Theorem 2atlt 40103

Description: Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)

Hypotheses

Ref	Expression
2atomslt.b	⊢ 𝐵 = (Base‘𝐾)
2atomslt.s	⊢ < = (lt‘𝐾)
2atomslt.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
2atlt	⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 < 𝑋))

Distinct variable groups:   𝐴,𝑞   𝐵,𝑞   𝐾,𝑞   𝑃,𝑞   < ,𝑞   𝑋,𝑞

Proof of Theorem 2atlt

Step	Hyp	Ref	Expression
1		2atomslt.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		2atomslt.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	atbase 39953	. . 3 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
4		eqid 2769	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
5		2atomslt.s	. . . 4 ⊢ < = (lt‘𝐾)
6		eqid 2769	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
7	1, 4, 5, 6, 2	hlrelat 40066	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞 ∈ 𝐴 (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋))
8	3, 7	syl3anl2 1438	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞 ∈ 𝐴 (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋))
9		simp3l 1218	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃 < (𝑃(join‘𝐾)𝑞))
10		simp1l1 1283	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝐾 ∈ HL)
11		simp1l2 1284	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃 ∈ 𝐴)
12		simp2 1153	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞 ∈ 𝐴)
13		eqid 2769	. . . . . . . . . 10 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
14	5, 6, 2, 13	atltcvr 40099	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴)) → (𝑃 < (𝑃(join‘𝐾)𝑞) ↔ 𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞)))
15	10, 11, 11, 12, 14	syl13anc 1397	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑃 < (𝑃(join‘𝐾)𝑞) ↔ 𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞)))
16	9, 15	mpbid 235	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞))
17	6, 13, 2	atcvr1 40081	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑞 ∈ 𝐴) → (𝑃 ≠ 𝑞 ↔ 𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞)))
18	10, 11, 12, 17	syl3anc 1396	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑃 ≠ 𝑞 ↔ 𝑃( ⋖ ‘𝐾)(𝑃(join‘𝐾)𝑞)))
19	16, 18	mpbird 260	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃 ≠ 𝑞)
20	19	necomd 3019	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞 ≠ 𝑃)
21	5, 6, 2	atlt 40101	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑞 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑞 < (𝑞(join‘𝐾)𝑃) ↔ 𝑞 ≠ 𝑃))
22	10, 12, 11, 21	syl3anc 1396	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑞 < (𝑞(join‘𝐾)𝑃) ↔ 𝑞 ≠ 𝑃))
23	20, 22	mpbird 260	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞 < (𝑞(join‘𝐾)𝑃))
24	10	hllatd 40028	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝐾 ∈ Lat)
25	11, 3	syl 18	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑃 ∈ 𝐵)
26	1, 2	atbase 39953	. . . . . . . . 9 ⊢ (𝑞 ∈ 𝐴 → 𝑞 ∈ 𝐵)
27	26	3ad2ant2 1150	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞 ∈ 𝐵)
28	1, 6	latjcom 18503	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑃(join‘𝐾)𝑞) = (𝑞(join‘𝐾)𝑃))
29	24, 25, 27, 28	syl3anc 1396	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑃(join‘𝐾)𝑞) = (𝑞(join‘𝐾)𝑃))
30	23, 29	breqtrrd 5143	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞 < (𝑃(join‘𝐾)𝑞))
31		simp3r 1219	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)
32		hlpos 40030	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
33	10, 32	syl 18	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝐾 ∈ Poset)
34	1, 6	latjcl 18495	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑃(join‘𝐾)𝑞) ∈ 𝐵)
35	24, 25, 27, 34	syl3anc 1396	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑃(join‘𝐾)𝑞) ∈ 𝐵)
36		simp1l3 1285	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑋 ∈ 𝐵)
37	1, 4, 5	pltletr 18397	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑞 ∈ 𝐵 ∧ (𝑃(join‘𝐾)𝑞) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑞 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑞 < 𝑋))
38	33, 27, 35, 36, 37	syl13anc 1397	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → ((𝑞 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑞 < 𝑋))
39	30, 31, 38	mp2and 711	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → 𝑞 < 𝑋)
40	20, 39	jca 520	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) ∧ 𝑞 ∈ 𝐴 ∧ (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋)) → (𝑞 ≠ 𝑃 ∧ 𝑞 < 𝑋))
41	40	3exp 1135	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) → (𝑞 ∈ 𝐴 → ((𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋) → (𝑞 ≠ 𝑃 ∧ 𝑞 < 𝑋))))
42	41	reximdvai 3182	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) → (∃𝑞 ∈ 𝐴 (𝑃 < (𝑃(join‘𝐾)𝑞) ∧ (𝑃(join‘𝐾)𝑞)(le‘𝐾)𝑋) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 < 𝑋)))
43	8, 42	mpd 16	1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑃 < 𝑋) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 < 𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  lecple 17317  Posetcpo 18363  ltcplt 18364  joincjn 18367  Latclat 18487   ⋖ ccvr 39926  Atomscatm 39927  HLchlt 40014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-proset 18350  df-poset 18369  df-plt 18384  df-lub 18400  df-glb 18401  df-join 18402  df-meet 18403  df-p0 18479  df-lat 18488  df-clat 18555  df-oposet 39840  df-ol 39842  df-oml 39843  df-covers 39930  df-ats 39931  df-atl 39962  df-cvlat 39986  df-hlat 40015

This theorem is referenced by:  cdlemb  40458  lhpexle1  40672