Theorem dibglbN 41168

Description: Partial isomorphism B of a lattice glb. (Contributed by NM, 9-Mar-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dibglb.g	⊢ 𝐺 = (glb‘𝐾)
dibglb.h	⊢ 𝐻 = (LHyp‘𝐾)
dibglb.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)

Assertion

Ref	Expression
dibglbN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))

Distinct variable groups:   𝑥,𝐺   𝑥,𝐻   𝑥,𝐾   𝑥,𝑆   𝑥,𝑊

Allowed substitution hint:   𝐼(𝑥)

Proof of Theorem dibglbN

Dummy variables 𝑓 𝑠 ℎ 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simprl 771	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ dom 𝐼)
3		eqid 2737	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2737	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
5		dibglb.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
6		dibglb.i	. . . . . 6 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
7	3, 4, 5, 6	dibdmN 41159	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
8	7	sseq2d 4016	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑆 ⊆ dom 𝐼 ↔ 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊}))
9	8	adantr 480	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝑆 ⊆ dom 𝐼 ↔ 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊}))
10	2, 9	mpbid 232	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
11		simprr 773	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ≠ ∅)
12	5, 6	dibvalrel 41165	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘(𝐺‘𝑆)))
13	12	adantr 480	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → Rel (𝐼‘(𝐺‘𝑆)))
14		n0 4353	. . . . . . . 8 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆)
15	14	biimpi 216	. . . . . . 7 ⊢ (𝑆 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝑆)
16	15	ad2antll 729	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → ∃𝑥 𝑥 ∈ 𝑆)
17	5, 6	dibvalrel 41165	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑥))
18	17	adantr 480	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → Rel (𝐼‘𝑥))
19	18	a1d 25	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝑥 ∈ 𝑆 → Rel (𝐼‘𝑥)))
20	19	ancld 550	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝑥 ∈ 𝑆 → (𝑥 ∈ 𝑆 ∧ Rel (𝐼‘𝑥))))
21	20	eximdv 1917	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (∃𝑥 𝑥 ∈ 𝑆 → ∃𝑥(𝑥 ∈ 𝑆 ∧ Rel (𝐼‘𝑥))))
22	16, 21	mpd 15	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → ∃𝑥(𝑥 ∈ 𝑆 ∧ Rel (𝐼‘𝑥)))
23		df-rex 3071	. . . . 5 ⊢ (∃𝑥 ∈ 𝑆 Rel (𝐼‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑆 ∧ Rel (𝐼‘𝑥)))
24	22, 23	sylibr 234	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → ∃𝑥 ∈ 𝑆 Rel (𝐼‘𝑥))
25		reliin 5827	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 Rel (𝐼‘𝑥) → Rel ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))
26	24, 25	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → Rel ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))
27		id 22	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)))
28		simpl 482	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
29		simprl 771	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
30		eqid 2737	. . . . . . . . . . . . 13 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
31	3, 4, 5, 30	diadm 41037	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom ((DIsoA‘𝐾)‘𝑊) = {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
32	31	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → dom ((DIsoA‘𝐾)‘𝑊) = {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
33	29, 32	sseqtrrd 4021	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ dom ((DIsoA‘𝐾)‘𝑊))
34		simprr 773	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝑆 ≠ ∅)
35		dibglb.g	. . . . . . . . . . 11 ⊢ 𝐺 = (glb‘𝐾)
36	35, 5, 30	diaglbN 41057	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom ((DIsoA‘𝐾)‘𝑊) ∧ 𝑆 ≠ ∅)) → (((DIsoA‘𝐾)‘𝑊)‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (((DIsoA‘𝐾)‘𝑊)‘𝑥))
37	28, 33, 34, 36	syl12anc 837	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (((DIsoA‘𝐾)‘𝑊)‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (((DIsoA‘𝐾)‘𝑊)‘𝑥))
38	37	eleq2d 2827	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝐺‘𝑆)) ↔ 𝑓 ∈ ∩ 𝑥 ∈ 𝑆 (((DIsoA‘𝐾)‘𝑊)‘𝑥)))
39		vex 3484	. . . . . . . . 9 ⊢ 𝑓 ∈ V
40		eliin 4996	. . . . . . . . 9 ⊢ (𝑓 ∈ V → (𝑓 ∈ ∩ 𝑥 ∈ 𝑆 (((DIsoA‘𝐾)‘𝑊)‘𝑥) ↔ ∀𝑥 ∈ 𝑆 𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑥)))
41	39, 40	ax-mp 5	. . . . . . . 8 ⊢ (𝑓 ∈ ∩ 𝑥 ∈ 𝑆 (((DIsoA‘𝐾)‘𝑊)‘𝑥) ↔ ∀𝑥 ∈ 𝑆 𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑥))
42	38, 41	bitrdi 287	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝐺‘𝑆)) ↔ ∀𝑥 ∈ 𝑆 𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑥)))
43	42	anbi1d 631	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝐺‘𝑆)) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))) ↔ (∀𝑥 ∈ 𝑆 𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑥) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))))
44		r19.27zv 4506	. . . . . . 7 ⊢ (𝑆 ≠ ∅ → (∀𝑥 ∈ 𝑆 (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑥) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))) ↔ (∀𝑥 ∈ 𝑆 𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑥) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))))
45	44	ad2antll 729	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (∀𝑥 ∈ 𝑆 (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑥) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))) ↔ (∀𝑥 ∈ 𝑆 𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑥) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))))
46	43, 45	bitr4d 282	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝐺‘𝑆)) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))) ↔ ∀𝑥 ∈ 𝑆 (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑥) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))))
47		hlclat 39359	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
48	47	ad2antrr 726	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝐾 ∈ CLat)
49		ssrab2 4080	. . . . . . . 8 ⊢ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ⊆ (Base‘𝐾)
50	29, 49	sstrdi 3996	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ (Base‘𝐾))
51	3, 35	clatglbcl 18550	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾)) → (𝐺‘𝑆) ∈ (Base‘𝐾))
52	48, 50, 51	syl2anc 584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐺‘𝑆) ∈ (Base‘𝐾))
53		hllat 39364	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
54	53	ad3antrrr 730	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝐾 ∈ Lat)
55	47	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝐾 ∈ CLat)
56		simplrl 777	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
57	56, 49	sstrdi 3996	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐾))
58	55, 57, 51	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆) ∈ (Base‘𝐾))
59	50	sselda 3983	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐾))
60	3, 5	lhpbase 40000	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
61	60	ad3antlr 731	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑊 ∈ (Base‘𝐾))
62		simpr 484	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
63	3, 4, 35	clatglble 18562	. . . . . . . . 9 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆)(le‘𝐾)𝑥)
64	55, 57, 62, 63	syl3anc 1373	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆)(le‘𝐾)𝑥)
65	29	sselda 3983	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
66		breq1 5146	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦(le‘𝐾)𝑊 ↔ 𝑥(le‘𝐾)𝑊))
67	66	elrab 3692	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊))
68	65, 67	sylib 218	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊))
69	68	simprd 495	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑥(le‘𝐾)𝑊)
70	3, 4, 54, 58, 59, 61, 64, 69	lattrd 18491	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆)(le‘𝐾)𝑊)
71	16, 70	exlimddv 1935	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐺‘𝑆)(le‘𝐾)𝑊)
72		eqid 2737	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
73		eqid 2737	. . . . . . 7 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))
74	3, 4, 5, 72, 73, 30, 6	dibopelval2 41147	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐺‘𝑆) ∈ (Base‘𝐾) ∧ (𝐺‘𝑆)(le‘𝐾)𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝐺‘𝑆)) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝐺‘𝑆)) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))))
75	28, 52, 71, 74	syl12anc 837	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝐺‘𝑆)) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝐺‘𝑆)) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))))
76		opex 5469	. . . . . . 7 ⊢ ⟨𝑓, 𝑠⟩ ∈ V
77		eliin 4996	. . . . . . 7 ⊢ (⟨𝑓, 𝑠⟩ ∈ V → (⟨𝑓, 𝑠⟩ ∈ ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥) ↔ ∀𝑥 ∈ 𝑆 ⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑥)))
78	76, 77	ax-mp 5	. . . . . 6 ⊢ (⟨𝑓, 𝑠⟩ ∈ ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥) ↔ ∀𝑥 ∈ 𝑆 ⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑥))
79		simpll 767	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
80	3, 4, 5, 72, 73, 30, 6	dibopelval2 41147	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑥) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑥) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))))
81	79, 68, 80	syl2anc 584	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑥) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑥) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))))
82	81	ralbidva 3176	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (∀𝑥 ∈ 𝑆 ⟨𝑓, 𝑠⟩ ∈ (𝐼‘𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑥) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))))
83	78, 82	bitrid 283	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (⟨𝑓, 𝑠⟩ ∈ ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑥) ∧ 𝑠 = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))))
84	46, 75, 83	3bitr4d 311	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (⟨𝑓, 𝑠⟩ ∈ (𝐼‘(𝐺‘𝑆)) ↔ ⟨𝑓, 𝑠⟩ ∈ ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥)))
85	84	eqrelrdv2 5805	. . 3 ⊢ (((Rel (𝐼‘(𝐺‘𝑆)) ∧ Rel ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥)) ∧ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅))) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))
86	13, 26, 27, 85	syl21anc 838	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))
87	1, 10, 11, 86	syl12anc 837	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480   ⊆ wss 3951  ∅c0 4333  ⟨cop 4632  ∩ ciin 4992   class class class wbr 5143   ↦ cmpt 5225   I cid 5577  dom cdm 5685   ↾ cres 5687  Rel wrel 5690  ‘cfv 6561  Basecbs 17247  lecple 17304  glbcglb 18356  Latclat 18476  CLatccla 18543  HLchlt 39351  LHypclh 39986  LTrncltrn 40103  DIsoAcdia 41030  DIsoBcdib 41140

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-lhyp 39990  df-laut 39991  df-ldil 40106  df-ltrn 40107  df-trl 40161  df-disoa 41031  df-dib 41141

This theorem is referenced by:  dibintclN  41169  dihglblem3N  41297  dihmeetlem2N  41301