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Theorem atcvrj2b 33535
Description: Condition for an atom to be covered by the join of two others. (Contributed by NM, 7-Feb-2012.)
Hypotheses
Ref Expression
atcvrj1x.l = (le‘𝐾)
atcvrj1x.j = (join‘𝐾)
atcvrj1x.c 𝐶 = ( ⋖ ‘𝐾)
atcvrj1x.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
atcvrj2b ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑄𝑅𝑃 (𝑄 𝑅)) ↔ 𝑃𝐶(𝑄 𝑅)))

Proof of Theorem atcvrj2b
StepHypRef Expression
1 simpl3l 1108 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑄𝑅)
21necomd 2832 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝑄)
3 simpl1 1056 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝐾 ∈ HL)
4 simpl23 1133 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝐴)
5 simpl22 1132 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑄𝐴)
6 atcvrj1x.j . . . . . . . 8 = (join‘𝐾)
7 atcvrj1x.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
8 atcvrj1x.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
96, 7, 8atcvr2 33521 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑄𝐴) → (𝑅𝑄𝑅𝐶(𝑄 𝑅)))
103, 4, 5, 9syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → (𝑅𝑄𝑅𝐶(𝑄 𝑅)))
112, 10mpbid 220 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝐶(𝑄 𝑅))
12 breq1 4576 . . . . . 6 (𝑃 = 𝑅 → (𝑃𝐶(𝑄 𝑅) ↔ 𝑅𝐶(𝑄 𝑅)))
1312adantl 480 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → (𝑃𝐶(𝑄 𝑅) ↔ 𝑅𝐶(𝑄 𝑅)))
1411, 13mpbird 245 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑃𝐶(𝑄 𝑅))
15 simpl1 1056 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝐾 ∈ HL)
16 simpl2 1057 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → (𝑃𝐴𝑄𝐴𝑅𝐴))
17 simpr 475 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝑃𝑅)
18 simpl3r 1109 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝑃 (𝑄 𝑅))
19 atcvrj1x.l . . . . . 6 = (le‘𝐾)
2019, 6, 7, 8atcvrj1 33534 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑅𝑃 (𝑄 𝑅))) → 𝑃𝐶(𝑄 𝑅))
2115, 16, 17, 18, 20syl112anc 1321 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝑃𝐶(𝑄 𝑅))
2214, 21pm2.61dane 2864 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) → 𝑃𝐶(𝑄 𝑅))
23223expia 1258 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑄𝑅𝑃 (𝑄 𝑅)) → 𝑃𝐶(𝑄 𝑅)))
24 hlatl 33464 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2524ad2antrr 757 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝐾 ∈ AtLat)
26 simplr1 1095 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃𝐴)
27 eqid 2605 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
2827, 8atn0 33412 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃 ≠ (0.‘𝐾))
2925, 26, 28syl2anc 690 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 ≠ (0.‘𝐾))
30 simpll 785 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝐾 ∈ HL)
31 eqid 2605 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3231, 8atbase 33393 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3326, 32syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 ∈ (Base‘𝐾))
34 simplr2 1096 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄𝐴)
35 simplr3 1097 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑅𝐴)
36 simpr 475 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃𝐶(𝑄 𝑅))
3731, 6, 27, 7, 8atcvrj0 33531 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄𝐴𝑅𝐴) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅))
3830, 33, 34, 35, 36, 37syl131anc 1330 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅))
3938necon3bid 2821 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑃 ≠ (0.‘𝐾) ↔ 𝑄𝑅))
4029, 39mpbid 220 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄𝑅)
41 hllat 33467 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ Lat)
4241ad2antrr 757 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝐾 ∈ Lat)
4331, 8atbase 33393 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
4434, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄 ∈ (Base‘𝐾))
4531, 8atbase 33393 . . . . . . . 8 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
4635, 45syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑅 ∈ (Base‘𝐾))
4731, 6latjcl 16816 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑄 𝑅) ∈ (Base‘𝐾))
4842, 44, 46, 47syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑄 𝑅) ∈ (Base‘𝐾))
4930, 33, 483jca 1234 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)))
5031, 19, 7cvrle 33382 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 (𝑄 𝑅))
5149, 50sylancom 697 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 (𝑄 𝑅))
5240, 51jca 552 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑄𝑅𝑃 (𝑄 𝑅)))
5352ex 448 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃𝐶(𝑄 𝑅) → (𝑄𝑅𝑃 (𝑄 𝑅))))
5423, 53impbid 200 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑄𝑅𝑃 (𝑄 𝑅)) ↔ 𝑃𝐶(𝑄 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775   class class class wbr 4573  cfv 5786  (class class class)co 6523  Basecbs 15637  lecple 15717  joincjn 16709  0.cp0 16802  Latclat 16810  ccvr 33366  Atomscatm 33367  AtLatcal 33368  HLchlt 33454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-preset 16693  df-poset 16711  df-plt 16723  df-lub 16739  df-glb 16740  df-join 16741  df-meet 16742  df-p0 16804  df-lat 16811  df-clat 16873  df-oposet 33280  df-ol 33282  df-oml 33283  df-covers 33370  df-ats 33371  df-atl 33402  df-cvlat 33426  df-hlat 33455
This theorem is referenced by:  atcvrj2  33536
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