Theorem atlatmstc 36457

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 30141 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 1188	. . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ CLat)
2		ssrab2 4058	. . . . 5 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵
3		atlatmstc.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
4		atlatmstc.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	atssbase 36428	. . . . . 6 ⊢ 𝐴 ⊆ 𝐵
6		rabss2 4056	. . . . . 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 17733	. . . . 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 1449	. . . 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 36439	. . . . 5 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset)
14	13	3ad2ant3 1131	. . . 4 ⊢ ((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) → 𝐾 ∈ Poset)
15		simpl 485	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Poset)
16		simpr 487	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
17	3, 8, 9, 15, 16	lubid 17602	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}) = 𝑋)
18	14, 17	sylan 582	. . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}) = 𝑋)
19	12, 18	breqtrd 5094	. 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ≤ 𝑋)
20		breq1 5071	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ 𝑋 ↔ 𝑥 ≤ 𝑋))
21	20	elrab 3682	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑋))
22		simpll2 1209	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) → 𝐾 ∈ CLat)
23		ssrab2 4058	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐴
24	23, 5	sstri 3978	. . . . . . . . . . . 12 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵
25	3, 8, 9	lubel 17734	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ CLat ∧ 𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ∧ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵) → 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))
26	24, 25	mp3an3 1446	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ CLat ∧ 𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) → 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))
27	22, 26	sylancom 590	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) → 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))
28	27	ex 415	. . . . . . . . 9 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} → 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})))
29	21, 28	syl5bir 245	. . . . . . . 8 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑋) → 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})))
30	29	expdimp 455	. . . . . . 7 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥 ≤ 𝑋 → 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})))
31		simpll3 1210	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐾 ∈ AtLat)
32		eqid 2823	. . . . . . . . . . . 12 ⊢ (0.‘𝐾) = (0.‘𝐾)
33	32, 4	atn0 36446	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ AtLat ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ (0.‘𝐾))
34	31, 33	sylancom 590	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≠ (0.‘𝐾))
35	34	adantr 483	. . . . . . . . 9 ⊢ (((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 ≤ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) → 𝑥 ≠ (0.‘𝐾))
36		simpl3 1189	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ AtLat)
37		atllat 36438	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
38	36, 37	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat)
39	38	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐾 ∈ Lat)
40	3, 4	atbase 36427	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)
41	40	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
42	3, 9	clatlubcl 17724	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ CLat ∧ {𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∈ 𝐵)
43	1, 24, 42	sylancl 588	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∈ 𝐵)
44	43	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∈ 𝐵)
45		simpl1 1187	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OML)
46		omlop 36379	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ OML → 𝐾 ∈ OP)
47	45, 46	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP)
48		eqid 2823	. . . . . . . . . . . . . . . . 17 ⊢ (oc‘𝐾) = (oc‘𝐾)
49	3, 48	opoccl 36332	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ OP ∧ ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) ∈ 𝐵) → ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) ∈ 𝐵)
50	47, 43, 49	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) ∈ 𝐵)
51	50	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋})) ∈ 𝐵)
52		eqid 2823	. . . . . . . . . . . . . . 15 ⊢ (meet‘𝐾) = (meet‘𝐾)
53	3, 8, 52	latlem12 17690	. . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . 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 36346	. . . . . . . . . . . . . . . 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 586	. . . . . . . . . . . . . . 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 5080	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . 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 36325	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ 𝐵) → (𝑥 ≤ (0.‘𝐾) ↔ 𝑥 = (0.‘𝐾)))
60	47, 40, 59	syl2an 597	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → (𝑥 ≤ (0.‘𝐾) ↔ 𝑥 = (0.‘𝐾)))
61	54, 58, 60	3bitrd 307	. . . . . . . . . . . 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 479	. . . . . . . . . . 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 459	. . . . . . . . . 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 3031	. . . . . . . . 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 415	. . . . . . 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 402	. . . . . 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 220	. . . . 5 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → ¬ (𝑥 ≤ 𝑋 ∧ 𝑥 ≤ ((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))))
70		simplr 767	. . . . . 6 ⊢ ((((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑋 ∈ 𝐵)
71	3, 8, 52	latlem12 17690	. . . . . 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 1368	. . . . 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 326	. . . 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 3272	. . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ¬ ∃𝑥 ∈ 𝐴 𝑥 ≤ (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))))
75		simpll3 1210	. . . . . 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 487	. . . . . . . 8 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
77	3, 52	latmcl 17664	. . . . . . . 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 1367	. . . . . . 7 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → (𝑋(meet‘𝐾)((oc‘𝐾)‘( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}))) ∈ 𝐵)
79	78	adantr 483	. . . . . 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 487	. . . . . 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 36454	. . . . . 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 1367	. . . . 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 415	. . . 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 3036	. . 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 36383	. . 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 1367	. 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 697	1 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → ( 1 ‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∃wrex 3141  {crab 3144   ⊆ wss 3938   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  lecple 16574  occoc 16575  Posetcpo 17552  lubclub 17554  meetcmee 17557  0.cp0 17649  Latclat 17657  CLatccla 17719  OPcops 36310  OMLcoml 36313  Atomscatm 36401  AtLatcal 36402

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-proset 17540  df-poset 17558  df-plt 17570  df-lub 17586  df-glb 17587  df-join 17588  df-meet 17589  df-p0 17651  df-lat 17658  df-clat 17720  df-oposet 36314  df-ol 36316  df-oml 36317  df-covers 36404  df-ats 36405  df-atl 36436

This theorem is referenced by:  atlatle  36458  hlatmstcOLDN  36535  pmaple  36899  pol1N  37048  polpmapN  37050  pmaplubN  37062