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Theorem atcvrj2b 37091
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 1229 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑄𝑅)
21necomd 2989 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝑄)
3 simpl1 1192 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝐾 ∈ HL)
4 simpl23 1254 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝐴)
5 simpl22 1253 . . . . . . 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 37077 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑄𝐴) → (𝑅𝑄𝑅𝐶(𝑄 𝑅)))
103, 4, 5, 9syl3anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → (𝑅𝑄𝑅𝐶(𝑄 𝑅)))
112, 10mpbid 235 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑅𝐶(𝑄 𝑅))
12 breq1 5033 . . . . . 6 (𝑃 = 𝑅 → (𝑃𝐶(𝑄 𝑅) ↔ 𝑅𝐶(𝑄 𝑅)))
1312adantl 485 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → (𝑃𝐶(𝑄 𝑅) ↔ 𝑅𝐶(𝑄 𝑅)))
1411, 13mpbird 260 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃 = 𝑅) → 𝑃𝐶(𝑄 𝑅))
15 simpl1 1192 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝐾 ∈ HL)
16 simpl2 1193 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → (𝑃𝐴𝑄𝐴𝑅𝐴))
17 simpr 488 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝑃𝑅)
18 simpl3r 1230 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝑃 (𝑄 𝑅))
19 atcvrj1x.l . . . . . 6 = (le‘𝐾)
2019, 6, 7, 8atcvrj1 37090 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑅𝑃 (𝑄 𝑅))) → 𝑃𝐶(𝑄 𝑅))
2115, 16, 17, 18, 20syl112anc 1375 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) ∧ 𝑃𝑅) → 𝑃𝐶(𝑄 𝑅))
2214, 21pm2.61dane 3021 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑃 (𝑄 𝑅))) → 𝑃𝐶(𝑄 𝑅))
23223expia 1122 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑄𝑅𝑃 (𝑄 𝑅)) → 𝑃𝐶(𝑄 𝑅)))
24 hlatl 37019 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2524ad2antrr 726 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝐾 ∈ AtLat)
26 simplr1 1216 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃𝐴)
27 eqid 2738 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
2827, 8atn0 36967 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃 ≠ (0.‘𝐾))
2925, 26, 28syl2anc 587 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 ≠ (0.‘𝐾))
30 simpll 767 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝐾 ∈ HL)
31 eqid 2738 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
3231, 8atbase 36948 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3326, 32syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 ∈ (Base‘𝐾))
34 simplr2 1217 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄𝐴)
35 simplr3 1218 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑅𝐴)
36 simpr 488 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃𝐶(𝑄 𝑅))
3731, 6, 27, 7, 8atcvrj0 37087 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄𝐴𝑅𝐴) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅))
3830, 33, 34, 35, 36, 37syl131anc 1384 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅))
3938necon3bid 2978 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑃 ≠ (0.‘𝐾) ↔ 𝑄𝑅))
4029, 39mpbid 235 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄𝑅)
41 hllat 37022 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ Lat)
4241ad2antrr 726 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝐾 ∈ Lat)
4331, 8atbase 36948 . . . . . . . 8 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
4434, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑄 ∈ (Base‘𝐾))
4531, 8atbase 36948 . . . . . . . 8 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
4635, 45syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑅 ∈ (Base‘𝐾))
4731, 6latjcl 17779 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑄 𝑅) ∈ (Base‘𝐾))
4842, 44, 46, 47syl3anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑄 𝑅) ∈ (Base‘𝐾))
4930, 33, 483jca 1129 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)))
5031, 19, 7cvrle 36937 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 𝑅) ∈ (Base‘𝐾)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 (𝑄 𝑅))
5149, 50sylancom 591 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → 𝑃 (𝑄 𝑅))
5240, 51jca 515 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ 𝑃𝐶(𝑄 𝑅)) → (𝑄𝑅𝑃 (𝑄 𝑅)))
5352ex 416 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃𝐶(𝑄 𝑅) → (𝑄𝑅𝑃 (𝑄 𝑅))))
5423, 53impbid 215 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑄𝑅𝑃 (𝑄 𝑅)) ↔ 𝑃𝐶(𝑄 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wne 2934   class class class wbr 5030  cfv 6339  (class class class)co 7172  Basecbs 16588  lecple 16677  joincjn 17672  0.cp0 17765  Latclat 17773  ccvr 36921  Atomscatm 36922  AtLatcal 36923  HLchlt 37009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7129  df-ov 7175  df-oprab 7176  df-proset 17656  df-poset 17674  df-plt 17686  df-lub 17702  df-glb 17703  df-join 17704  df-meet 17705  df-p0 17767  df-lat 17774  df-clat 17836  df-oposet 36835  df-ol 36837  df-oml 36838  df-covers 36925  df-ats 36926  df-atl 36957  df-cvlat 36981  df-hlat 37010
This theorem is referenced by:  atcvrj2  37092
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