dibintclN - Mathbox for Norm Megill

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Theorem dibintclN 41613

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 41607	. . . . . . 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 6781	. . . . . 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 6047	. . . . 5 ⊢ (^◡𝐼 “ 𝑆) ⊆ dom 𝐼
8		fnssres 6621	. . . . 5 ⊢ ((𝐼 Fn dom 𝐼 ∧ (^◡𝐼 “ 𝑆) ⊆ dom 𝐼) → (𝐼 ↾ (^◡𝐼 “ 𝑆)) Fn (^◡𝐼 “ 𝑆))
9	6, 7, 8	sylancl 587	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼 ↾ (^◡𝐼 “ 𝑆)) Fn (^◡𝐼 “ 𝑆))
10		fniinfv 6918	. . . 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 5644	. . . . 5 ⊢ (𝐼 “ (^◡𝐼 “ 𝑆)) = ran (𝐼 ↾ (^◡𝐼 “ 𝑆))
13		f1ofo 6787	. . . . . . . 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 771	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ ran 𝐼)
17		foimacnv 6797	. . . . . 6 ⊢ ((𝐼:dom 𝐼–onto→ran 𝐼 ∧ 𝑆 ⊆ ran 𝐼) → (𝐼 “ (^◡𝐼 “ 𝑆)) = 𝑆)
18	15, 16, 17	syl2anc 585	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼 “ (^◡𝐼 “ 𝑆)) = 𝑆)
19	12, 18	eqtr3id 2785	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ran (𝐼 ↾ (^◡𝐼 “ 𝑆)) = 𝑆)
20	19	inteqd 4894	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ ran (𝐼 ↾ (^◡𝐼 “ 𝑆)) = ∩ 𝑆)
21	11, 20	eqtrd 2771	. 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 773	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ≠ ∅)
25		n0 4293	. . . . . . 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 3921	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ran 𝐼)
28	3	ad2antrr 727	. . . . . . . . . 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 6900	. . . . . . . . 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 6782	. . . . . . . . . . . . . . . 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 7005	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐼 ∧ 𝑥 ∈ dom 𝐼) → ((𝐼‘𝑥) ∈ 𝑆 ↔ 𝑥 ∈ (^◡𝐼 “ 𝑆)))
37	35, 36	sylan 581	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ dom 𝐼) → ((𝐼‘𝑥) ∈ 𝑆 ↔ 𝑥 ∈ (^◡𝐼 “ 𝑆)))
38		ne0i 4281	. . . . . . . . . . . . 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 2824	. . . . . . . . . . . . 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 3136	. . . . . . 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 1937	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ≠ ∅)
50		eqid 2736	. . . . . 6 ⊢ (glb‘𝐾) = (glb‘𝐾)
51	50, 1, 2	dibglbN 41612	. . . . 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 837	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) = ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)(𝐼‘𝑦))
53		fvres 6859	. . . . 5 ⊢ (𝑦 ∈ (^◡𝐼 “ 𝑆) → ((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) = (𝐼‘𝑦))
54	53	iineq2i 4956	. . . 4 ⊢ ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) = ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)(𝐼‘𝑦)
55	52, 54	eqtr4di 2789	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) = ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦))
56		hlclat 39804	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
57	56	ad2antrr 727	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → 𝐾 ∈ CLat)
58		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
59		eqid 2736	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
60	58, 59, 1, 2	dibdmN 41603	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊})
61		ssrab2 4020	. . . . . . . . 9 ⊢ {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊} ⊆ (Base‘𝐾)
62	60, 61	eqsstrdi 3966	. . . . . . . 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 3933	. . . . . 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 585	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ (Base‘𝐾))
67		n0 4293	. . . . . . 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 39809	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
70	69	ad3antrrr 731	. . . . . . 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 3921	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝑦 ∈ (Base‘𝐾))
73	58, 1	lhpbase 40444	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
74	73	ad3antlr 732	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → 𝑊 ∈ (Base‘𝐾))
75	56	ad3antrrr 731	. . . . . . . 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 3964	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (^◡𝐼 “ 𝑆) ⊆ {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊})
78	77, 61	sstrdi 3934	. . . . . . . . 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 1374	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) ∧ 𝑦 ∈ (^◡𝐼 “ 𝑆)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑦)
83	7	sseli 3917	. . . . . . . . . 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 41604	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑦 ∈ dom 𝐼 ↔ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)))
86	85	ad2antrr 727	. . . . . . . . 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 1937	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆))(le‘𝐾)𝑊)
91	58, 59, 1, 2	dibeldmN 41604	. . . . . 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 714	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ dom 𝐼)
94	1, 2	dibclN 41608	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((glb‘𝐾)‘(^◡𝐼 “ 𝑆)) ∈ dom 𝐼) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) ∈ ran 𝐼)
95	93, 94	syldan 592	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘((glb‘𝐾)‘(^◡𝐼 “ 𝑆))) ∈ ran 𝐼)
96	55, 95	eqeltrrd 2837	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑦 ∈ (^◡𝐼 “ 𝑆)((𝐼 ↾ (^◡𝐼 “ 𝑆))‘𝑦) ∈ ran 𝐼)
97	21, 96	eqeltrrd 2837	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 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2932 ∃wrex 3061 {crab 3389 ⊆ wss 3889 ∅c0 4273 ∩ cint 4889 ∩ ciin 4934 class class class wbr 5085 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 ↾ cres 5633 “ cima 5634 Fun wfun 6492 Fn wfn 6493 –onto→wfo 6496 –1-1-onto→wf1o 6497 ‘cfv 6498 Basecbs 17179 lecple 17227 glbcglb 18276 Latclat 18397 CLatccla 18464 HLchlt 39796 LHypclh 40430 DIsoBcdib 41584

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-riotaBAD 39399

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-undef 8223 df-map 8775 df-proset 18260 df-poset 18279 df-plt 18294 df-lub 18310 df-glb 18311 df-join 18312 df-meet 18313 df-p0 18389 df-p1 18390 df-lat 18398 df-clat 18465 df-oposet 39622 df-ol 39624 df-oml 39625 df-covers 39712 df-ats 39713 df-atl 39744 df-cvlat 39768 df-hlat 39797 df-llines 39944 df-lplanes 39945 df-lvols 39946 df-lines 39947 df-psubsp 39949 df-pmap 39950 df-padd 40242 df-lhyp 40434 df-laut 40435 df-ldil 40550 df-ltrn 40551 df-trl 40605 df-disoa 41475 df-dib 41585

This theorem is referenced by: (None)

W3C validator