Theorem llncmp 38198

Description: If two lattice lines are comparable, they are equal. (Contributed by NM, 19-Jun-2012.)

Hypotheses

Ref	Expression
llncmp.l	⊢ ≤ = (le‘𝐾)
llncmp.n	⊢ 𝑁 = (LLines‘𝐾)

Assertion

Ref	Expression
llncmp	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Proof of Theorem llncmp

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1137	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → 𝑋 ∈ 𝑁)
2		simp1 1136	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → 𝐾 ∈ HL)
3		eqid 2731	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		llncmp.n	. . . . . . 7 ⊢ 𝑁 = (LLines‘𝐾)
5	3, 4	llnbase 38185	. . . . . 6 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝐾))
6	5	3ad2ant2 1134	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐾))
7		eqid 2731	. . . . . 6 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
8		eqid 2731	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
9	3, 7, 8, 4	islln4 38183	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾)) → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋))
10	2, 6, 9	syl2anc 584	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋))
11	1, 10	mpbid 231	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋)
12		simpr3 1196	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ≤ 𝑌)
13		hlpos 38041	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
14	13	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → 𝐾 ∈ Poset)
15	14	adantr 481	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ Poset)
16	6	adantr 481	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ∈ (Base‘𝐾))
17		simpl3 1193	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ 𝑁)
18	3, 4	llnbase 38185	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑁 → 𝑌 ∈ (Base‘𝐾))
19	17, 18	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ (Base‘𝐾))
20		simpr1 1194	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑝 ∈ (Atoms‘𝐾))
21	3, 8	atbase 37964	. . . . . . . 8 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
22	20, 21	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑝 ∈ (Base‘𝐾))
23		simpr2 1195	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑋)
24		simpl1 1191	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ HL)
25		llncmp.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
26	3, 25, 7	cvrle 37953	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑝( ⋖ ‘𝐾)𝑋) → 𝑝 ≤ 𝑋)
27	24, 22, 16, 23, 26	syl31anc 1373	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑝 ≤ 𝑋)
28	3, 25	postr 18255	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ (𝑝 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑝 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) → 𝑝 ≤ 𝑌))
29	15, 22, 16, 19, 28	syl13anc 1372	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → ((𝑝 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) → 𝑝 ≤ 𝑌))
30	27, 12, 29	mp2and 697	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑝 ≤ 𝑌)
31	25, 7, 8, 4	atcvrlln2 38195	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑌 ∈ 𝑁) ∧ 𝑝 ≤ 𝑌) → 𝑝( ⋖ ‘𝐾)𝑌)
32	24, 20, 17, 30, 31	syl31anc 1373	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑌)
33	3, 25, 7	cvrcmp 37958	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) ∧ (𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑝( ⋖ ‘𝐾)𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
34	15, 16, 19, 22, 23, 32, 33	syl132anc 1388	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
35	12, 34	mpbid 231	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 = 𝑌)
36	35	3exp2 1354	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑝 ∈ (Atoms‘𝐾) → (𝑝( ⋖ ‘𝐾)𝑋 → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))))
37	36	rexlimdv 3152	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋 → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌)))
38	11, 37	mpd 15	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))
39	3, 25	posref 18253	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ (Base‘𝐾)) → 𝑋 ≤ 𝑋)
40	14, 6, 39	syl2anc 584	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → 𝑋 ≤ 𝑋)
41		breq2 5145	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
42	40, 41	syl5ibcom 244	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
43	38, 42	impbid 211	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3069   class class class wbr 5141  ‘cfv 6532  Basecbs 17126  lecple 17186  Posetcpo 18242   ⋖ ccvr 37937  Atomscatm 37938  HLchlt 38025  LLinesclln 38167

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-proset 18230  df-poset 18248  df-plt 18265  df-lub 18281  df-glb 18282  df-join 18283  df-meet 18284  df-p0 18360  df-lat 18367  df-clat 18434  df-oposet 37851  df-ol 37853  df-oml 37854  df-covers 37941  df-ats 37942  df-atl 37973  df-cvlat 37997  df-hlat 38026  df-llines 38174

This theorem is referenced by:  llnnlt  38199  2llnmat  38200  llnmlplnN  38215  dalem16  38355  dalem60  38408  llnexchb2  38545