Theorem glbconN 40211

Description: De Morgan's law for GLB and LUB. This holds in any complete ortholattice, although we assume HL for convenience. (Contributed by NM, 17-Jan-2012.) New df-riota 7376. (Revised by SN, 3-Jan-2025.) (New usage is discouraged.)

Hypotheses

Ref	Expression
glbcon.b	⊢ 𝐵 = (Base‘𝐾)
glbcon.u	⊢ 𝑈 = (lub‘𝐾)
glbcon.g	⊢ 𝐺 = (glb‘𝐾)
glbcon.o	⊢ ⊥ = (oc‘𝐾)

Assertion

Ref	Expression
glbconN	⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) = ( ⊥ ‘(𝑈‘{𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆})))

Distinct variable groups:   𝑥,𝐵   𝑥, ⊥   𝑥,𝑆

Allowed substitution hints:   𝑈(𝑥)   𝐺(𝑥)   𝐾(𝑥)

Proof of Theorem glbconN

Dummy variables 𝑢 𝑡 𝑣 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseqin2 4176	. . . . 5 ⊢ (𝑆 ⊆ 𝐵 ↔ (𝐵 ∩ 𝑆) = 𝑆)
2	1	biimpi 219	. . . 4 ⊢ (𝑆 ⊆ 𝐵 → (𝐵 ∩ 𝑆) = 𝑆)
3		dfin5 3914	. . . 4 ⊢ (𝐵 ∩ 𝑆) = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}
4	2, 3	eqtr3di 2815	. . 3 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆})
5	4	fveq2d 6889	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝐺‘𝑆) = (𝐺‘{𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}))
6		glbcon.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
7		eqid 2765	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
8		glbcon.g	. . . 4 ⊢ 𝐺 = (glb‘𝐾)
9		biid 264	. . . 4 ⊢ ((∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑦(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)𝑦)) ↔ (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑦(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)𝑦)))
10		id 23	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ HL)
11		ssrab2 4035	. . . . 5 ⊢ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ⊆ 𝐵
12	11	a1i 11	. . . 4 ⊢ (𝐾 ∈ HL → {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ⊆ 𝐵)
13	6, 7, 8, 9, 10, 12	glbval 18447	. . 3 ⊢ (𝐾 ∈ HL → (𝐺‘{𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}) = (℩𝑦 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑦(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)𝑦))))
14		hlop 40196	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
15		hlclat 40192	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
16	6, 8	clatglbcl2 18586	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ⊆ 𝐵) → {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ∈ dom 𝐺)
17	15, 12, 16	syl2anc 596	. . . . 5 ⊢ (𝐾 ∈ HL → {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ∈ dom 𝐺)
18	6, 7, 8, 9, 10, 17	glbeu 18446	. . . 4 ⊢ (𝐾 ∈ HL → ∃!𝑦 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑦(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)𝑦)))
19		glbcon.o	. . . . 5 ⊢ ⊥ = (oc‘𝐾)
20		breq1 5114	. . . . . . 7 ⊢ (𝑦 = ( ⊥ ‘𝑣) → (𝑦(le‘𝐾)𝑧 ↔ ( ⊥ ‘𝑣)(le‘𝐾)𝑧))
21	20	ralbidv 3190	. . . . . 6 ⊢ (𝑦 = ( ⊥ ‘𝑣) → (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑦(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑣)(le‘𝐾)𝑧))
22		breq2 5115	. . . . . . . 8 ⊢ (𝑦 = ( ⊥ ‘𝑣) → (𝑤(le‘𝐾)𝑦 ↔ 𝑤(le‘𝐾)( ⊥ ‘𝑣)))
23	22	imbi2d 343	. . . . . . 7 ⊢ (𝑦 = ( ⊥ ‘𝑣) → ((∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)𝑦) ↔ (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)( ⊥ ‘𝑣))))
24	23	ralbidv 3190	. . . . . 6 ⊢ (𝑦 = ( ⊥ ‘𝑣) → (∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)𝑦) ↔ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)( ⊥ ‘𝑣))))
25	21, 24	anbi12d 644	. . . . 5 ⊢ (𝑦 = ( ⊥ ‘𝑣) → ((∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑦(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)𝑦)) ↔ (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑣)(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)( ⊥ ‘𝑣)))))
26	6, 19, 25	riotaocN 40043	. . . 4 ⊢ ((𝐾 ∈ OP ∧ ∃!𝑦 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑦(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)𝑦))) → (℩𝑦 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑦(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)𝑦))) = ( ⊥ ‘(℩𝑣 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑣)(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)( ⊥ ‘𝑣))))))
27	14, 18, 26	syl2anc 596	. . 3 ⊢ (𝐾 ∈ HL → (℩𝑦 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑦(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)𝑦))) = ( ⊥ ‘(℩𝑣 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑣)(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)( ⊥ ‘𝑣))))))
28	14	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝐾 ∈ OP)
29	6, 19	opoccl 40028	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑢 ∈ 𝐵) → ( ⊥ ‘𝑢) ∈ 𝐵)
30	28, 29	sylancom 600	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → ( ⊥ ‘𝑢) ∈ 𝐵)
31	14	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝐾 ∈ OP)
32	6, 19	opoccl 40028	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OP ∧ 𝑧 ∈ 𝐵) → ( ⊥ ‘𝑧) ∈ 𝐵)
33	31, 32	sylancom 600	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ( ⊥ ‘𝑧) ∈ 𝐵)
34	6, 19	opococ 40029	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ OP ∧ 𝑧 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑧)) = 𝑧)
35	31, 34	sylancom 600	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑧)) = 𝑧)
36	35	eqcomd 2771	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 = ( ⊥ ‘( ⊥ ‘𝑧)))
37		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑢 = ( ⊥ ‘𝑧) → ( ⊥ ‘𝑢) = ( ⊥ ‘( ⊥ ‘𝑧)))
38	37	rspceeqv 3606	. . . . . . . . . . 11 ⊢ ((( ⊥ ‘𝑧) ∈ 𝐵 ∧ 𝑧 = ( ⊥ ‘( ⊥ ‘𝑧))) → ∃𝑢 ∈ 𝐵 𝑧 = ( ⊥ ‘𝑢))
39	33, 36, 38	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ∃𝑢 ∈ 𝐵 𝑧 = ( ⊥ ‘𝑢))
40		eleq1 2853	. . . . . . . . . . . 12 ⊢ (𝑧 = ( ⊥ ‘𝑢) → (𝑧 ∈ 𝑆 ↔ ( ⊥ ‘𝑢) ∈ 𝑆))
41		breq2 5115	. . . . . . . . . . . 12 ⊢ (𝑧 = ( ⊥ ‘𝑢) → (( ⊥ ‘𝑣)(le‘𝐾)𝑧 ↔ ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢)))
42	40, 41	imbi12d 347	. . . . . . . . . . 11 ⊢ (𝑧 = ( ⊥ ‘𝑢) → ((𝑧 ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)𝑧) ↔ (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢))))
43	42	adantl 487	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑧 = ( ⊥ ‘𝑢)) → ((𝑧 ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)𝑧) ↔ (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢))))
44	30, 39, 43	ralxfrd 5381	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 (𝑧 ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)𝑧) ↔ ∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢))))
45		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐵)
46		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝑣 ∈ 𝐵)
47	6, 7, 19	oplecon3b 40034	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OP ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(le‘𝐾)𝑣 ↔ ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢)))
48	28, 45, 46, 47	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → (𝑢(le‘𝐾)𝑣 ↔ ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢)))
49	48	imbi2d 343	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → ((( ⊥ ‘𝑢) ∈ 𝑆 → 𝑢(le‘𝐾)𝑣) ↔ (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢))))
50	49	ralbidva 3188	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) → (∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → 𝑢(le‘𝐾)𝑣) ↔ ∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢))))
51	44, 50	bitr4d 285	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 (𝑧 ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)𝑧) ↔ ∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → 𝑢(le‘𝐾)𝑣)))
52		eleq1 2853	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑆 ↔ 𝑧 ∈ 𝑆))
53	52	ralrab 3659	. . . . . . . 8 ⊢ (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑣)(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ 𝐵 (𝑧 ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)𝑧))
54		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑢))
55	54	eleq1d 2850	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (( ⊥ ‘𝑥) ∈ 𝑆 ↔ ( ⊥ ‘𝑢) ∈ 𝑆))
56	55	ralrab 3659	. . . . . . . 8 ⊢ (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑣 ↔ ∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → 𝑢(le‘𝐾)𝑣))
57	51, 53, 56	3bitr4g 317	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) → (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑣)(le‘𝐾)𝑧 ↔ ∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑣))
58	14	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → 𝐾 ∈ OP)
59	6, 19	opoccl 40028	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑡 ∈ 𝐵) → ( ⊥ ‘𝑡) ∈ 𝐵)
60	58, 59	sylancom 600	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → ( ⊥ ‘𝑡) ∈ 𝐵)
61	14	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵) → 𝐾 ∈ OP)
62	6, 19	opoccl 40028	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑤 ∈ 𝐵) → ( ⊥ ‘𝑤) ∈ 𝐵)
63	61, 62	sylancom 600	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵) → ( ⊥ ‘𝑤) ∈ 𝐵)
64	6, 19	opococ 40029	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OP ∧ 𝑤 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑤)) = 𝑤)
65	61, 64	sylancom 600	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑤)) = 𝑤)
66	65	eqcomd 2771	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵) → 𝑤 = ( ⊥ ‘( ⊥ ‘𝑤)))
67		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑡 = ( ⊥ ‘𝑤) → ( ⊥ ‘𝑡) = ( ⊥ ‘( ⊥ ‘𝑤)))
68	67	rspceeqv 3606	. . . . . . . . . 10 ⊢ ((( ⊥ ‘𝑤) ∈ 𝐵 ∧ 𝑤 = ( ⊥ ‘( ⊥ ‘𝑤))) → ∃𝑡 ∈ 𝐵 𝑤 = ( ⊥ ‘𝑡))
69	63, 66, 68	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵) → ∃𝑡 ∈ 𝐵 𝑤 = ( ⊥ ‘𝑡))
70		breq1 5114	. . . . . . . . . . . 12 ⊢ (𝑤 = ( ⊥ ‘𝑡) → (𝑤(le‘𝐾)𝑧 ↔ ( ⊥ ‘𝑡)(le‘𝐾)𝑧))
71	70	ralbidv 3190	. . . . . . . . . . 11 ⊢ (𝑤 = ( ⊥ ‘𝑡) → (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑡)(le‘𝐾)𝑧))
72		breq1 5114	. . . . . . . . . . 11 ⊢ (𝑤 = ( ⊥ ‘𝑡) → (𝑤(le‘𝐾)( ⊥ ‘𝑣) ↔ ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑣)))
73	71, 72	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑤 = ( ⊥ ‘𝑡) → ((∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)( ⊥ ‘𝑣)) ↔ (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑡)(le‘𝐾)𝑧 → ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑣))))
74	73	adantl 487	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 = ( ⊥ ‘𝑡)) → ((∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)( ⊥ ‘𝑣)) ↔ (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑡)(le‘𝐾)𝑧 → ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑣))))
75	60, 69, 74	ralxfrd 5381	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) → (∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)( ⊥ ‘𝑣)) ↔ ∀𝑡 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑡)(le‘𝐾)𝑧 → ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑣))))
76	14	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝐾 ∈ OP)
77		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐵)
78		simplr 781	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝑡 ∈ 𝐵)
79	6, 7, 19	oplecon3b 40034	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ OP ∧ 𝑢 ∈ 𝐵 ∧ 𝑡 ∈ 𝐵) → (𝑢(le‘𝐾)𝑡 ↔ ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑢)))
80	76, 77, 78, 79	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → (𝑢(le‘𝐾)𝑡 ↔ ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑢)))
81	80	imbi2d 343	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → ((( ⊥ ‘𝑢) ∈ 𝑆 → 𝑢(le‘𝐾)𝑡) ↔ (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑢))))
82	81	ralbidva 3188	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → (∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → 𝑢(le‘𝐾)𝑡) ↔ ∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑢))))
83	76, 29	sylancom 600	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → ( ⊥ ‘𝑢) ∈ 𝐵)
84	14	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝐾 ∈ OP)
85	84, 32	sylancom 600	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ( ⊥ ‘𝑧) ∈ 𝐵)
86	84, 34	sylancom 600	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑧)) = 𝑧)
87	86	eqcomd 2771	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 = ( ⊥ ‘( ⊥ ‘𝑧)))
88	85, 87, 38	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ∃𝑢 ∈ 𝐵 𝑧 = ( ⊥ ‘𝑢))
89		breq2 5115	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ( ⊥ ‘𝑢) → (( ⊥ ‘𝑡)(le‘𝐾)𝑧 ↔ ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑢)))
90	40, 89	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ( ⊥ ‘𝑢) → ((𝑧 ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)𝑧) ↔ (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑢))))
91	90	adantl 487	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑧 = ( ⊥ ‘𝑢)) → ((𝑧 ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)𝑧) ↔ (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑢))))
92	83, 88, 91	ralxfrd 5381	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 (𝑧 ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)𝑧) ↔ ∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑢))))
93	82, 92	bitr4d 285	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → (∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → 𝑢(le‘𝐾)𝑡) ↔ ∀𝑧 ∈ 𝐵 (𝑧 ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)𝑧)))
94	55	ralrab 3659	. . . . . . . . . . 11 ⊢ (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑡 ↔ ∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → 𝑢(le‘𝐾)𝑡))
95	52	ralrab 3659	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑡)(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ 𝐵 (𝑧 ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)𝑧))
96	93, 94, 95	3bitr4g 317	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑡 ↔ ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑡)(le‘𝐾)𝑧))
97		simplr 781	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → 𝑣 ∈ 𝐵)
98		simpr 490	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝐵)
99	6, 7, 19	oplecon3b 40034	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑣 ∈ 𝐵 ∧ 𝑡 ∈ 𝐵) → (𝑣(le‘𝐾)𝑡 ↔ ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑣)))
100	58, 97, 98, 99	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → (𝑣(le‘𝐾)𝑡 ↔ ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑣)))
101	96, 100	imbi12d 347	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → ((∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑡 → 𝑣(le‘𝐾)𝑡) ↔ (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑡)(le‘𝐾)𝑧 → ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑣))))
102	101	ralbidva 3188	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) → (∀𝑡 ∈ 𝐵 (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑡 → 𝑣(le‘𝐾)𝑡) ↔ ∀𝑡 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑡)(le‘𝐾)𝑧 → ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑣))))
103	75, 102	bitr4d 285	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) → (∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)( ⊥ ‘𝑣)) ↔ ∀𝑡 ∈ 𝐵 (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑡 → 𝑣(le‘𝐾)𝑡)))
104	57, 103	anbi12d 644	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) → ((∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑣)(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)( ⊥ ‘𝑣))) ↔ (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑣 ∧ ∀𝑡 ∈ 𝐵 (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑡 → 𝑣(le‘𝐾)𝑡))))
105	104	riotabidva 7395	. . . . 5 ⊢ (𝐾 ∈ HL → (℩𝑣 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑣)(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)( ⊥ ‘𝑣)))) = (℩𝑣 ∈ 𝐵 (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑣 ∧ ∀𝑡 ∈ 𝐵 (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑡 → 𝑣(le‘𝐾)𝑡))))
106		ssrab2 4035	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆} ⊆ 𝐵
107		glbcon.u	. . . . . . 7 ⊢ 𝑈 = (lub‘𝐾)
108		biid 264	. . . . . . 7 ⊢ ((∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑣 ∧ ∀𝑡 ∈ 𝐵 (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑡 → 𝑣(le‘𝐾)𝑡)) ↔ (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑣 ∧ ∀𝑡 ∈ 𝐵 (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑡 → 𝑣(le‘𝐾)𝑡)))
109		simpl 488	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆} ⊆ 𝐵) → 𝐾 ∈ HL)
110		simpr 490	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆} ⊆ 𝐵) → {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆} ⊆ 𝐵)
111	6, 7, 107, 108, 109, 110	lubval 18434	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆} ⊆ 𝐵) → (𝑈‘{𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}) = (℩𝑣 ∈ 𝐵 (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑣 ∧ ∀𝑡 ∈ 𝐵 (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑡 → 𝑣(le‘𝐾)𝑡))))
112	106, 111	mpan2 704	. . . . 5 ⊢ (𝐾 ∈ HL → (𝑈‘{𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}) = (℩𝑣 ∈ 𝐵 (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑣 ∧ ∀𝑡 ∈ 𝐵 (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑡 → 𝑣(le‘𝐾)𝑡))))
113	105, 112	eqtr4d 2803	. . . 4 ⊢ (𝐾 ∈ HL → (℩𝑣 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑣)(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)( ⊥ ‘𝑣)))) = (𝑈‘{𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}))
114	113	fveq2d 6889	. . 3 ⊢ (𝐾 ∈ HL → ( ⊥ ‘(℩𝑣 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑣)(le‘𝐾)𝑧 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 → 𝑤(le‘𝐾)( ⊥ ‘𝑣))))) = ( ⊥ ‘(𝑈‘{𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆})))
115	13, 27, 114	3eqtrd 2804	. 2 ⊢ (𝐾 ∈ HL → (𝐺‘{𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}) = ( ⊥ ‘(𝑈‘{𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆})))
116	5, 115	sylan9eqr 2822	1 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) = ( ⊥ ‘(𝑈‘{𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  ∃!wreu 3369  {crab 3418   ∩ cin 3905   ⊆ wss 3906   class class class wbr 5111  dom cdm 5663  ‘cfv 6540  ℩crio 7375  Basecbs 17293  lecple 17341  occoc 17342  lubclub 18389  glbcglb 18390  CLatccla 18578  OPcops 40006  HLchlt 40184

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-lub 18424  df-glb 18425  df-clat 18579  df-oposet 40010  df-ol 40012  df-oml 40013  df-hlat 40185

This theorem is used by:  glbconxN  40212