Theorem dibglbN 40865

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 481	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simprl 769	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ dom 𝐼)
3		eqid 2726	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2726	. . . . . 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 40856	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
8	7	sseq2d 4012	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑆 ⊆ dom 𝐼 ↔ 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊}))
9	8	adantr 479	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝑆 ⊆ dom 𝐼 ↔ 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊}))
10	2, 9	mpbid 231	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
11		simprr 771	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → 𝑆 ≠ ∅)
12	5, 6	dibvalrel 40862	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘(𝐺‘𝑆)))
13	12	adantr 479	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → Rel (𝐼‘(𝐺‘𝑆)))
14		n0 4349	. . . . . . . 8 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆)
15	14	biimpi 215	. . . . . . 7 ⊢ (𝑆 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝑆)
16	15	ad2antll 727	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → ∃𝑥 𝑥 ∈ 𝑆)
17	5, 6	dibvalrel 40862	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘𝑥))
18	17	adantr 479	. . . . . . . . 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 549	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝑥 ∈ 𝑆 → (𝑥 ∈ 𝑆 ∧ Rel (𝐼‘𝑥))))
21	20	eximdv 1913	. . . . . 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 3061	. . . . 5 ⊢ (∃𝑥 ∈ 𝑆 Rel (𝐼‘𝑥) ↔ ∃𝑥(𝑥 ∈ 𝑆 ∧ Rel (𝐼‘𝑥)))
24	22, 23	sylibr 233	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → ∃𝑥 ∈ 𝑆 Rel (𝐼‘𝑥))
25		reliin 5823	. . . 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 481	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
29		simprl 769	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
30		eqid 2726	. . . . . . . . . . . . 13 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
31	3, 4, 5, 30	diadm 40734	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → dom ((DIsoA‘𝐾)‘𝑊) = {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
32	31	adantr 479	. . . . . . . . . . 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 771	. . . . . . . . . 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 40754	. . . . . . . . . 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 835	. . . . . . . . 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 2812	. . . . . . . 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 3466	. . . . . . . . 9 ⊢ 𝑓 ∈ V
40		eliin 5006	. . . . . . . . 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 286	. . . . . . 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 629	. . . . . 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 4510	. . . . . . 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 727	. . . . . 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 281	. . . . 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 39056	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
48	47	ad2antrr 724	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝐾 ∈ CLat)
49		ssrab2 4076	. . . . . . . 8 ⊢ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ⊆ (Base‘𝐾)
50	29, 49	sstrdi 3992	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → 𝑆 ⊆ (Base‘𝐾))
51	3, 35	clatglbcl 18530	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾)) → (𝐺‘𝑆) ∈ (Base‘𝐾))
52	48, 50, 51	syl2anc 582	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐺‘𝑆) ∈ (Base‘𝐾))
53		hllat 39061	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
54	53	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝐾 ∈ Lat)
55	47	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝐾 ∈ CLat)
56		simplrl 775	. . . . . . . . . 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 3992	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐾))
58	55, 57, 51	syl2anc 582	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆) ∈ (Base‘𝐾))
59	50	sselda 3979	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝐾))
60	3, 5	lhpbase 39697	. . . . . . . . 9 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
61	60	ad3antlr 729	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑊 ∈ (Base‘𝐾))
62		simpr 483	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
63	3, 4, 35	clatglble 18542	. . . . . . . . 9 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ (Base‘𝐾) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆)(le‘𝐾)𝑥)
64	55, 57, 62, 63	syl3anc 1368	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑆)(le‘𝐾)𝑥)
65	29	sselda 3979	. . . . . . . . . 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 5156	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦(le‘𝐾)𝑊 ↔ 𝑥(le‘𝐾)𝑊))
67	66	elrab 3681	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊))
68	65, 67	sylib 217	. . . . . . . . 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 494	. . . . . . . 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 18471	. . . . . . 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 1931	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ∧ 𝑆 ≠ ∅)) → (𝐺‘𝑆)(le‘𝐾)𝑊)
72		eqid 2726	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
73		eqid 2726	. . . . . . 7 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))
74	3, 4, 5, 72, 73, 30, 6	dibopelval2 40844	. . . . . 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 835	. . . . 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 5470	. . . . . . 7 ⊢ ⟨𝑓, 𝑠⟩ ∈ V
77		eliin 5006	. . . . . . 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 765	. . . . . . . 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 40844	. . . . . . . 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 582	. . . . . . 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 3166	. . . . . 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 282	. . . . 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 310	. . . 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 5801	. . 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 836	. 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 835	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  {crab 3419  Vcvv 3462   ⊆ wss 3947  ∅c0 4325  ⟨cop 4639  ∩ ciin 5002   class class class wbr 5153   ↦ cmpt 5236   I cid 5579  dom cdm 5682   ↾ cres 5684  Rel wrel 5687  ‘cfv 6554  Basecbs 17213  lecple 17273  glbcglb 18335  Latclat 18456  CLatccla 18523  HLchlt 39048  LHypclh 39683  LTrncltrn 39800  DIsoAcdia 40727  DIsoBcdib 40837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8857  df-proset 18320  df-poset 18338  df-plt 18355  df-lub 18371  df-glb 18372  df-join 18373  df-meet 18374  df-p0 18450  df-p1 18451  df-lat 18457  df-clat 18524  df-oposet 38874  df-ol 38876  df-oml 38877  df-covers 38964  df-ats 38965  df-atl 38996  df-cvlat 39020  df-hlat 39049  df-lhyp 39687  df-laut 39688  df-ldil 39803  df-ltrn 39804  df-trl 39858  df-disoa 40728  df-dib 40838

This theorem is referenced by:  dibintclN  40866  dihglblem3N  40994  dihmeetlem2N  40998