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Theorem atcvrj2b 35500
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 1305 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑄𝑅)
21necomd 3054 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝑄)
3 simpl1 1246 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝐾 ∈ HL)
4 simpl23 1343 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝐴)
5 simpl22 1341 . . . . . . 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 35486 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑄𝐴) → (𝑅𝑄𝑅𝐶(𝑄 𝑅)))
103, 4, 5, 9syl3anc 1494 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → (𝑅𝑄𝑅𝐶(𝑄 𝑅)))
112, 10mpbid 224 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝐶(𝑄 𝑅))
12 breq1 4876 . . . . . 6 (𝑃 = 𝑅 → (𝑃𝐶(𝑄 𝑅) ↔ 𝑅𝐶(𝑄 𝑅)))
1312adantl 475 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → (𝑃𝐶(𝑄 𝑅) ↔ 𝑅𝐶(𝑄 𝑅)))
1411, 13mpbird 249 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑃𝐶(𝑄 𝑅))
15 simpl1 1246 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝐾 ∈ HL)
16 simpl2 1248 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → (𝑃𝐴𝑄𝐴𝑅𝐴))
17 simpr 479 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝑃𝑅)
18 simpl3r 1307 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝑃 (𝑄 𝑅))
19 atcvrj1x.l . . . . . 6 = (le‘𝐾)
2019, 6, 7, 8atcvrj1 35499 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑅𝑃 (𝑄 𝑅))) → 𝑃𝐶(𝑄 𝑅))
2115, 16, 17, 18, 20syl112anc 1497 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝑃𝐶(𝑄 𝑅))
2214, 21pm2.61dane 3086 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) → 𝑃𝐶(𝑄 𝑅))
23223expia 1154 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑄𝑅𝑃 (𝑄 𝑅)) → 𝑃𝐶(𝑄 𝑅)))
24 hlatl 35428 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2524ad2antrr 717 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝐾 ∈ AtLat)
26 simplr1 1279 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃𝐴)
27 eqid 2825 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
2827, 8atn0 35376 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃 ≠ (0.‘𝐾))
2925, 26, 28syl2anc 579 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 ≠ (0.‘𝐾))
30 simpll 783 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝐾 ∈ HL)
31 eqid 2825 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3231, 8atbase 35357 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3326, 32syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 ∈ (Base‘𝐾))
34 simplr2 1281 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄𝐴)
35 simplr3 1283 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑅𝐴)
36 simpr 479 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃𝐶(𝑄 𝑅))
3731, 6, 27, 7, 8atcvrj0 35496 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄𝐴𝑅𝐴) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅))
3830, 33, 34, 35, 36, 37syl131anc 1506 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅))
3938necon3bid 3043 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑃 ≠ (0.‘𝐾) ↔ 𝑄𝑅))
4029, 39mpbid 224 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄𝑅)
41 hllat 35431 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ Lat)
4241ad2antrr 717 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝐾 ∈ Lat)
4331, 8atbase 35357 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
4434, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄 ∈ (Base‘𝐾))
4531, 8atbase 35357 . . . . . . . 8 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
4635, 45syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑅 ∈ (Base‘𝐾))
4731, 6latjcl 17404 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑄 𝑅) ∈ (Base‘𝐾))
4842, 44, 46, 47syl3anc 1494 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑄 𝑅) ∈ (Base‘𝐾))
4930, 33, 483jca 1162 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)))
5031, 19, 7cvrle 35346 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 (𝑄 𝑅))
5149, 50sylancom 582 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 (𝑄 𝑅))
5240, 51jca 507 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑄𝑅𝑃 (𝑄 𝑅)))
5352ex 403 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃𝐶(𝑄 𝑅) → (𝑄𝑅𝑃 (𝑄 𝑅))))
5423, 53impbid 204 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑄𝑅𝑃 (𝑄 𝑅)) ↔ 𝑃𝐶(𝑄 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999   class class class wbr 4873  cfv 6123  (class class class)co 6905  Basecbs 16222  lecple 16312  joincjn 17297  0.cp0 17390  Latclat 17398  ccvr 35330  Atomscatm 35331  AtLatcal 35332  HLchlt 35418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-proset 17281  df-poset 17299  df-plt 17311  df-lub 17327  df-glb 17328  df-join 17329  df-meet 17330  df-p0 17392  df-lat 17399  df-clat 17461  df-oposet 35244  df-ol 35246  df-oml 35247  df-covers 35334  df-ats 35335  df-atl 35366  df-cvlat 35390  df-hlat 35419
This theorem is referenced by:  atcvrj2  35501
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