → (^◡𝐼 “ 𝑆) ≠ ∅)))
41		eleq1 2817	. . . . . . . . . . . . 13 ⊢ ((𝐼‘𝑥) = 𝑦 → ((𝐼‘𝑥) ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
42	41	biimprd 248	. . . . . . . . . . . 12 ⊢ ((𝐼‘𝑥) = 𝑦 → (𝑦 ∈ 𝑆 → (𝐼‘𝑥) ∈ 𝑆))
43	42	imim1d 82	. . . . . . . . . . 11 ⊢ ((𝐼‘𝑥) = 𝑦 → (((𝐼‘𝑥) ∈ 𝑆 → (^◡𝐼 “ 𝑆) ≠ ∅) → (𝑦 ∈ 𝑆 → (^◡𝐼 “ 𝑆) ≠ ∅)))
44	40, 43	syl9 77	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ((𝐼‘𝑥) = 𝑦 → (𝑥 ∈ dom 𝐼 → (𝑦 ∈ 𝑆 → (^◡𝐼 “ 𝑆) ≠ ∅))))
45	44	com24 95	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝑦 ∈ 𝑆 → (𝑥 ∈ dom 𝐼 → ((𝐼‘𝑥) = 𝑦 → (^◡𝐼 “ 𝑆) ≠ ∅))))
46	45	imp 406	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → (𝑥 ∈ dom 𝐼 → ((𝐼‘𝑥) = 𝑦 → (^◡𝐼 “ 𝑆) ≠ ∅)))
47	46	rexlimdv 3133	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → (∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝑦 → (^◡𝐼 “ 𝑆) ≠ ∅))
48	32, 47	mpd 15	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → (^◡𝐼 “ 𝑆) ≠ ∅)
49	26, 48	exlimddv 1935	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ≠ ∅)
50		eqid 2730	. . . . . 6 ⊢ (glb‘𝐾) = (glb‘𝐾)
51	50, 1, 2	diaglbN 41056	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((^◡𝐼 “ 𝑆) ⊆ dom 𝐼 ∧ (^◡𝐼 “ 𝑆) ≠ ∅)) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) = ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)(𝐼‘𝑦))
52	22, 23, 49, 51	syl12anc 836	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) = ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)(𝐼‘𝑦))
53		fvres 6880	. . . . 5 ⊢ (𝑦 ∈ (^◡𝐼 “ 𝑆) → ((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) = (𝐼‘𝑦))
54	53	iineq2i 4981	. . . 4 ⊢ ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) = ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)(𝐼‘𝑦)
55	52, 54	eqtr4di 2783	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) = ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦))
56		hlclat 39358	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
57	56	ad2antrr 726	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝐾 ∈ CLat)
58		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
59		eqid 2730	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
60	58, 59, 1, 2	diadm 41036	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊})
61		ssrab2 4046	. . . . . . . . 9 ⊢ {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊} ⊆ (Base‘𝐾)
62	60, 61	eqsstrdi 3994	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom 𝐼 ⊆ (Base‘𝐾))
63	62	adantr 480	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → dom 𝐼 ⊆ (Base‘𝐾))
64	7, 63	sstrid 3961	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ⊆ (Base‘𝐾))
65	58, 50	clatglbcl 18471	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ (^◡𝐼 “ 𝑆) ⊆ (Base‘𝐾)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ (Base‘𝐾))
66	57, 64, 65	syl2anc 584	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ (Base‘𝐾))
67		n0 4319	. . . . . . 7 ⊢ ((^◡𝐼 “ 𝑆) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (^◡𝐼 “ 𝑆))
68	49, 67	sylib 218	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∃𝑦 𝑦 ∈ (^◡𝐼 “ 𝑆))
69		hllat 39363	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
70	69	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝐾 ∈ Lat)
71	66	adantr 480	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ (Base‘𝐾))
72	64	sselda 3949	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝑦 ∈ (Base‘𝐾))
73	58, 1	lhpbase 39999	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
74	73	ad3antlr 731	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝑊 ∈ (Base‘𝐾))
75	56	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝐾 ∈ CLat)
76	60	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → dom 𝐼 = {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊})
77	7, 76	sseqtrid 3992	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ⊆ {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊})
78	77, 61	sstrdi 3962	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ⊆ (Base‘𝐾))
79	78	adantr 480	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → (^◡𝐼 “ 𝑆) ⊆ (Base‘𝐾))
80		simpr 484	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝑦 ∈ (^◡𝐼 “ 𝑆))
81	58, 59, 50	clatglble 18483	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ (^◡𝐼 “ 𝑆) ⊆ (Base‘𝐾) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑦)
82	75, 79, 80, 81	syl3anc 1373	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑦)
83	7	sseli 3945	. . . . . . . . . 10 ⊢ (𝑦 ∈ (^◡𝐼 “ 𝑆) → 𝑦 ∈ dom 𝐼)
84	83	adantl 481	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝑦 ∈ dom 𝐼)
85	58, 59, 1, 2	diaeldm 41037	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑦 ∈ dom 𝐼 ↔ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)))
86	85	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → (𝑦 ∈ dom 𝐼 ↔ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)))
87	84, 86	mpbid 232	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊))
88	87	simprd 495	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝑦(le‘𝐾)𝑊)
89	58, 59, 70, 71, 72, 74, 82, 88	lattrd 18412	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑊)
90	68, 89	exlimddv 1935	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑊)
91	58, 59, 1, 2	diaeldm 41037	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ dom 𝐼 ↔ (((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ (Base‘𝐾) ∧ ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑊)))
92	91	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ dom 𝐼 ↔ (((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ (Base‘𝐾) ∧ ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑊)))
93	66, 90, 92	mpbir2and 713	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ dom 𝐼)
94	1, 2	diaclN 41051	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ dom 𝐼) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) ∈ ran 𝐼)
95	93, 94	syldan 591	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) ∈ ran 𝐼)
96	55, 95	eqeltrrd 2830	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) ∈ ran 𝐼)
97	21, 96	eqeltrrd 2830	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑆 ∈ ran 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2926 ∃wrex 3054 {crab 3408 ⊆ wss 3917 ∅c0 4299 ∩ cint 4913 ∩ ciin 4959 class class class wbr 5110 ^◡ccnv 5640 dom cdm 5641 ran crn 5642 ↾ cres 5643 “ cima 5644 Fun wfun 6508 Fn wfn 6509 –onto→wfo 6512 –1-1-onto→wf1o 6513 ‘cfv 6514 Basecbs 17186 lecple 17234 glbcglb 18278 Latclat 18397 CLatccla 18464 HLchlt 39350 LHypclh 39985 DIsoAcdia 41029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-riotaBAD 38953

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-undef 8255 df-map 8804 df-proset 18262 df-poset 18281 df-plt 18296 df-lub 18312 df-glb 18313 df-join 18314 df-meet 18315 df-p0 18391 df-p1 18392 df-lat 18398 df-clat 18465 df-oposet 39176 df-ol 39178 df-oml 39179 df-covers 39266 df-ats 39267 df-atl 39298 df-cvlat 39322 df-hlat 39351 df-llines 39499 df-lplanes 39500 df-lvols 39501 df-lines 39502 df-psubsp 39504 df-pmap 39505 df-padd 39797 df-lhyp 39989 df-laut 39990 df-ldil 40105 df-ltrn 40106 df-trl 40160 df-disoa 41030

This theorem is referenced by: docaclN 41125

W3C validator