diaintclN - Mathbox for Norm Megill

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Theorem diaintclN 41683

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 41674	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼)
4	3	adantr 484	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼)
5		f1ofn 6808	. . . . . 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 6072	. . . . 5 ⊢ (^◡𝐼 “ 𝑆) ⊆ dom 𝐼
8		fnssres 6645	. . . . 5 ⊢ ((𝐼 Fn dom 𝐼 ∧ (^◡𝐼 “ 𝑆) ⊆ dom 𝐼) → (𝐼 ↾ (^◡𝐼 “ 𝑆)) Fn (^◡𝐼 “ 𝑆))
9	6, 7, 8	sylancl 595	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼 ↾ (^◡𝐼 “ 𝑆)) Fn (^◡𝐼 “ 𝑆))
10		fniinfv 6946	. . . 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 5661	. . . . 5 ⊢ (𝐼 “ (^◡𝐼 “ 𝑆)) = ran (𝐼 ↾ (^◡𝐼 “ 𝑆))
13		f1ofo 6815	. . . . . . . 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 484	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝐼:dom 𝐼–onto→ran 𝐼)
16		simprl 780	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ ran 𝐼)
17		foimacnv 6825	. . . . . 6 ⊢ ((𝐼:dom 𝐼–onto→ran 𝐼 ∧ 𝑆 ⊆ ran 𝐼) → (𝐼 “ (^◡𝐼 “ 𝑆)) = 𝑆)
18	15, 16, 17	syl2anc 593	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼 “ (^◡𝐼 “ 𝑆)) = 𝑆)
19	12, 18	eqtr3id 2812	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ran (𝐼 ↾ (^◡𝐼 “ 𝑆)) = 𝑆)
20	19	inteqd 4911	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ ran (𝐼 ↾ (^◡𝐼 “ 𝑆)) = ∩ 𝑆)
21	11, 20	eqtrd 2798	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) = ∩ 𝑆)
22		simpl 486	. . . . 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 782	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ≠ ∅)
25		n0 4306	. . . . . . 7 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑆)
26	24, 25	sylib 220	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∃𝑦 𝑦 ∈ 𝑆)
27	16	sselda 3937	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ran 𝐼)
28	3	ad2antrr 736	. . . . . . . . . 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 6928	. . . . . . . . 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 234	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → ∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝑦)
33		f1ofun 6809	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → Fun 𝐼)
36		fvimacnv 7035	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐼 ∧ 𝑥 ∈ dom 𝐼) → ((𝐼‘𝑥) ∈ 𝑆 ↔ 𝑥 ∈ (^◡𝐼 “ 𝑆)))
37	35, 36	sylan 589	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ dom 𝐼) → ((𝐼‘𝑥) ∈ 𝑆 ↔ 𝑥 ∈ (^◡𝐼 “ 𝑆)))
38		ne0i 4294	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (^◡𝐼 “ 𝑆) → (^◡𝐼 “ 𝑆) ≠ ∅)
39	37, 38	biimtrdi 255	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ dom 𝐼) → ((𝐼‘𝑥) ∈ 𝑆 → (^◡𝐼 “ 𝑆) ≠ ∅))
40	39	ex 416	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝑥 ∈ dom 𝐼 → ((𝐼‘𝑥) ∈ 𝑆 → (^◡𝐼 “ 𝑆) ≠ ∅)))
41		eleq1 2851	. . . . . . . . . . . . 13 ⊢ ((𝐼‘𝑥) = 𝑦 → ((𝐼‘𝑥) ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
42	41	biimprd 250	. . . . . . . . . . . 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 410	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → (𝑥 ∈ dom 𝐼 → ((𝐼‘𝑥) = 𝑦 → (^◡𝐼 “ 𝑆) ≠ ∅)))
47	46	rexlimdv 3162	. . . . . . 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 1956	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ≠ ∅)
50		eqid 2763	. . . . . 6 ⊢ (glb‘𝐾) = (glb‘𝐾)
51	50, 1, 2	diaglbN 41680	. . . . 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 847	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) = ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)(𝐼‘𝑦))
53		fvres 6887	. . . . 5 ⊢ (𝑦 ∈ (^◡𝐼 “ 𝑆) → ((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) = (𝐼‘𝑦))
54	53	iineq2i 4973	. . . 4 ⊢ ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) = ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)(𝐼‘𝑦)
55	52, 54	eqtr4di 2816	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) = ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦))
56		hlclat 39983	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
57	56	ad2antrr 736	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝐾 ∈ CLat)
58		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
59		eqid 2763	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
60	58, 59, 1, 2	diadm 41660	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊})
61		ssrab2 4034	. . . . . . . . 9 ⊢ {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊} ⊆ (Base‘𝐾)
62	60, 61	eqsstrdi 3981	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom 𝐼 ⊆ (Base‘𝐾))
63	62	adantr 484	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → dom 𝐼 ⊆ (Base‘𝐾))
64	7, 63	sstrid 3948	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ⊆ (Base‘𝐾))
65	58, 50	clatglbcl 18538	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ (^◡𝐼 “ 𝑆) ⊆ (Base‘𝐾)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ (Base‘𝐾))
66	57, 64, 65	syl2anc 593	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ (Base‘𝐾))
67		n0 4306	. . . . . . 7 ⊢ ((^◡𝐼 “ 𝑆) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (^◡𝐼 “ 𝑆))
68	49, 67	sylib 220	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∃𝑦 𝑦 ∈ (^◡𝐼 “ 𝑆))
69		hllat 39988	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
70	69	ad3antrrr 740	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝐾 ∈ Lat)
71	66	adantr 484	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ (Base‘𝐾))
72	64	sselda 3937	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝑦 ∈ (Base‘𝐾))
73	58, 1	lhpbase 40623	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
74	73	ad3antlr 741	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝑊 ∈ (Base‘𝐾))
75	56	ad3antrrr 740	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝐾 ∈ CLat)
76	60	adantr 484	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → dom 𝐼 = {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊})
77	7, 76	sseqtrid 3979	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ⊆ {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊})
78	77, 61	sstrdi 3949	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ⊆ (Base‘𝐾))
79	78	adantr 484	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → (^◡𝐼 “ 𝑆) ⊆ (Base‘𝐾))
80		simpr 488	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝑦 ∈ (^◡𝐼 “ 𝑆))
81	58, 59, 50	clatglble 18550	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ (^◡𝐼 “ 𝑆) ⊆ (Base‘𝐾) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑦)
82	75, 79, 80, 81	syl3anc 1391	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑦)
83	7	sseli 3933	. . . . . . . . . 10 ⊢ (𝑦 ∈ (^◡𝐼 “ 𝑆) → 𝑦 ∈ dom 𝐼)
84	83	adantl 485	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝑦 ∈ dom 𝐼)
85	58, 59, 1, 2	diaeldm 41661	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑦 ∈ dom 𝐼 ↔ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)))
86	85	ad2antrr 736	. . . . . . . . 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 234	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊))
88	87	simprd 499	. . . . . . 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 18479	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑊)
90	68, 89	exlimddv 1956	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑊)
91	58, 59, 1, 2	diaeldm 41661	. . . . . 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 484	. . . . 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 723	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ dom 𝐼)
94	1, 2	diaclN 41675	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ dom 𝐼) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) ∈ ran 𝐼)
95	93, 94	syldan 600	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) ∈ ran 𝐼)
96	55, 95	eqeltrrd 2864	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) ∈ ran 𝐼)
97	21, 96	eqeltrrd 2864	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑆 ∈ ran 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1561 ∃wex 1800 ∈ wcel 2143 ≠ wne 2958 ∃wrex 3087 {crab 3415 ⊆ wss 3905 ∅c0 4286 ∩ cint 4906 ∩ ciin 4951 class class class wbr 5101 ^◡ccnv 5647 dom cdm 5648 ran crn 5649 ↾ cres 5650 “ cima 5651 Fun wfun 6516 Fn wfn 6517 –onto→wfo 6520 –1-1-onto→wf1o 6521 ‘cfv 6522 Basecbs 17246 lecple 17294 glbcglb 18343 Latclat 18464 CLatccla 18531 HLchlt 39975 LHypclh 40609 DIsoAcdia 41653

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-riotaBAD 39578

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5102 df-opab 5164 df-mpt 5183 df-id 5543 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-1st 7971 df-2nd 7972 df-undef 8254 df-map 8811 df-proset 18327 df-poset 18346 df-plt 18361 df-lub 18377 df-glb 18378 df-join 18379 df-meet 18380 df-p0 18456 df-p1 18457 df-lat 18465 df-clat 18532 df-oposet 39801 df-ol 39803 df-oml 39804 df-covers 39891 df-ats 39892 df-atl 39923 df-cvlat 39947 df-hlat 39976 df-llines 40123 df-lplanes 40124 df-lvols 40125 df-lines 40126 df-psubsp 40128 df-pmap 40129 df-padd 40421 df-lhyp 40613 df-laut 40614 df-ldil 40729 df-ltrn 40730 df-trl 40784 df-disoa 41654

This theorem is referenced by: docaclN 41749

W3C validator