Theorem dibglbN 42042

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 488	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simprl 783	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ dom 𝐼)
3		eqid 2760	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2760	. . . . . 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 42033	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
8	7	sseq2d 3963	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑆 ⊆ dom 𝐼 ↔ 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊}))
9	8	adantr 486	. . 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 785	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ≠ ∅)
12	5, 6	dibvalrel 42039	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘(𝐺‘𝑆)))
13	12	adantr 486	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → Rel (𝐼‘(𝐺‘𝑆)))
14		n0 4300	. . . . . . . 8 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆)
15	14	biimpi 219	. . . . . . 7 ⊢ (𝑆 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝑆)
16	15	ad2antll 742	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → ∃𝑥 𝑥 ∈ 𝑆)
17	5, 6	dibvalrel 42039	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑥))
18	17	adantr 486	. . . . . . . . 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 560	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝑥 ∈ 𝑆 → (𝑥 ∈ 𝑆 ∧ Rel (𝐼‘𝑥))))
21	20	eximdv 1950	. . . . . 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 3087	. . . . 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 5798	. . . 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 488	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
29		simprl 783	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
30		eqid 2760	. . . . . . . . . . . . 13 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
31	3, 4, 5, 30	diadm 41911	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom ((DIsoA‘𝐾)‘𝑊) = {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
32	31	adantr 486	. . . . . . . . . . 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 3968	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ dom ((DIsoA‘𝐾)‘𝑊))
34		simprr 785	. . . . . . . . . 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 41931	. . . . . . . . . 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 850	. . . . . . . . 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 2846	. . . . . . . 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 3454	. . . . . . . . 9 ⊢ 𝑓 ∈ V
40		eliin 4956	. . . . . . . . 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 643	. . . . . 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 4467	. . . . . . 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 742	. . . . . 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 40234	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
48	47	ad2antrr 739	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝐾 ∈ CLat)
49		ssrab2 4028	. . . . . . . 8 ⊢ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ⊆ (Base‘𝐾)
50	29, 49	sstrdi 3943	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ (Base‘𝐾))
51	3, 35	clatglbcl 18596	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾)) → (𝐺‘𝑆) ∈ (Base‘𝐾))
52	48, 50, 51	syl2anc 596	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐺‘𝑆) ∈ (Base‘𝐾))
53		hllat 40239	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
54	53	ad3antrrr 743	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝐾 ∈ Lat)
55	47	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝐾 ∈ CLat)
56		simplrl 789	. . . . . . . . . 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 3943	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐾))
58	55, 57, 51	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆) ∈ (Base‘𝐾))
59	50	sselda 3931	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐾))
60	3, 5	lhpbase 40874	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
61	60	ad3antlr 744	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑊 ∈ (Base‘𝐾))
62		simpr 490	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
63	3, 4, 35	clatglble 18608	. . . . . . . . 9 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆)(le‘𝐾)𝑥)
64	55, 57, 62, 63	syl3anc 1398	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆)(le‘𝐾)𝑥)
65	29	sselda 3931	. . . . . . . . . 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 5106	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦(le‘𝐾)𝑊 ↔ 𝑥(le‘𝐾)𝑊))
67	66	elrab 3645	. . . . . . . . . 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 501	. . . . . . . 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 18537	. . . . . . 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 1968	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐺‘𝑆)(le‘𝐾)𝑊)
72		eqid 2760	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
73		eqid 2760	. . . . . . 7 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))
74	3, 4, 5, 72, 73, 30, 6	dibopelval2 42021	. . . . . 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 850	. . . . 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 5439	. . . . . . 7 ⊢ ⟨𝑓, 𝑠⟩ ∈ V
77		eliin 4956	. . . . . . 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 779	. . . . . . . 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 42021	. . . . . . . 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 596	. . . . . . 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 3183	. . . . . 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 5775	. . 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 851	. 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 850	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  ⟨cop 4590  ∩ ciin 4952   class class class wbr 5103   ↦ cmpt 5186   I cid 5549  dom cdm 5655   ↾ cres 5657  Rel wrel 5660  ‘cfv 6533  Basecbs 17304  lecple 17352  glbcglb 18401  Latclat 18522  CLatccla 18589  HLchlt 40226  LHypclh 40860  LTrncltrn 40977  DIsoAcdia 41904  DIsoBcdib 42014

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8831  df-proset 18385  df-poset 18404  df-plt 18419  df-lub 18435  df-glb 18436  df-join 18437  df-meet 18438  df-p0 18514  df-p1 18515  df-lat 18523  df-clat 18590  df-oposet 40052  df-ol 40054  df-oml 40055  df-covers 40142  df-ats 40143  df-atl 40174  df-cvlat 40198  df-hlat 40227  df-lhyp 40864  df-laut 40865  df-ldil 40980  df-ltrn 40981  df-trl 41035  df-disoa 41905  df-dib 42015

This theorem is used by:  dibintclN  42043  dihglblem3N  42171  dihmeetlem2N  42175