Theorem dibglbN 41864

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 487	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simprl 782	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ dom 𝐼)
3		eqid 2769	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2769	. . . . . 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 41855	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
8	7	sseq2d 3977	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑆 ⊆ dom 𝐼 ↔ 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊}))
9	8	adantr 485	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝑆 ⊆ dom 𝐼 ↔ 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊}))
10	2, 9	mpbid 235	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
11		simprr 784	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ≠ ∅)
12	5, 6	dibvalrel 41861	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘(𝐺‘𝑆)))
13	12	adantr 485	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → Rel (𝐼‘(𝐺‘𝑆)))
14		n0 4315	. . . . . . . 8 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆)
15	14	biimpi 219	. . . . . . 7 ⊢ (𝑆 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝑆)
16	15	ad2antll 741	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → ∃𝑥 𝑥 ∈ 𝑆)
17	5, 6	dibvalrel 41861	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑥))
18	17	adantr 485	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → Rel (𝐼‘𝑥))
19	18	a1d 26	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝑥 ∈ 𝑆 → Rel (𝐼‘𝑥)))
20	19	ancld 559	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝑥 ∈ 𝑆 → (𝑥 ∈ 𝑆 ∧ Rel (𝐼‘𝑥))))
21	20	eximdv 1944	. . . . . 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 16	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → ∃𝑥(𝑥 ∈ 𝑆 ∧ Rel (𝐼‘𝑥)))
23		df-rex 3096	. . . . 5 ⊢ (∃𝑥 ∈ 𝑆 Rel (𝐼‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑆 ∧ Rel (𝐼‘𝑥)))
24	22, 23	sylibr 237	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → ∃𝑥 ∈ 𝑆 Rel (𝐼‘𝑥))
25		reliin 5805	. . . 4 ⊢ (∃𝑥 ∈ 𝑆 Rel (𝐼‘𝑥) → Rel ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))
26	24, 25	syl 18	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → Rel ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))
27		id 23	. . 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 487	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
29		simprl 782	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
30		eqid 2769	. . . . . . . . . . . . 13 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
31	3, 4, 5, 30	diadm 41733	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom ((DIsoA‘𝐾)‘𝑊) = {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
32	31	adantr 485	. . . . . . . . . . 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 3982	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ dom ((DIsoA‘𝐾)‘𝑊))
34		simprr 784	. . . . . . . . . 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 41753	. . . . . . . . . 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 849	. . . . . . . . 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 2855	. . . . . . . 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 3467	. . . . . . . . 9 ⊢ 𝑓 ∈ V
40		eliin 4965	. . . . . . . . 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 290	. . . . . . 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 642	. . . . . 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 4477	. . . . . . 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 741	. . . . . 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 285	. . . . 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 40056	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
48	47	ad2antrr 738	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝐾 ∈ CLat)
49		ssrab2 4042	. . . . . . . 8 ⊢ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ⊆ (Base‘𝐾)
50	29, 49	sstrdi 3957	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ (Base‘𝐾))
51	3, 35	clatglbcl 18561	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾)) → (𝐺‘𝑆) ∈ (Base‘𝐾))
52	48, 50, 51	syl2anc 595	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐺‘𝑆) ∈ (Base‘𝐾))
53		hllat 40061	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
54	53	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝐾 ∈ Lat)
55	47	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝐾 ∈ CLat)
56		simplrl 788	. . . . . . . . . 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 3957	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐾))
58	55, 57, 51	syl2anc 595	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆) ∈ (Base‘𝐾))
59	50	sselda 3945	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐾))
60	3, 5	lhpbase 40696	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
61	60	ad3antlr 743	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑊 ∈ (Base‘𝐾))
62		simpr 489	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
63	3, 4, 35	clatglble 18573	. . . . . . . . 9 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆)(le‘𝐾)𝑥)
64	55, 57, 62, 63	syl3anc 1396	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆)(le‘𝐾)𝑥)
65	29	sselda 3945	. . . . . . . . . 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 5116	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦(le‘𝐾)𝑊 ↔ 𝑥(le‘𝐾)𝑊))
67	66	elrab 3659	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊))
68	65, 67	sylib 221	. . . . . . . . 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 500	. . . . . . . 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 18502	. . . . . . 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 1962	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐺‘𝑆)(le‘𝐾)𝑊)
72		eqid 2769	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
73		eqid 2769	. . . . . . 7 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))
74	3, 4, 5, 72, 73, 30, 6	dibopelval2 41843	. . . . . 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 849	. . . . 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 5446	. . . . . . 7 ⊢ ⟨𝑓, 𝑠⟩ ∈ V
77		eliin 4965	. . . . . . 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 778	. . . . . . . 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 41843	. . . . . . . 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 595	. . . . . . 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 3192	. . . . . 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 286	. . . . 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 314	. . . 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 5782	. . 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 850	. 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 849	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  ⟨cop 4600  ∩ ciin 4961   class class class wbr 5113   ↦ cmpt 5196   I cid 5556  dom cdm 5662   ↾ cres 5664  Rel wrel 5667  ‘cfv 6537  Basecbs 17269  lecple 17317  glbcglb 18366  Latclat 18487  CLatccla 18554  HLchlt 40048  LHypclh 40682  LTrncltrn 40799  DIsoAcdia 41726  DIsoBcdib 41836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8826  df-proset 18350  df-poset 18369  df-plt 18384  df-lub 18400  df-glb 18401  df-join 18402  df-meet 18403  df-p0 18479  df-p1 18480  df-lat 18488  df-clat 18555  df-oposet 39874  df-ol 39876  df-oml 39877  df-covers 39964  df-ats 39965  df-atl 39996  df-cvlat 40020  df-hlat 40049  df-lhyp 40686  df-laut 40687  df-ldil 40802  df-ltrn 40803  df-trl 40857  df-disoa 41727  df-dib 41837

This theorem is referenced by:  dibintclN  41865  dihglblem3N  41993  dihmeetlem2N  41997