Theorem glbconN 39647

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 7315. (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 4175	. . . . 5 ⊢ (𝑆 ⊆ 𝐵 ↔ (𝐵 ∩ 𝑆) = 𝑆)
2	1	biimpi 216	. . . 4 ⊢ (𝑆 ⊆ 𝐵 → (𝐵 ∩ 𝑆) = 𝑆)
3		dfin5 3909	. . . 4 ⊢ (𝐵 ∩ 𝑆) = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}
4	2, 3	eqtr3di 2786	. . 3 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆})
5	4	fveq2d 6838	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝐺‘𝑆) = (𝐺‘{𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}))
6		glbcon.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
7		eqid 2736	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
8		glbcon.g	. . . 4 ⊢ 𝐺 = (glb‘𝐾)
9		biid 261	. . . 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 22	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ HL)
11		ssrab2 4032	. . . . 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 18290	. . 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 39632	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
15		hlclat 39628	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
16	6, 8	clatglbcl2 18429	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ⊆ 𝐵) → {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ∈ dom 𝐺)
17	15, 12, 16	syl2anc 584	. . . . 5 ⊢ (𝐾 ∈ HL → {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ∈ dom 𝐺)
18	6, 7, 8, 9, 10, 17	glbeu 18289	. . . 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 5101	. . . . . . 7 ⊢ (𝑦 = ( ⊥ ‘𝑣) → (𝑦(le‘𝐾)𝑧 ↔ ( ⊥ ‘𝑣)(le‘𝐾)𝑧))
21	20	ralbidv 3159	. . . . . 6 ⊢ (𝑦 = ( ⊥ ‘𝑣) → (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑦(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑣)(le‘𝐾)𝑧))
22		breq2 5102	. . . . . . . 8 ⊢ (𝑦 = ( ⊥ ‘𝑣) → (𝑤(le‘𝐾)𝑦 ↔ 𝑤(le‘𝐾)( ⊥ ‘𝑣)))
23	22	imbi2d 340	. . . . . . 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 3159	. . . . . 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 632	. . . . 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 39479	. . . 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 584	. . 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 726	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝐾 ∈ OP)
29	6, 19	opoccl 39464	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑢 ∈ 𝐵) → ( ⊥ ‘𝑢) ∈ 𝐵)
30	28, 29	sylancom 588	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → ( ⊥ ‘𝑢) ∈ 𝐵)
31	14	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝐾 ∈ OP)
32	6, 19	opoccl 39464	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OP ∧ 𝑧 ∈ 𝐵) → ( ⊥ ‘𝑧) ∈ 𝐵)
33	31, 32	sylancom 588	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ( ⊥ ‘𝑧) ∈ 𝐵)
34	6, 19	opococ 39465	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ OP ∧ 𝑧 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑧)) = 𝑧)
35	31, 34	sylancom 588	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑧)) = 𝑧)
36	35	eqcomd 2742	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 = ( ⊥ ‘( ⊥ ‘𝑧)))
37		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑢 = ( ⊥ ‘𝑧) → ( ⊥ ‘𝑢) = ( ⊥ ‘( ⊥ ‘𝑧)))
38	37	rspceeqv 3599	. . . . . . . . . . 11 ⊢ ((( ⊥ ‘𝑧) ∈ 𝐵 ∧ 𝑧 = ( ⊥ ‘( ⊥ ‘𝑧))) → ∃𝑢 ∈ 𝐵 𝑧 = ( ⊥ ‘𝑢))
39	33, 36, 38	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ∃𝑢 ∈ 𝐵 𝑧 = ( ⊥ ‘𝑢))
40		eleq1 2824	. . . . . . . . . . . 12 ⊢ (𝑧 = ( ⊥ ‘𝑢) → (𝑧 ∈ 𝑆 ↔ ( ⊥ ‘𝑢) ∈ 𝑆))
41		breq2 5102	. . . . . . . . . . . 12 ⊢ (𝑧 = ( ⊥ ‘𝑢) → (( ⊥ ‘𝑣)(le‘𝐾)𝑧 ↔ ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢)))
42	40, 41	imbi12d 344	. . . . . . . . . . 11 ⊢ (𝑧 = ( ⊥ ‘𝑢) → ((𝑧 ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)𝑧) ↔ (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢))))
43	42	adantl 481	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑧 = ( ⊥ ‘𝑢)) → ((𝑧 ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)𝑧) ↔ (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢))))
44	30, 39, 43	ralxfrd 5353	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 (𝑧 ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)𝑧) ↔ ∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢))))
45		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐵)
46		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝑣 ∈ 𝐵)
47	6, 7, 19	oplecon3b 39470	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OP ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢(le‘𝐾)𝑣 ↔ ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢)))
48	28, 45, 46, 47	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → (𝑢(le‘𝐾)𝑣 ↔ ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢)))
49	48	imbi2d 340	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → ((( ⊥ ‘𝑢) ∈ 𝑆 → 𝑢(le‘𝐾)𝑣) ↔ (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)( ⊥ ‘𝑢))))
50	49	ralbidva 3157	. . . . . . . . 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 282	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 (𝑧 ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)𝑧) ↔ ∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → 𝑢(le‘𝐾)𝑣)))
52		eleq1 2824	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑆 ↔ 𝑧 ∈ 𝑆))
53	52	ralrab 3652	. . . . . . . 8 ⊢ (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑣)(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ 𝐵 (𝑧 ∈ 𝑆 → ( ⊥ ‘𝑣)(le‘𝐾)𝑧))
54		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ( ⊥ ‘𝑥) = ( ⊥ ‘𝑢))
55	54	eleq1d 2821	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (( ⊥ ‘𝑥) ∈ 𝑆 ↔ ( ⊥ ‘𝑢) ∈ 𝑆))
56	55	ralrab 3652	. . . . . . . 8 ⊢ (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑣 ↔ ∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → 𝑢(le‘𝐾)𝑣))
57	51, 53, 56	3bitr4g 314	. . . . . . 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 726	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → 𝐾 ∈ OP)
59	6, 19	opoccl 39464	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑡 ∈ 𝐵) → ( ⊥ ‘𝑡) ∈ 𝐵)
60	58, 59	sylancom 588	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → ( ⊥ ‘𝑡) ∈ 𝐵)
61	14	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵) → 𝐾 ∈ OP)
62	6, 19	opoccl 39464	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑤 ∈ 𝐵) → ( ⊥ ‘𝑤) ∈ 𝐵)
63	61, 62	sylancom 588	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵) → ( ⊥ ‘𝑤) ∈ 𝐵)
64	6, 19	opococ 39465	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ OP ∧ 𝑤 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑤)) = 𝑤)
65	61, 64	sylancom 588	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑤)) = 𝑤)
66	65	eqcomd 2742	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵) → 𝑤 = ( ⊥ ‘( ⊥ ‘𝑤)))
67		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑡 = ( ⊥ ‘𝑤) → ( ⊥ ‘𝑡) = ( ⊥ ‘( ⊥ ‘𝑤)))
68	67	rspceeqv 3599	. . . . . . . . . 10 ⊢ ((( ⊥ ‘𝑤) ∈ 𝐵 ∧ 𝑤 = ( ⊥ ‘( ⊥ ‘𝑤))) → ∃𝑡 ∈ 𝐵 𝑤 = ( ⊥ ‘𝑡))
69	63, 66, 68	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵) → ∃𝑡 ∈ 𝐵 𝑤 = ( ⊥ ‘𝑡))
70		breq1 5101	. . . . . . . . . . . 12 ⊢ (𝑤 = ( ⊥ ‘𝑡) → (𝑤(le‘𝐾)𝑧 ↔ ( ⊥ ‘𝑡)(le‘𝐾)𝑧))
71	70	ralbidv 3159	. . . . . . . . . . 11 ⊢ (𝑤 = ( ⊥ ‘𝑡) → (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}𝑤(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑡)(le‘𝐾)𝑧))
72		breq1 5101	. . . . . . . . . . 11 ⊢ (𝑤 = ( ⊥ ‘𝑡) → (𝑤(le‘𝐾)( ⊥ ‘𝑣) ↔ ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑣)))
73	71, 72	imbi12d 344	. . . . . . . . . 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 481	. . . . . . . . 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 5353	. . . . . . . 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 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝐾 ∈ OP)
77		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐵)
78		simplr 768	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝑡 ∈ 𝐵)
79	6, 7, 19	oplecon3b 39470	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ OP ∧ 𝑢 ∈ 𝐵 ∧ 𝑡 ∈ 𝐵) → (𝑢(le‘𝐾)𝑡 ↔ ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑢)))
80	76, 77, 78, 79	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → (𝑢(le‘𝐾)𝑡 ↔ ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑢)))
81	80	imbi2d 340	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → ((( ⊥ ‘𝑢) ∈ 𝑆 → 𝑢(le‘𝐾)𝑡) ↔ (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑢))))
82	81	ralbidva 3157	. . . . . . . . . . . 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 588	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → ( ⊥ ‘𝑢) ∈ 𝐵)
84	14	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝐾 ∈ OP)
85	84, 32	sylancom 588	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ( ⊥ ‘𝑧) ∈ 𝐵)
86	84, 34	sylancom 588	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑧)) = 𝑧)
87	86	eqcomd 2742	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → 𝑧 = ( ⊥ ‘( ⊥ ‘𝑧)))
88	85, 87, 38	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ∃𝑢 ∈ 𝐵 𝑧 = ( ⊥ ‘𝑢))
89		breq2 5102	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ( ⊥ ‘𝑢) → (( ⊥ ‘𝑡)(le‘𝐾)𝑧 ↔ ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑢)))
90	40, 89	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ( ⊥ ‘𝑢) → ((𝑧 ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)𝑧) ↔ (( ⊥ ‘𝑢) ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑢))))
91	90	adantl 481	. . . . . . . . . . . . 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 5353	. . . . . . . . . . . 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 282	. . . . . . . . . . 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 3652	. . . . . . . . . . 11 ⊢ (∀𝑢 ∈ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆}𝑢(le‘𝐾)𝑡 ↔ ∀𝑢 ∈ 𝐵 (( ⊥ ‘𝑢) ∈ 𝑆 → 𝑢(le‘𝐾)𝑡))
95	52	ralrab 3652	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆} ( ⊥ ‘𝑡)(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ 𝐵 (𝑧 ∈ 𝑆 → ( ⊥ ‘𝑡)(le‘𝐾)𝑧))
96	93, 94, 95	3bitr4g 314	. . . . . . . . . 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 768	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → 𝑣 ∈ 𝐵)
98		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝐵)
99	6, 7, 19	oplecon3b 39470	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OP ∧ 𝑣 ∈ 𝐵 ∧ 𝑡 ∈ 𝐵) → (𝑣(le‘𝐾)𝑡 ↔ ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑣)))
100	58, 97, 98, 99	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑣 ∈ 𝐵) ∧ 𝑡 ∈ 𝐵) → (𝑣(le‘𝐾)𝑡 ↔ ( ⊥ ‘𝑡)(le‘𝐾)( ⊥ ‘𝑣)))
101	96, 100	imbi12d 344	. . . . . . . . 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 3157	. . . . . . . 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 282	. . . . . . 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 632	. . . . . 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 7334	. . . . 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 4032	. . . . . 6 ⊢ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆} ⊆ 𝐵
107		glbcon.u	. . . . . . 7 ⊢ 𝑈 = (lub‘𝐾)
108		biid 261	. . . . . . 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 482	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆} ⊆ 𝐵) → 𝐾 ∈ HL)
110		simpr 484	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆} ⊆ 𝐵) → {𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆} ⊆ 𝐵)
111	6, 7, 107, 108, 109, 110	lubval 18277	. . . . . 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 691	. . . . 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 2774	. . . 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 6838	. . 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 2775	. 2 ⊢ (𝐾 ∈ HL → (𝐺‘{𝑥 ∈ 𝐵 ∣ 𝑥 ∈ 𝑆}) = ( ⊥ ‘(𝑈‘{𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆})))
116	5, 115	sylan9eqr 2793	1 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) = ( ⊥ ‘(𝑈‘{𝑥 ∈ 𝐵 ∣ ( ⊥ ‘𝑥) ∈ 𝑆})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  ∃!wreu 3348  {crab 3399   ∩ cin 3900   ⊆ wss 3901   class class class wbr 5098  dom cdm 5624  ‘cfv 6492  ℩crio 7314  Basecbs 17136  lecple 17184  occoc 17185  lubclub 18232  glbcglb 18233  CLatccla 18421  OPcops 39442  HLchlt 39620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-lub 18267  df-glb 18268  df-clat 18422  df-oposet 39446  df-ol 39448  df-oml 39449  df-hlat 39621

This theorem is referenced by:  glbconxN  39648