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Theorem 2atm2atN 38656
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 38232 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
21adantr 482 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ OP)
3 simpr3 1197 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅 ∈ 𝐴)
4 2atm2at.z . . . . 5 0 = (0.β€˜πΎ)
5 eqid 2733 . . . . 5 (ltβ€˜πΎ) = (ltβ€˜πΎ)
6 2atm2at.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
74, 5, 60ltat 38161 . . . 4 ((𝐾 ∈ OP ∧ 𝑅 ∈ 𝐴) β†’ 0 (ltβ€˜πΎ)𝑅)
82, 3, 7syl2anc 585 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 0 (ltβ€˜πΎ)𝑅)
9 simpl 484 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
10 simpr1 1195 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
11 eqid 2733 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
12 2atm2at.j . . . . . 6 ∨ = (joinβ€˜πΎ)
1311, 12, 6hlatlej1 38245 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) β†’ 𝑅(leβ€˜πΎ)(𝑅 ∨ 𝑃))
149, 3, 10, 13syl3anc 1372 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅(leβ€˜πΎ)(𝑅 ∨ 𝑃))
15 simpr2 1196 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
1611, 12, 6hlatlej1 38245 . . . . 5 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑅(leβ€˜πΎ)(𝑅 ∨ 𝑄))
179, 3, 15, 16syl3anc 1372 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅(leβ€˜πΎ)(𝑅 ∨ 𝑄))
18 hllat 38233 . . . . . 6 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1918adantr 482 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
20 eqid 2733 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2120, 6atbase 38159 . . . . . 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 38237 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) β†’ (𝑅 ∨ 𝑃) ∈ (Baseβ€˜πΎ))
249, 3, 10, 23syl3anc 1372 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑅 ∨ 𝑃) ∈ (Baseβ€˜πΎ))
2520, 12, 6hlatjcl 38237 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑅 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
269, 3, 15, 25syl3anc 1372 . . . . 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 18419 . . . . 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 1373 . . . 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 711 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅(leβ€˜πΎ)((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)))
31 hlpos 38236 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ Poset)
3231adantr 482 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ Poset)
3320, 4op0cl 38054 . . . . 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 18393 . . . . 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 1372 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)) ∈ (Baseβ€˜πΎ))
3720, 11, 5pltletr 18296 . . . 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 1373 . . 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 698 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 0 (ltβ€˜πΎ)((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)))
4020, 5, 4opltn0 38060 . . 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 585 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  Posetcpo 18260  ltcplt 18261  joincjn 18264  meetcmee 18265  0.cp0 18376  Latclat 18384  OPcops 38042  Atomscatm 38133  HLchlt 38220
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-lat 18385  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221
This theorem is referenced by: (None)
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