dibintclN - Mathbox for Norm Megill

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Theorem dibintclN 39108

Description: The intersection of partial isomorphism B closed subspaces is a closed subspace. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

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

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

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

Step	Hyp	Ref	Expression
1		dibintcl.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
2		dibintcl.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
3	1, 2	dibf11N 39102	. . . . . . 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 6701	. . . . . 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 5978	. . . . 5 ⊢ (^◡𝐼 “ 𝑆) ⊆ dom 𝐼
8		fnssres 6539	. . . . 5 ⊢ ((𝐼 Fn dom 𝐼 ∧ (^◡𝐼 “ 𝑆) ⊆ dom 𝐼) → (𝐼 ↾ (^◡𝐼 “ 𝑆)) Fn (^◡𝐼 “ 𝑆))
9	6, 7, 8	sylancl 585	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼 ↾ (^◡𝐼 “ 𝑆)) Fn (^◡𝐼 “ 𝑆))
10		fniinfv 6828	. . . 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 5593	. . . . 5 ⊢ (𝐼 “ (^◡𝐼 “ 𝑆)) = ran (𝐼 ↾ (^◡𝐼 “ 𝑆))
13		f1ofo 6707	. . . . . . . 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 767	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ ran 𝐼)
17		foimacnv 6717	. . . . . 6 ⊢ ((𝐼:dom 𝐼–onto→ran 𝐼 ∧ 𝑆 ⊆ ran 𝐼) → (𝐼 “ (^◡𝐼 “ 𝑆)) = 𝑆)
18	15, 16, 17	syl2anc 583	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼 “ (^◡𝐼 “ 𝑆)) = 𝑆)
19	12, 18	eqtr3id 2793	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ran (𝐼 ↾ (^◡𝐼 “ 𝑆)) = 𝑆)
20	19	inteqd 4881	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ ran (𝐼 ↾ (^◡𝐼 “ 𝑆)) = ∩ 𝑆)
21	11, 20	eqtrd 2778	. 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 769	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ≠ ∅)
25		n0 4277	. . . . . . 7 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝑆)
26	24, 25	sylib 217	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∃𝑦 𝑦 ∈ 𝑆)
27	16	sselda 3917	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ran 𝐼)
28	3	ad2antrr 722	. . . . . . . . . 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 6812	. . . . . . . . 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 231	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → ∃𝑥 ∈ dom 𝐼(𝐼‘𝑥) = 𝑦)
33		f1ofun 6702	. . . . . . . . . . . . . . . 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 6912	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐼 ∧ 𝑥 ∈ dom 𝐼) → ((𝐼‘𝑥) ∈ 𝑆 ↔ 𝑥 ∈ (^◡𝐼 “ 𝑆)))
37	35, 36	sylan 579	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ dom 𝐼) → ((𝐼‘𝑥) ∈ 𝑆 ↔ 𝑥 ∈ (^◡𝐼 “ 𝑆)))
38		ne0i 4265	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (^◡𝐼 “ 𝑆) → (^◡𝐼 “ 𝑆) ≠ ∅)
39	37, 38	syl6bi 252	. . . . . . . . . . . 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 2826	. . . . . . . . . . . . 13 ⊢ ((𝐼‘𝑥) = 𝑦 → ((𝐼‘𝑥) ∈ 𝑆 ↔ 𝑦 ∈ 𝑆))
42	41	biimprd 247	. . . . . . . . . . . 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 3211	. . . . . . 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 1939	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ≠ ∅)
50		eqid 2738	. . . . . 6 ⊢ (glb‘𝐾) = (glb‘𝐾)
51	50, 1, 2	dibglbN 39107	. . . . 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 833	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) = ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)(𝐼‘𝑦))
53		fvres 6775	. . . . 5 ⊢ (𝑦 ∈ (^◡𝐼 “ 𝑆) → ((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) = (𝐼‘𝑦))
54	53	iineq2i 4943	. . . 4 ⊢ ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) = ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)(𝐼‘𝑦)
55	52, 54	eqtr4di 2797	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) = ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦))
56		hlclat 37299	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
57	56	ad2antrr 722	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝐾 ∈ CLat)
58		eqid 2738	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
59		eqid 2738	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
60	58, 59, 1, 2	dibdmN 39098	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊})
61		ssrab2 4009	. . . . . . . . 9 ⊢ {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊} ⊆ (Base‘𝐾)
62	60, 61	eqsstrdi 3971	. . . . . . . 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 3928	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ⊆ (Base‘𝐾))
65	58, 50	clatglbcl 18138	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ (^◡𝐼 “ 𝑆) ⊆ (Base‘𝐾)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ (Base‘𝐾))
66	57, 64, 65	syl2anc 583	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ (Base‘𝐾))
67		n0 4277	. . . . . . 7 ⊢ ((^◡𝐼 “ 𝑆) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (^◡𝐼 “ 𝑆))
68	49, 67	sylib 217	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∃𝑦 𝑦 ∈ (^◡𝐼 “ 𝑆))
69		hllat 37304	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
70	69	ad3antrrr 726	. . . . . . 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 3917	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝑦 ∈ (Base‘𝐾))
73	58, 1	lhpbase 37939	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
74	73	ad3antlr 727	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝑊 ∈ (Base‘𝐾))
75	56	ad3antrrr 726	. . . . . . . 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 3969	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ⊆ {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊})
78	77, 61	sstrdi 3929	. . . . . . . . 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 18150	. . . . . . . 8 ⊢ ((𝐾 ∈ CLat ∧ (^◡𝐼 “ 𝑆) ⊆ (Base‘𝐾) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑦)
82	75, 79, 80, 81	syl3anc 1369	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑦)
83	7	sseli 3913	. . . . . . . . . 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	dibeldmN 39099	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑦 ∈ dom 𝐼 ↔ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)))
86	85	ad2antrr 722	. . . . . . . . 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 231	. . . . . . . 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 18079	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑊)
90	68, 89	exlimddv 1939	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑊)
91	58, 59, 1, 2	dibeldmN 39099	. . . . . 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 709	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ dom 𝐼)
94	1, 2	dibclN 39103	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ dom 𝐼) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) ∈ ran 𝐼)
95	93, 94	syldan 590	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) ∈ ran 𝐼)
96	55, 95	eqeltrrd 2840	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) ∈ ran 𝐼)
97	21, 96	eqeltrrd 2840	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑆 ∈ ran 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∃wex 1783 ∈ wcel 2108 ≠ wne 2942 ∃wrex 3064 {crab 3067 ⊆ wss 3883 ∅c0 4253 ∩ cint 4876 ∩ ciin 4922 class class class wbr 5070 ^◡ccnv 5579 dom cdm 5580 ran crn 5581 ↾ cres 5582 “ cima 5583 Fun wfun 6412 Fn wfn 6413 –onto→wfo 6416 –1-1-onto→wf1o 6417 ‘cfv 6418 Basecbs 16840 lecple 16895 glbcglb 17943 Latclat 18064 CLatccla 18131 HLchlt 37291 LHypclh 37925 DIsoBcdib 39079

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-undef 8060 df-map 8575 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-disoa 38970 df-dib 39080

This theorem is referenced by: (None)

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