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Theorem atcvrj2b 38816
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 ≀ = (leβ€˜πΎ)
atcvrj1x.j ∨ = (joinβ€˜πΎ)
atcvrj1x.c 𝐢 = ( β‹– β€˜πΎ)
atcvrj1x.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
atcvrj2b ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅)) ↔ 𝑃𝐢(𝑄 ∨ 𝑅)))

Proof of Theorem atcvrj2b
StepHypRef Expression
1 simpl3l 1225 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) β†’ 𝑄 β‰  𝑅)
21necomd 2990 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) β†’ 𝑅 β‰  𝑄)
3 simpl1 1188 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) β†’ 𝐾 ∈ HL)
4 simpl23 1250 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) β†’ 𝑅 ∈ 𝐴)
5 simpl22 1249 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) β†’ 𝑄 ∈ 𝐴)
6 atcvrj1x.j . . . . . . . 8 ∨ = (joinβ€˜πΎ)
7 atcvrj1x.c . . . . . . . 8 𝐢 = ( β‹– β€˜πΎ)
8 atcvrj1x.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
96, 7, 8atcvr2 38802 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑅 β‰  𝑄 ↔ 𝑅𝐢(𝑄 ∨ 𝑅)))
103, 4, 5, 9syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) β†’ (𝑅 β‰  𝑄 ↔ 𝑅𝐢(𝑄 ∨ 𝑅)))
112, 10mpbid 231 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) β†’ 𝑅𝐢(𝑄 ∨ 𝑅))
12 breq1 5144 . . . . . 6 (𝑃 = 𝑅 β†’ (𝑃𝐢(𝑄 ∨ 𝑅) ↔ 𝑅𝐢(𝑄 ∨ 𝑅)))
1312adantl 481 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) β†’ (𝑃𝐢(𝑄 ∨ 𝑅) ↔ 𝑅𝐢(𝑄 ∨ 𝑅)))
1411, 13mpbird 257 . . . 4 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 = 𝑅) β†’ 𝑃𝐢(𝑄 ∨ 𝑅))
15 simpl1 1188 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 β‰  𝑅) β†’ 𝐾 ∈ HL)
16 simpl2 1189 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 β‰  𝑅) β†’ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴))
17 simpr 484 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 β‰  𝑅) β†’ 𝑃 β‰  𝑅)
18 simpl3r 1226 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 β‰  𝑅) β†’ 𝑃 ≀ (𝑄 ∨ 𝑅))
19 atcvrj1x.l . . . . . 6 ≀ = (leβ€˜πΎ)
2019, 6, 7, 8atcvrj1 38815 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) β†’ 𝑃𝐢(𝑄 ∨ 𝑅))
2115, 16, 17, 18, 20syl112anc 1371 . . . 4 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) ∧ 𝑃 β‰  𝑅) β†’ 𝑃𝐢(𝑄 ∨ 𝑅))
2214, 21pm2.61dane 3023 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))) β†’ 𝑃𝐢(𝑄 ∨ 𝑅))
23223expia 1118 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ 𝑃𝐢(𝑄 ∨ 𝑅)))
24 hlatl 38743 . . . . . . 7 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
2524ad2antrr 723 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝐾 ∈ AtLat)
26 simplr1 1212 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑃 ∈ 𝐴)
27 eqid 2726 . . . . . . 7 (0.β€˜πΎ) = (0.β€˜πΎ)
2827, 8atn0 38691 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 𝑃 β‰  (0.β€˜πΎ))
2925, 26, 28syl2anc 583 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑃 β‰  (0.β€˜πΎ))
30 simpll 764 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝐾 ∈ HL)
31 eqid 2726 . . . . . . . . 9 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3231, 8atbase 38672 . . . . . . . 8 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
3326, 32syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
34 simplr2 1213 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑄 ∈ 𝐴)
35 simplr3 1214 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑅 ∈ 𝐴)
36 simpr 484 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑃𝐢(𝑄 ∨ 𝑅))
3731, 6, 27, 7, 8atcvrj0 38812 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ (𝑃 = (0.β€˜πΎ) ↔ 𝑄 = 𝑅))
3830, 33, 34, 35, 36, 37syl131anc 1380 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ (𝑃 = (0.β€˜πΎ) ↔ 𝑄 = 𝑅))
3938necon3bid 2979 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ (𝑃 β‰  (0.β€˜πΎ) ↔ 𝑄 β‰  𝑅))
4029, 39mpbid 231 . . . 4 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑄 β‰  𝑅)
41 hllat 38746 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
4241ad2antrr 723 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝐾 ∈ Lat)
4331, 8atbase 38672 . . . . . . . 8 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
4434, 43syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
4531, 8atbase 38672 . . . . . . . 8 (𝑅 ∈ 𝐴 β†’ 𝑅 ∈ (Baseβ€˜πΎ))
4635, 45syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑅 ∈ (Baseβ€˜πΎ))
4731, 6latjcl 18404 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ)) β†’ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ))
4842, 44, 46, 47syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ))
4930, 33, 483jca 1125 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ (𝐾 ∈ HL ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ)))
5031, 19, 7cvrle 38661 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑃 ≀ (𝑄 ∨ 𝑅))
5149, 50sylancom 587 . . . 4 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ 𝑃 ≀ (𝑄 ∨ 𝑅))
5240, 51jca 511 . . 3 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ 𝑃𝐢(𝑄 ∨ 𝑅)) β†’ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅)))
5352ex 412 . 2 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃𝐢(𝑄 ∨ 𝑅) β†’ (𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅))))
5423, 53impbid 211 1 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑄 β‰  𝑅 ∧ 𝑃 ≀ (𝑄 ∨ 𝑅)) ↔ 𝑃𝐢(𝑄 ∨ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  0.cp0 18388  Latclat 18396   β‹– ccvr 38645  Atomscatm 38646  AtLatcal 38647  HLchlt 38733
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734
This theorem is referenced by:  atcvrj2  38817
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