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Theorem cdleme0fN 39602
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme0.l ≀ = (leβ€˜πΎ)
cdleme0.j ∨ = (joinβ€˜πΎ)
cdleme0.m ∧ = (meetβ€˜πΎ)
cdleme0.a 𝐴 = (Atomsβ€˜πΎ)
cdleme0.h 𝐻 = (LHypβ€˜πΎ)
cdleme0.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdleme0c.3 𝑉 = ((𝑃 ∨ 𝑅) ∧ π‘Š)
Assertion
Ref Expression
cdleme0fN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑉 β‰  𝑃)

Proof of Theorem cdleme0fN
StepHypRef Expression
1 cdleme0c.3 . . 3 𝑉 = ((𝑃 ∨ 𝑅) ∧ π‘Š)
2 simp1l 1194 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
32hllatd 38747 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
4 simp2l 1196 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
5 eqid 2726 . . . . . . 7 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
6 cdleme0.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
75, 6atbase 38672 . . . . . 6 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
84, 7syl 17 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
9 simp3r 1199 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅 ∈ 𝐴)
105, 6atbase 38672 . . . . . 6 (𝑅 ∈ 𝐴 β†’ 𝑅 ∈ (Baseβ€˜πΎ))
119, 10syl 17 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅 ∈ (Baseβ€˜πΎ))
12 cdleme0.j . . . . . 6 ∨ = (joinβ€˜πΎ)
135, 12latjcl 18404 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ 𝑅) ∈ (Baseβ€˜πΎ))
143, 8, 11, 13syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑅) ∈ (Baseβ€˜πΎ))
15 simp1r 1195 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ π‘Š ∈ 𝐻)
16 cdleme0.h . . . . . 6 𝐻 = (LHypβ€˜πΎ)
175, 16lhpbase 39382 . . . . 5 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ (Baseβ€˜πΎ))
1815, 17syl 17 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ π‘Š ∈ (Baseβ€˜πΎ))
19 cdleme0.l . . . . 5 ≀ = (leβ€˜πΎ)
20 cdleme0.m . . . . 5 ∧ = (meetβ€˜πΎ)
215, 19, 20latmle2 18430 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑅) ∈ (Baseβ€˜πΎ) ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑅) ∧ π‘Š) ≀ π‘Š)
223, 14, 18, 21syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑅) ∧ π‘Š) ≀ π‘Š)
231, 22eqbrtrid 5176 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑉 ≀ π‘Š)
24 simp2r 1197 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ Β¬ 𝑃 ≀ π‘Š)
25 nbrne2 5161 . 2 ((𝑉 ≀ π‘Š ∧ Β¬ 𝑃 ≀ π‘Š) β†’ 𝑉 β‰  𝑃)
2623, 24, 25syl2anc 583 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑉 β‰  𝑃)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ 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  meetcmee 18277  Latclat 18396  Atomscatm 38646  HLchlt 38733  LHypclh 39368
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-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-lat 18397  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-lhyp 39372
This theorem is referenced by: (None)
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