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Theorem atcvrj2b 36438
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 1222 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑄𝑅)
21necomd 3076 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝑄)
3 simpl1 1185 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝐾 ∈ HL)
4 simpl23 1247 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝐴)
5 simpl22 1246 . . . . . . 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 36424 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑄𝐴) → (𝑅𝑄𝑅𝐶(𝑄 𝑅)))
103, 4, 5, 9syl3anc 1365 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → (𝑅𝑄𝑅𝐶(𝑄 𝑅)))
112, 10mpbid 233 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝐶(𝑄 𝑅))
12 breq1 5066 . . . . . 6 (𝑃 = 𝑅 → (𝑃𝐶(𝑄 𝑅) ↔ 𝑅𝐶(𝑄 𝑅)))
1312adantl 482 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → (𝑃𝐶(𝑄 𝑅) ↔ 𝑅𝐶(𝑄 𝑅)))
1411, 13mpbird 258 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑃𝐶(𝑄 𝑅))
15 simpl1 1185 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝐾 ∈ HL)
16 simpl2 1186 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → (𝑃𝐴𝑄𝐴𝑅𝐴))
17 simpr 485 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝑃𝑅)
18 simpl3r 1223 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝑃 (𝑄 𝑅))
19 atcvrj1x.l . . . . . 6 = (le‘𝐾)
2019, 6, 7, 8atcvrj1 36437 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑅𝑃 (𝑄 𝑅))) → 𝑃𝐶(𝑄 𝑅))
2115, 16, 17, 18, 20syl112anc 1368 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝑃𝐶(𝑄 𝑅))
2214, 21pm2.61dane 3109 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) → 𝑃𝐶(𝑄 𝑅))
23223expia 1115 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑄𝑅𝑃 (𝑄 𝑅)) → 𝑃𝐶(𝑄 𝑅)))
24 hlatl 36366 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2524ad2antrr 722 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝐾 ∈ AtLat)
26 simplr1 1209 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃𝐴)
27 eqid 2826 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
2827, 8atn0 36314 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃 ≠ (0.‘𝐾))
2925, 26, 28syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 ≠ (0.‘𝐾))
30 simpll 763 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝐾 ∈ HL)
31 eqid 2826 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3231, 8atbase 36295 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3326, 32syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 ∈ (Base‘𝐾))
34 simplr2 1210 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄𝐴)
35 simplr3 1211 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑅𝐴)
36 simpr 485 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃𝐶(𝑄 𝑅))
3731, 6, 27, 7, 8atcvrj0 36434 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄𝐴𝑅𝐴) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅))
3830, 33, 34, 35, 36, 37syl131anc 1377 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅))
3938necon3bid 3065 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑃 ≠ (0.‘𝐾) ↔ 𝑄𝑅))
4029, 39mpbid 233 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄𝑅)
41 hllat 36369 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ Lat)
4241ad2antrr 722 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝐾 ∈ Lat)
4331, 8atbase 36295 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
4434, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄 ∈ (Base‘𝐾))
4531, 8atbase 36295 . . . . . . . 8 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
4635, 45syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑅 ∈ (Base‘𝐾))
4731, 6latjcl 17651 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑄 𝑅) ∈ (Base‘𝐾))
4842, 44, 46, 47syl3anc 1365 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑄 𝑅) ∈ (Base‘𝐾))
4930, 33, 483jca 1122 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)))
5031, 19, 7cvrle 36284 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 (𝑄 𝑅))
5149, 50sylancom 588 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 (𝑄 𝑅))
5240, 51jca 512 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑄𝑅𝑃 (𝑄 𝑅)))
5352ex 413 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃𝐶(𝑄 𝑅) → (𝑄𝑅𝑃 (𝑄 𝑅))))
5423, 53impbid 213 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑄𝑅𝑃 (𝑄 𝑅)) ↔ 𝑃𝐶(𝑄 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021   class class class wbr 5063  cfv 6352  (class class class)co 7148  Basecbs 16473  lecple 16562  joincjn 17544  0.cp0 17637  Latclat 17645  ccvr 36268  Atomscatm 36269  AtLatcal 36270  HLchlt 36356
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-proset 17528  df-poset 17546  df-plt 17558  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-p0 17639  df-lat 17646  df-clat 17708  df-oposet 36182  df-ol 36184  df-oml 36185  df-covers 36272  df-ats 36273  df-atl 36304  df-cvlat 36328  df-hlat 36357
This theorem is referenced by:  atcvrj2  36439
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