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Theorem atltcvr 37944
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 7365 . . . . . 6 (𝑄 = 𝑅 β†’ (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑅))
2 simpr3 1197 . . . . . . 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 37877 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴) β†’ (𝑅 ∨ 𝑅) = 𝑅)
62, 5syldan 592 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑅 ∨ 𝑅) = 𝑅)
71, 6sylan9eqr 2795 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑄 = 𝑅) β†’ (𝑄 ∨ 𝑅) = 𝑅)
87breq2d 5118 . . . 4 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑄 = 𝑅) β†’ (𝑃 < (𝑄 ∨ 𝑅) ↔ 𝑃 < 𝑅))
9 hlatl 37868 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
109adantr 482 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ AtLat)
11 simpr1 1195 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
12 atltcvr.s . . . . . . . 8 < = (ltβ€˜πΎ)
1312, 4atnlt 37821 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) β†’ Β¬ 𝑃 < 𝑅)
1410, 11, 2, 13syl3anc 1372 . . . . . 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 482 . . . 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 484 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
19 hllat 37871 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
2019adantr 482 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
21 simpr2 1196 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
22 eqid 2733 . . . . . . . . 9 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2322, 4atbase 37797 . . . . . . . 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 37797 . . . . . . . 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 18333 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ)) β†’ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ))
2820, 24, 26, 27syl3anc 1372 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ))
29 eqid 2733 . . . . . . 7 (leβ€˜πΎ) = (leβ€˜πΎ)
3029, 12pltle 18227 . . . . . 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 1372 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃 < (𝑄 ∨ 𝑅) β†’ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅)))
3231adantr 482 . . . 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 766 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (𝑄 β‰  𝑅 ∧ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑅))) β†’ 𝐾 ∈ HL)
34 simplr 768 . . . . . . . 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 486 . . . . . . . 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 1129 . . . . . . 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 469 . . . . . 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 37942 . . . . . 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 414 . . . 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 3029 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃 < (𝑄 ∨ 𝑅) β†’ 𝑃𝐢(𝑄 ∨ 𝑅)))
4422, 4atbase 37797 . . . 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 37778 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑃 < (𝑄 ∨ 𝑅))
4746ex 414 . . 3 ((𝐾 ∈ HL ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ)) β†’ (𝑃𝐢(𝑄 ∨ 𝑅) β†’ 𝑃 < (𝑄 ∨ 𝑅)))
4818, 45, 28, 47syl3anc 1372 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  ltcplt 18202  joincjn 18205  Latclat 18325   β‹– ccvr 37770  Atomscatm 37771  AtLatcal 37772  HLchlt 37858
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-proset 18189  df-poset 18207  df-plt 18224  df-lub 18240  df-glb 18241  df-join 18242  df-meet 18243  df-p0 18319  df-lat 18326  df-clat 18393  df-oposet 37684  df-ol 37686  df-oml 37687  df-covers 37774  df-ats 37775  df-atl 37806  df-cvlat 37830  df-hlat 37859
This theorem is referenced by:  atlt  37946  2atlt  37948  atexchltN  37950
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