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Theorem atltcvr 38940
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 7433 . . . . . 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 38873 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴) β†’ (𝑅 ∨ 𝑅) = 𝑅)
62, 5syldan 589 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑅 ∨ 𝑅) = 𝑅)
71, 6sylan9eqr 2790 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑄 = 𝑅) β†’ (𝑄 ∨ 𝑅) = 𝑅)
87breq2d 5164 . . . 4 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑄 = 𝑅) β†’ (𝑃 < (𝑄 ∨ 𝑅) ↔ 𝑃 < 𝑅))
9 hlatl 38864 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
109adantr 479 . . . . . . 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 38817 . . . . . . 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 479 . . . 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 481 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
19 hllat 38867 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
2019adantr 479 . . . . . . 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 2728 . . . . . . . . 9 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2322, 4atbase 38793 . . . . . . . 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 38793 . . . . . . . 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 18438 . . . . . . 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 2728 . . . . . . 7 (leβ€˜πΎ) = (leβ€˜πΎ)
3029, 12pltle 18332 . . . . . 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 479 . . . 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 765 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑄 β‰  𝑅 ∧ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅))) β†’ 𝐾 ∈ HL)
34 simplr 767 . . . . . . . 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 483 . . . . . . . 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 466 . . . . . 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 38938 . . . . . 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 411 . . . 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 3026 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃 < (𝑄 ∨ 𝑅) β†’ 𝑃𝐢(𝑄 ∨ 𝑅)))
4422, 4atbase 38793 . . . 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 38774 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑃 < (𝑄 ∨ 𝑅))
4746ex 411 . . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2937   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  lecple 17247  ltcplt 18307  joincjn 18310  Latclat 18430   β‹– ccvr 38766  Atomscatm 38767  AtLatcal 38768  HLchlt 38854
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855
This theorem is referenced by:  atlt  38942  2atlt  38944  atexchltN  38946
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