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Theorem atltcvr 38819
Description: An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)
Hypotheses
Ref Expression
atltcvr.s < = (ltβ€˜πΎ)
atltcvr.j ∨ = (joinβ€˜πΎ)
atltcvr.a 𝐴 = (Atomsβ€˜πΎ)
atltcvr.c 𝐢 = ( β‹– β€˜πΎ)
Assertion
Ref Expression
atltcvr ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃 < (𝑄 ∨ 𝑅) ↔ 𝑃𝐢(𝑄 ∨ 𝑅)))

Proof of Theorem atltcvr
StepHypRef Expression
1 oveq1 7412 . . . . . 6 (𝑄 = 𝑅 β†’ (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑅))
2 simpr3 1193 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅 ∈ 𝐴)
3 atltcvr.j . . . . . . . 8 ∨ = (joinβ€˜πΎ)
4 atltcvr.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
53, 4hlatjidm 38752 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴) β†’ (𝑅 ∨ 𝑅) = 𝑅)
62, 5syldan 590 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑅 ∨ 𝑅) = 𝑅)
71, 6sylan9eqr 2788 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑄 = 𝑅) β†’ (𝑄 ∨ 𝑅) = 𝑅)
87breq2d 5153 . . . 4 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑄 = 𝑅) β†’ (𝑃 < (𝑄 ∨ 𝑅) ↔ 𝑃 < 𝑅))
9 hlatl 38743 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
109adantr 480 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ AtLat)
11 simpr1 1191 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
12 atltcvr.s . . . . . . . 8 < = (ltβ€˜πΎ)
1312, 4atnlt 38696 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) β†’ Β¬ 𝑃 < 𝑅)
1410, 11, 2, 13syl3anc 1368 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ Β¬ 𝑃 < 𝑅)
1514pm2.21d 121 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃 < 𝑅 β†’ 𝑃𝐢(𝑄 ∨ 𝑅)))
1615adantr 480 . . . 4 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑄 = 𝑅) β†’ (𝑃 < 𝑅 β†’ 𝑃𝐢(𝑄 ∨ 𝑅)))
178, 16sylbid 239 . . 3 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑄 = 𝑅) β†’ (𝑃 < (𝑄 ∨ 𝑅) β†’ 𝑃𝐢(𝑄 ∨ 𝑅)))
18 simpl 482 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
19 hllat 38746 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
2019adantr 480 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
21 simpr2 1192 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
22 eqid 2726 . . . . . . . . 9 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2322, 4atbase 38672 . . . . . . . 8 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
2421, 23syl 17 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
2522, 4atbase 38672 . . . . . . . 8 (𝑅 ∈ 𝐴 β†’ 𝑅 ∈ (Baseβ€˜πΎ))
262, 25syl 17 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅 ∈ (Baseβ€˜πΎ))
2722, 3latjcl 18404 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ)) β†’ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ))
2820, 24, 26, 27syl3anc 1368 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ))
29 eqid 2726 . . . . . . 7 (leβ€˜πΎ) = (leβ€˜πΎ)
3029, 12pltle 18298 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ)) β†’ (𝑃 < (𝑄 ∨ 𝑅) β†’ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅)))
3118, 11, 28, 30syl3anc 1368 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃 < (𝑄 ∨ 𝑅) β†’ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅)))
3231adantr 480 . . . 4 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑄 β‰  𝑅) β†’ (𝑃 < (𝑄 ∨ 𝑅) β†’ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅)))
33 simpll 764 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑄 β‰  𝑅 ∧ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅))) β†’ 𝐾 ∈ HL)
34 simplr 766 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑄 β‰  𝑅 ∧ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅))) β†’ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴))
35 simpr 484 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑄 β‰  𝑅 ∧ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅))) β†’ (𝑄 β‰  𝑅 ∧ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅)))
3633, 34, 353jca 1125 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑄 β‰  𝑅 ∧ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅))) β†’ (𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅))))
3736anassrs 467 . . . . . 6 ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑄 β‰  𝑅) ∧ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅)) β†’ (𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅))))
38 atltcvr.c . . . . . . 7 𝐢 = ( β‹– β€˜πΎ)
3929, 3, 38, 4atcvrj2 38817 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅))) β†’ 𝑃𝐢(𝑄 ∨ 𝑅))
4037, 39syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑄 β‰  𝑅) ∧ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅)) β†’ 𝑃𝐢(𝑄 ∨ 𝑅))
4140ex 412 . . . 4 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑄 β‰  𝑅) β†’ (𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅) β†’ 𝑃𝐢(𝑄 ∨ 𝑅)))
4232, 41syld 47 . . 3 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑄 β‰  𝑅) β†’ (𝑃 < (𝑄 ∨ 𝑅) β†’ 𝑃𝐢(𝑄 ∨ 𝑅)))
4317, 42pm2.61dane 3023 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃 < (𝑄 ∨ 𝑅) β†’ 𝑃𝐢(𝑄 ∨ 𝑅)))
4422, 4atbase 38672 . . . 4 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
4511, 44syl 17 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
4622, 12, 38cvrlt 38653 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑃 < (𝑄 ∨ 𝑅))
4746ex 412 . . 3 ((𝐾 ∈ HL ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ)) β†’ (𝑃𝐢(𝑄 ∨ 𝑅) β†’ 𝑃 < (𝑄 ∨ 𝑅)))
4818, 45, 28, 47syl3anc 1368 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃𝐢(𝑄 ∨ 𝑅) β†’ 𝑃 < (𝑄 ∨ 𝑅)))
4943, 48impbid 211 1 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃 < (𝑄 ∨ 𝑅) ↔ 𝑃𝐢(𝑄 ∨ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  ltcplt 18273  joincjn 18276  Latclat 18396   β‹– ccvr 38645  Atomscatm 38646  AtLatcal 38647  HLchlt 38733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  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 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734
This theorem is referenced by:  atlt  38821  2atlt  38823  atexchltN  38825
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