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Theorem llncmp 38014
Description: If two lattice lines are comparable, they are equal. (Contributed by NM, 19-Jun-2012.)
Hypotheses
Ref Expression
llncmp.l ≀ = (leβ€˜πΎ)
llncmp.n 𝑁 = (LLinesβ€˜πΎ)
Assertion
Ref Expression
llncmp ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 = π‘Œ))

Proof of Theorem llncmp
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 simp2 1138 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) β†’ 𝑋 ∈ 𝑁)
2 simp1 1137 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) β†’ 𝐾 ∈ HL)
3 eqid 2737 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4 llncmp.n . . . . . . 7 𝑁 = (LLinesβ€˜πΎ)
53, 4llnbase 38001 . . . . . 6 (𝑋 ∈ 𝑁 β†’ 𝑋 ∈ (Baseβ€˜πΎ))
653ad2ant2 1135 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) β†’ 𝑋 ∈ (Baseβ€˜πΎ))
7 eqid 2737 . . . . . 6 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
8 eqid 2737 . . . . . 6 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
93, 7, 8, 4islln4 37999 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋 ∈ (Baseβ€˜πΎ)) β†’ (𝑋 ∈ 𝑁 ↔ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)𝑝( β‹– β€˜πΎ)𝑋))
102, 6, 9syl2anc 585 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) β†’ (𝑋 ∈ 𝑁 ↔ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)𝑝( β‹– β€˜πΎ)𝑋))
111, 10mpbid 231 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) β†’ βˆƒπ‘ ∈ (Atomsβ€˜πΎ)𝑝( β‹– β€˜πΎ)𝑋)
12 simpr3 1197 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑋 ≀ π‘Œ)
13 hlpos 37857 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ Poset)
14133ad2ant1 1134 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) β†’ 𝐾 ∈ Poset)
1514adantr 482 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝐾 ∈ Poset)
166adantr 482 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑋 ∈ (Baseβ€˜πΎ))
17 simpl3 1194 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ π‘Œ ∈ 𝑁)
183, 4llnbase 38001 . . . . . . . 8 (π‘Œ ∈ 𝑁 β†’ π‘Œ ∈ (Baseβ€˜πΎ))
1917, 18syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ π‘Œ ∈ (Baseβ€˜πΎ))
20 simpr1 1195 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑝 ∈ (Atomsβ€˜πΎ))
213, 8atbase 37780 . . . . . . . 8 (𝑝 ∈ (Atomsβ€˜πΎ) β†’ 𝑝 ∈ (Baseβ€˜πΎ))
2220, 21syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑝 ∈ (Baseβ€˜πΎ))
23 simpr2 1196 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑝( β‹– β€˜πΎ)𝑋)
24 simpl1 1192 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝐾 ∈ HL)
25 llncmp.l . . . . . . . . . . 11 ≀ = (leβ€˜πΎ)
263, 25, 7cvrle 37769 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Baseβ€˜πΎ) ∧ 𝑋 ∈ (Baseβ€˜πΎ)) ∧ 𝑝( β‹– β€˜πΎ)𝑋) β†’ 𝑝 ≀ 𝑋)
2724, 22, 16, 23, 26syl31anc 1374 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑝 ≀ 𝑋)
283, 25postr 18216 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ (𝑝 ∈ (Baseβ€˜πΎ) ∧ 𝑋 ∈ (Baseβ€˜πΎ) ∧ π‘Œ ∈ (Baseβ€˜πΎ))) β†’ ((𝑝 ≀ 𝑋 ∧ 𝑋 ≀ π‘Œ) β†’ 𝑝 ≀ π‘Œ))
2915, 22, 16, 19, 28syl13anc 1373 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ ((𝑝 ≀ 𝑋 ∧ 𝑋 ≀ π‘Œ) β†’ 𝑝 ≀ π‘Œ))
3027, 12, 29mp2and 698 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑝 ≀ π‘Œ)
3125, 7, 8, 4atcvrlln2 38011 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑝 ∈ (Atomsβ€˜πΎ) ∧ π‘Œ ∈ 𝑁) ∧ 𝑝 ≀ π‘Œ) β†’ 𝑝( β‹– β€˜πΎ)π‘Œ)
3224, 20, 17, 30, 31syl31anc 1374 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑝( β‹– β€˜πΎ)π‘Œ)
333, 25, 7cvrcmp 37774 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑋 ∈ (Baseβ€˜πΎ) ∧ π‘Œ ∈ (Baseβ€˜πΎ) ∧ 𝑝 ∈ (Baseβ€˜πΎ)) ∧ (𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑝( β‹– β€˜πΎ)π‘Œ)) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 = π‘Œ))
3415, 16, 19, 22, 23, 32, 33syl132anc 1389 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 = π‘Œ))
3512, 34mpbid 231 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ 𝑝( β‹– β€˜πΎ)𝑋 ∧ 𝑋 ≀ π‘Œ)) β†’ 𝑋 = π‘Œ)
36353exp2 1355 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) β†’ (𝑝 ∈ (Atomsβ€˜πΎ) β†’ (𝑝( β‹– β€˜πΎ)𝑋 β†’ (𝑋 ≀ π‘Œ β†’ 𝑋 = π‘Œ))))
3736rexlimdv 3151 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) β†’ (βˆƒπ‘ ∈ (Atomsβ€˜πΎ)𝑝( β‹– β€˜πΎ)𝑋 β†’ (𝑋 ≀ π‘Œ β†’ 𝑋 = π‘Œ)))
3811, 37mpd 15 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) β†’ (𝑋 ≀ π‘Œ β†’ 𝑋 = π‘Œ))
393, 25posref 18214 . . . 4 ((𝐾 ∈ Poset ∧ 𝑋 ∈ (Baseβ€˜πΎ)) β†’ 𝑋 ≀ 𝑋)
4014, 6, 39syl2anc 585 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) β†’ 𝑋 ≀ 𝑋)
41 breq2 5114 . . 3 (𝑋 = π‘Œ β†’ (𝑋 ≀ 𝑋 ↔ 𝑋 ≀ π‘Œ))
4240, 41syl5ibcom 244 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) β†’ (𝑋 = π‘Œ β†’ 𝑋 ≀ π‘Œ))
4338, 42impbid 211 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) β†’ (𝑋 ≀ π‘Œ ↔ 𝑋 = π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074   class class class wbr 5110  β€˜cfv 6501  Basecbs 17090  lecple 17147  Posetcpo 18203   β‹– ccvr 37753  Atomscatm 37754  HLchlt 37841  LLinesclln 37983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  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 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-proset 18191  df-poset 18209  df-plt 18226  df-lub 18242  df-glb 18243  df-join 18244  df-meet 18245  df-p0 18321  df-lat 18328  df-clat 18395  df-oposet 37667  df-ol 37669  df-oml 37670  df-covers 37757  df-ats 37758  df-atl 37789  df-cvlat 37813  df-hlat 37842  df-llines 37990
This theorem is referenced by:  llnnlt  38015  2llnmat  38016  llnmlplnN  38031  dalem16  38171  dalem60  38224  llnexchb2  38361
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