Theorem llncmp 39516

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 2729	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		llncmp.n	. . . . . . 7 ⊢ 𝑁 = (LLines‘𝐾)
5	3, 4	llnbase 39503	. . . . . 6 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝐾))
6	5	3ad2ant2 1134	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → 𝑋 ∈ (Base‘𝐾))
7		eqid 2729	. . . . . 6 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
8		eqid 2729	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
9	3, 7, 8, 4	islln4 39501	. . . . 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 232	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → ∃𝑝 ∈ (Atoms‘𝐾)𝑝( ⋖ ‘𝐾)𝑋)
12		simpr3 1197	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ≤ 𝑌)
13		hlpos 39359	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
14	13	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → 𝐾 ∈ Poset)
15	14	adantr 480	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝐾 ∈ Poset)
16	6	adantr 480	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 ∈ (Base‘𝐾))
17		simpl3 1194	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑌 ∈ 𝑁)
18	3, 4	llnbase 39503	. . . . . . . 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 1195	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑝 ∈ (Atoms‘𝐾))
21	3, 8	atbase 39282	. . . . . . . 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 1196	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑋)
24		simpl1 1192	. . . . . . . 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 39271	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑝 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾)) ∧ 𝑝( ⋖ ‘𝐾)𝑋) → 𝑝 ≤ 𝑋)
27	24, 22, 16, 23, 26	syl31anc 1375	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑝 ≤ 𝑋)
28	3, 25	postr 18281	. . . . . . . . . 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 1374	. . . . . . . . 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 699	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑝 ≤ 𝑌)
31	25, 7, 8, 4	atcvrlln2 39513	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑌 ∈ 𝑁) ∧ 𝑝 ≤ 𝑌) → 𝑝( ⋖ ‘𝐾)𝑌)
32	24, 20, 17, 30, 31	syl31anc 1375	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑝( ⋖ ‘𝐾)𝑌)
33	3, 25, 7	cvrcmp 39276	. . . . . . 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 1390	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
35	12, 34	mpbid 232	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝( ⋖ ‘𝐾)𝑋 ∧ 𝑋 ≤ 𝑌)) → 𝑋 = 𝑌)
36	35	3exp2 1355	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑝 ∈ (Atoms‘𝐾) → (𝑝( ⋖ ‘𝐾)𝑋 → (𝑋 ≤ 𝑌 → 𝑋 = 𝑌))))
37	36	rexlimdv 3132	. . 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 18279	. . . 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 5111	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑌))
42	40, 41	syl5ibcom 245	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 = 𝑌 → 𝑋 ≤ 𝑌))
43	38, 42	impbid 212	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   class class class wbr 5107  ‘cfv 6511  Basecbs 17179  lecple 17227  Posetcpo 18268   ⋖ ccvr 39255  Atomscatm 39256  HLchlt 39343  LLinesclln 39485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-lat 18391  df-clat 18458  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344  df-llines 39492

This theorem is referenced by:  llnnlt  39517  2llnmat  39518  llnmlplnN  39533  dalem16  39673  dalem60  39726  llnexchb2  39863