Theorem atlatmstc 36615

Description: An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in [Kalmbach] p. 140; also remark in [BeltramettiCassinelli] p. 98. (hatomistici 30145 analog.) (Contributed by NM, 5-Nov-2012.)

Hypotheses

Ref	Expression
atlatmstc.b	⊢ 𝐵 = (Base‘𝐾)
atlatmstc.l	⊢ ≤ = (le‘𝐾)
atlatmstc.u	⊢ 1 = (lub‘𝐾)
atlatmstc.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
atlatmstc	⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋)

Distinct variable groups:   𝑦, ≤   𝑦,𝐴   𝑦,𝐵   𝑦,𝑋

Allowed substitution hints:   1 (𝑦)   𝐾(𝑦)

Proof of Theorem atlatmstc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1189	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ CLat)
2		ssrab2 4007	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵
3		atlatmstc.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
4		atlatmstc.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	atssbase 36586	. . . . . 6 ⊢ 𝐴 ⊆ 𝐵
6		rabss2 4005	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ⊆ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋})
7	5, 6	ax-mp 5	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ⊆ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}
8		atlatmstc.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
9		atlatmstc.u	. . . . . 6 ⊢ 1 = (lub‘𝐾)
10	3, 8, 9	lubss 17723	. . . . 5 ⊢ ((𝐾 ∈ CLat ∧ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵 ∧ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ⊆ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ≤ ( 1 ‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}))
11	2, 7, 10	mp3an23 1450	. . . 4 ⊢ (𝐾 ∈ CLat → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ≤ ( 1 ‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}))
12	1, 11	syl 17	. . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ≤ ( 1 ‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}))
13		atlpos 36597	. . . . 5 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset)
14	13	3ad2ant3 1132	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) → 𝐾 ∈ Poset)
15		simpl 486	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Poset)
16		simpr 488	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
17	3, 8, 9, 15, 16	lubid 17592	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}) = 𝑋)
18	14, 17	sylan 583	. . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}) = 𝑋)
19	12, 18	breqtrd 5056	. 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ≤ 𝑋)
20		breq1 5033	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ 𝑋 ↔ 𝑥 ≤ 𝑋))
21	20	elrab 3628	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑋))
22		simpll2 1210	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) → 𝐾 ∈ CLat)
23		ssrab2 4007	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐴
24	23, 5	sstri 3924	. . . . . . . . . . . 12 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵
25	3, 8, 9	lubel 17724	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ CLat ∧ 𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ∧ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵) → 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))
26	24, 25	mp3an3 1447	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ CLat ∧ 𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) → 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))
27	22, 26	sylancom 591	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) → 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))
28	27	ex 416	. . . . . . . . 9 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} → 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})))
29	21, 28	syl5bir 246	. . . . . . . 8 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑋) → 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})))
30	29	expdimp 456	. . . . . . 7 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥 ≤ 𝑋 → 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})))
31		simpll3 1211	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐾 ∈ AtLat)
32		eqid 2798	. . . . . . . . . . . 12 ⊢ (0.‘𝐾) = (0.‘𝐾)
33	32, 4	atn0 36604	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ AtLat ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ (0.‘𝐾))
34	31, 33	sylancom 591	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ (0.‘𝐾))
35	34	adantr 484	. . . . . . . . 9 ⊢ (((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) → 𝑥 ≠ (0.‘𝐾))
36		simpl3 1190	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ AtLat)
37		atllat 36596	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
38	36, 37	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat)
39	38	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐾 ∈ Lat)
40	3, 4	atbase 36585	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)
41	40	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
42	3, 9	clatlubcl 17714	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ CLat ∧ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∈ 𝐵)
43	1, 24, 42	sylancl 589	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∈ 𝐵)
44	43	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∈ 𝐵)
45		simpl1 1188	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OML)
46		omlop 36537	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
47	45, 46	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP)
48		eqid 2798	. . . . . . . . . . . . . . . . 17 ⊢ (oc‘𝐾) = (oc‘𝐾)
49	3, 48	opoccl 36490	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ OP ∧ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∈ 𝐵) → ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) ∈ 𝐵)
50	47, 43, 49	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) ∈ 𝐵)
51	50	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) ∈ 𝐵)
52		eqid 2798	. . . . . . . . . . . . . . 15 ⊢ (meet‘𝐾) = (meet‘𝐾)
53	3, 8, 52	latlem12 17680	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑥 ∈ 𝐵 ∧ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∈ 𝐵 ∧ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) ∈ 𝐵)) → ((𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∧ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ↔ 𝑥 ≤ (( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})))))
54	39, 41, 44, 51, 53	syl13anc 1369	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∧ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ↔ 𝑥 ≤ (( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})))))
55	3, 48, 52, 32	opnoncon 36504	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ OP ∧ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∈ 𝐵) → (( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) = (0.‘𝐾))
56	47, 43, 55	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → (( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) = (0.‘𝐾))
57	56	breq2d 5042	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → (𝑥 ≤ (( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ↔ 𝑥 ≤ (0.‘𝐾)))
58	57	adantr 484	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥 ≤ (( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ↔ 𝑥 ≤ (0.‘𝐾)))
59	3, 8, 32	ople0 36483	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → (𝑥 ≤ (0.‘𝐾) ↔ 𝑥 = (0.‘𝐾)))
60	47, 40, 59	syl2an 598	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥 ≤ (0.‘𝐾) ↔ 𝑥 = (0.‘𝐾)))
61	54, 58, 60	3bitrd 308	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∧ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ↔ 𝑥 = (0.‘𝐾)))
62	61	biimpa 480	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ (𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∧ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})))) → 𝑥 = (0.‘𝐾))
63	62	expr 460	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) → (𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) → 𝑥 = (0.‘𝐾)))
64	63	necon3ad 3000	. . . . . . . . 9 ⊢ (((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) → (𝑥 ≠ (0.‘𝐾) → ¬ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))))
65	35, 64	mpd 15	. . . . . . . 8 ⊢ (((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) → ¬ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})))
66	65	ex 416	. . . . . . 7 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) → ¬ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))))
67	30, 66	syld 47	. . . . . 6 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥 ≤ 𝑋 → ¬ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))))
68		imnan 403	. . . . . 6 ⊢ ((𝑥 ≤ 𝑋 → ¬ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ↔ ¬ (𝑥 ≤ 𝑋 ∧ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))))
69	67, 68	sylib 221	. . . . 5 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ¬ (𝑥 ≤ 𝑋 ∧ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))))
70		simplr 768	. . . . . 6 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑋 ∈ 𝐵)
71	3, 8, 52	latlem12 17680	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) ∈ 𝐵)) → ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ↔ 𝑥 ≤ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})))))
72	39, 41, 70, 51, 71	syl13anc 1369	. . . . 5 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ↔ 𝑥 ≤ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})))))
73	69, 72	mtbid 327	. . . 4 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ≤ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))))
74	73	nrexdv 3229	. . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ¬ ∃𝑥 ∈ 𝐴 𝑥 ≤ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))))
75		simpll3 1211	. . . . . 6 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ≠ (0.‘𝐾)) → 𝐾 ∈ AtLat)
76		simpr 488	. . . . . . . 8 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
77	3, 52	latmcl 17654	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) ∈ 𝐵) → (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ∈ 𝐵)
78	38, 76, 50, 77	syl3anc 1368	. . . . . . 7 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ∈ 𝐵)
79	78	adantr 484	. . . . . 6 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ≠ (0.‘𝐾)) → (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ∈ 𝐵)
80		simpr 488	. . . . . 6 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ≠ (0.‘𝐾)) → (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ≠ (0.‘𝐾))
81	3, 8, 32, 4	atlex 36612	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ∈ 𝐵 ∧ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ≠ (0.‘𝐾)) → ∃𝑥 ∈ 𝐴 𝑥 ≤ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))))
82	75, 79, 80, 81	syl3anc 1368	. . . . 5 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ≠ (0.‘𝐾)) → ∃𝑥 ∈ 𝐴 𝑥 ≤ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))))
83	82	ex 416	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ((𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ≠ (0.‘𝐾) → ∃𝑥 ∈ 𝐴 𝑥 ≤ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})))))
84	83	necon1bd 3005	. . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → (¬ ∃𝑥 ∈ 𝐴 𝑥 ≤ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) → (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) = (0.‘𝐾)))
85	74, 84	mpd 15	. 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) = (0.‘𝐾))
86	3, 8, 52, 48, 32	omllaw3 36541	. . 3 ⊢ ((𝐾 ∈ OML ∧ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ≤ 𝑋 ∧ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) = (0.‘𝐾)) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋))
87	45, 43, 76, 86	syl3anc 1368	. 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ((( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ≤ 𝑋 ∧ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) = (0.‘𝐾)) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋))
88	19, 85, 87	mp2and 698	1 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107  {crab 3110   ⊆ wss 3881   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  occoc 16565  Posetcpo 17542  lubclub 17544  meetcmee 17547  0.cp0 17639  Latclat 17647  CLatccla 17709  OPcops 36468  OMLcoml 36471  Atomscatm 36559  AtLatcal 36560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36472  df-ol 36474  df-oml 36475  df-covers 36562  df-ats 36563  df-atl 36594

This theorem is referenced by:  atlatle  36616  hlatmstcOLDN  36693  pmaple  37057  pol1N  37206  polpmapN  37208  pmaplubN  37220