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Theorem 2atm2atN 38959
Description: Two joins with a common atom have a nonzero meet. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
2atm2at.j ∨ = (joinβ€˜πΎ)
2atm2at.m ∧ = (meetβ€˜πΎ)
2atm2at.z 0 = (0.β€˜πΎ)
2atm2at.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
2atm2atN ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)) β‰  0 )

Proof of Theorem 2atm2atN
StepHypRef Expression
1 hlop 38535 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
21adantr 479 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ OP)
3 simpr3 1194 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅 ∈ 𝐴)
4 2atm2at.z . . . . 5 0 = (0.β€˜πΎ)
5 eqid 2730 . . . . 5 (ltβ€˜πΎ) = (ltβ€˜πΎ)
6 2atm2at.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
74, 5, 60ltat 38464 . . . 4 ((𝐾 ∈ OP ∧ 𝑅 ∈ 𝐴) β†’ 0 (ltβ€˜πΎ)𝑅)
82, 3, 7syl2anc 582 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 0 (ltβ€˜πΎ)𝑅)
9 simpl 481 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
10 simpr1 1192 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
11 eqid 2730 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
12 2atm2at.j . . . . . 6 ∨ = (joinβ€˜πΎ)
1311, 12, 6hlatlej1 38548 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) β†’ 𝑅(leβ€˜πΎ)(𝑅 ∨ 𝑃))
149, 3, 10, 13syl3anc 1369 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅(leβ€˜πΎ)(𝑅 ∨ 𝑃))
15 simpr2 1193 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
1611, 12, 6hlatlej1 38548 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑅(leβ€˜πΎ)(𝑅 ∨ 𝑄))
179, 3, 15, 16syl3anc 1369 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅(leβ€˜πΎ)(𝑅 ∨ 𝑄))
18 hllat 38536 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1918adantr 479 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
20 eqid 2730 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2120, 6atbase 38462 . . . . . 6 (𝑅 ∈ 𝐴 β†’ 𝑅 ∈ (Baseβ€˜πΎ))
223, 21syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅 ∈ (Baseβ€˜πΎ))
2320, 12, 6hlatjcl 38540 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) β†’ (𝑅 ∨ 𝑃) ∈ (Baseβ€˜πΎ))
249, 3, 10, 23syl3anc 1369 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑅 ∨ 𝑃) ∈ (Baseβ€˜πΎ))
2520, 12, 6hlatjcl 38540 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑅 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
269, 3, 15, 25syl3anc 1369 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑅 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
27 2atm2at.m . . . . . 6 ∧ = (meetβ€˜πΎ)
2820, 11, 27latlem12 18423 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Baseβ€˜πΎ) ∧ (𝑅 ∨ 𝑃) ∈ (Baseβ€˜πΎ) ∧ (𝑅 ∨ 𝑄) ∈ (Baseβ€˜πΎ))) β†’ ((𝑅(leβ€˜πΎ)(𝑅 ∨ 𝑃) ∧ 𝑅(leβ€˜πΎ)(𝑅 ∨ 𝑄)) ↔ 𝑅(leβ€˜πΎ)((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄))))
2919, 22, 24, 26, 28syl13anc 1370 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑅(leβ€˜πΎ)(𝑅 ∨ 𝑃) ∧ 𝑅(leβ€˜πΎ)(𝑅 ∨ 𝑄)) ↔ 𝑅(leβ€˜πΎ)((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄))))
3014, 17, 29mpbi2and 708 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅(leβ€˜πΎ)((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)))
31 hlpos 38539 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ Poset)
3231adantr 479 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ Poset)
3320, 4op0cl 38357 . . . . 5 (𝐾 ∈ OP β†’ 0 ∈ (Baseβ€˜πΎ))
342, 33syl 17 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 0 ∈ (Baseβ€˜πΎ))
3520, 27latmcl 18397 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑅 ∨ 𝑃) ∈ (Baseβ€˜πΎ) ∧ (𝑅 ∨ 𝑄) ∈ (Baseβ€˜πΎ)) β†’ ((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)) ∈ (Baseβ€˜πΎ))
3619, 24, 26, 35syl3anc 1369 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)) ∈ (Baseβ€˜πΎ))
3720, 11, 5pltletr 18300 . . . 4 ((𝐾 ∈ Poset ∧ ( 0 ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ) ∧ ((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)) ∈ (Baseβ€˜πΎ))) β†’ (( 0 (ltβ€˜πΎ)𝑅 ∧ 𝑅(leβ€˜πΎ)((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄))) β†’ 0 (ltβ€˜πΎ)((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄))))
3832, 34, 22, 36, 37syl13anc 1370 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (( 0 (ltβ€˜πΎ)𝑅 ∧ 𝑅(leβ€˜πΎ)((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄))) β†’ 0 (ltβ€˜πΎ)((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄))))
398, 30, 38mp2and 695 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 0 (ltβ€˜πΎ)((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)))
4020, 5, 4opltn0 38363 . . 3 ((𝐾 ∈ OP ∧ ((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)) ∈ (Baseβ€˜πΎ)) β†’ ( 0 (ltβ€˜πΎ)((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)) ↔ ((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)) β‰  0 ))
412, 36, 40syl2anc 582 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ( 0 (ltβ€˜πΎ)((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)) ↔ ((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)) β‰  0 ))
4239, 41mpbid 231 1 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)) β‰  0 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  lecple 17208  Posetcpo 18264  ltcplt 18265  joincjn 18268  meetcmee 18269  0.cp0 18380  Latclat 18388  OPcops 38345  Atomscatm 38436  HLchlt 38523
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-proset 18252  df-poset 18270  df-plt 18287  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-p0 18382  df-lat 18389  df-oposet 38349  df-ol 38351  df-oml 38352  df-covers 38439  df-ats 38440  df-atl 38471  df-cvlat 38495  df-hlat 38524
This theorem is referenced by: (None)
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