Theorem dibglbN 41542

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 41533	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
8	7	sseq2d 3968	. . . 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 41539	. . . 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 4307	. . . . . . . 8 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆)
15	14	biimpi 216	. . . . . . 7 ⊢ (𝑆 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝑆)
16	15	ad2antll 730	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → ∃𝑥 𝑥 ∈ 𝑆)
17	5, 6	dibvalrel 41539	. . . . . . . . . 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 1919	. . . . . 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 3063	. . . . 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 5774	. . . 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 41411	. . . . . . . . . . . 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 3973	. . . . . . . . . 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 41431	. . . . . . . . . 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 2823	. . . . . . . 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 3446	. . . . . . . . 9 ⊢ 𝑓 ∈ V
40		eliin 4953	. . . . . . . . 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 632	. . . . . 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 4466	. . . . . . 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 730	. . . . . 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 39734	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
48	47	ad2antrr 727	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝐾 ∈ CLat)
49		ssrab2 4034	. . . . . . . 8 ⊢ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ⊆ (Base‘𝐾)
50	29, 49	sstrdi 3948	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ (Base‘𝐾))
51	3, 35	clatglbcl 18440	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾)) → (𝐺‘𝑆) ∈ (Base‘𝐾))
52	48, 50, 51	syl2anc 585	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐺‘𝑆) ∈ (Base‘𝐾))
53		hllat 39739	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
54	53	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝐾 ∈ Lat)
55	47	ad3antrrr 731	. . . . . . . . 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 3948	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐾))
58	55, 57, 51	syl2anc 585	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆) ∈ (Base‘𝐾))
59	50	sselda 3935	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐾))
60	3, 5	lhpbase 40374	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
61	60	ad3antlr 732	. . . . . . . 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 18452	. . . . . . . . 9 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆)(le‘𝐾)𝑥)
64	55, 57, 62, 63	syl3anc 1374	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆)(le‘𝐾)𝑥)
65	29	sselda 3935	. . . . . . . . . 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 5103	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦(le‘𝐾)𝑊 ↔ 𝑥(le‘𝐾)𝑊))
67	66	elrab 3648	. . . . . . . . . 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 18381	. . . . . . 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 1937	. . . . . 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 41521	. . . . . 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 5419	. . . . . . 7 ⊢ ⟨𝑓, 𝑠⟩ ∈ V
77		eliin 4953	. . . . . . 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 41521	. . . . . . . 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 585	. . . . . . 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 3159	. . . . . 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 5752	. . 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 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442   ⊆ wss 3903  ∅c0 4287  ⟨cop 4588  ∩ ciin 4949   class class class wbr 5100   ↦ cmpt 5181   I cid 5526  dom cdm 5632   ↾ cres 5634  Rel wrel 5637  ‘cfv 6500  Basecbs 17148  lecple 17196  glbcglb 18245  Latclat 18366  CLatccla 18433  HLchlt 39726  LHypclh 40360  LTrncltrn 40477  DIsoAcdia 41404  DIsoBcdib 41514

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-map 8777  df-proset 18229  df-poset 18248  df-plt 18263  df-lub 18279  df-glb 18280  df-join 18281  df-meet 18282  df-p0 18358  df-p1 18359  df-lat 18367  df-clat 18434  df-oposet 39552  df-ol 39554  df-oml 39555  df-covers 39642  df-ats 39643  df-atl 39674  df-cvlat 39698  df-hlat 39727  df-lhyp 40364  df-laut 40365  df-ldil 40480  df-ltrn 40481  df-trl 40535  df-disoa 41405  df-dib 41515

This theorem is referenced by:  dibintclN  41543  dihglblem3N  41671  dihmeetlem2N  41675