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Theorem cdleme32a 39825
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 19-Feb-2013.)
Hypotheses
Ref Expression
cdleme32.b 𝐡 = (Baseβ€˜πΎ)
cdleme32.l ≀ = (leβ€˜πΎ)
cdleme32.j ∨ = (joinβ€˜πΎ)
cdleme32.m ∧ = (meetβ€˜πΎ)
cdleme32.a 𝐴 = (Atomsβ€˜πΎ)
cdleme32.h 𝐻 = (LHypβ€˜πΎ)
cdleme32.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdleme32.c 𝐢 = ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)))
cdleme32.d 𝐷 = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
cdleme32.e 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑑) ∧ π‘Š)))
cdleme32.i 𝐼 = (℩𝑦 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑦 = 𝐸))
cdleme32.n 𝑁 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐼, 𝐢)
cdleme32.o 𝑂 = (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (π‘₯ ∧ π‘Š)) = π‘₯) β†’ 𝑧 = (𝑁 ∨ (π‘₯ ∧ π‘Š))))
cdleme32.f 𝐹 = (π‘₯ ∈ 𝐡 ↦ if((𝑃 β‰  𝑄 ∧ Β¬ π‘₯ ≀ π‘Š), 𝑂, π‘₯))
Assertion
Ref Expression
cdleme32a ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑋 ∈ 𝐡 ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑋 ≀ π‘Š)) ∧ ((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋)) β†’ (πΉβ€˜π‘‹) = (𝑁 ∨ (𝑋 ∧ π‘Š)))
Distinct variable groups:   𝑑,𝑠,π‘₯,𝑦,𝑧,𝐴   𝐡,𝑠,𝑑,π‘₯,𝑦,𝑧   𝑦,𝐢   𝐷,𝑠,𝑦,𝑧   𝑦,𝐸   𝐻,𝑠,𝑑   ∨ ,𝑠,𝑑,π‘₯,𝑦,𝑧   𝐾,𝑠,𝑑   ≀ ,𝑠,𝑑,π‘₯,𝑦,𝑧   ∧ ,𝑠,𝑑,π‘₯,𝑦,𝑧   π‘₯,𝑁,𝑧   𝑃,𝑠,𝑑,π‘₯,𝑦,𝑧   𝑄,𝑠,𝑑,π‘₯,𝑦,𝑧   π‘ˆ,𝑠,𝑑,π‘₯,𝑦,𝑧   π‘Š,𝑠,𝑑,π‘₯,𝑦,𝑧   𝑋,𝑠,𝑑,π‘₯,𝑧   𝑦,𝐻   𝑦,𝐾   𝑧,𝐻   𝑧,𝐾
Allowed substitution hints:   𝐢(π‘₯,𝑧,𝑑,𝑠)   𝐷(π‘₯,𝑑)   𝐸(π‘₯,𝑧,𝑑,𝑠)   𝐹(π‘₯,𝑦,𝑧,𝑑,𝑠)   𝐻(π‘₯)   𝐼(π‘₯,𝑦,𝑧,𝑑,𝑠)   𝐾(π‘₯)   𝑁(𝑦,𝑑,𝑠)   𝑂(π‘₯,𝑦,𝑧,𝑑,𝑠)   𝑋(𝑦)

Proof of Theorem cdleme32a
StepHypRef Expression
1 cdleme32.b . . . 4 𝐡 = (Baseβ€˜πΎ)
21fvexi 6899 . . 3 𝐡 ∈ V
3 anass 468 . . . 4 (((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) ↔ (𝑠 ∈ 𝐴 ∧ (Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋)))
4 cdleme32.o . . . . . . 7 𝑂 = (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (π‘₯ ∧ π‘Š)) = π‘₯) β†’ 𝑧 = (𝑁 ∨ (π‘₯ ∧ π‘Š))))
5 cdleme32.f . . . . . . 7 𝐹 = (π‘₯ ∈ 𝐡 ↦ if((𝑃 β‰  𝑄 ∧ Β¬ π‘₯ ≀ π‘Š), 𝑂, π‘₯))
6 eqid 2726 . . . . . . 7 (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ 𝑧 = (𝑁 ∨ (𝑋 ∧ π‘Š)))) = (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ 𝑧 = (𝑁 ∨ (𝑋 ∧ π‘Š))))
74, 5, 6cdleme31fv1 39775 . . . . . 6 ((𝑋 ∈ 𝐡 ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑋 ≀ π‘Š)) β†’ (πΉβ€˜π‘‹) = (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ 𝑧 = (𝑁 ∨ (𝑋 ∧ π‘Š)))))
87adantl 481 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑋 ∈ 𝐡 ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑋 ≀ π‘Š))) β†’ (πΉβ€˜π‘‹) = (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ 𝑧 = (𝑁 ∨ (𝑋 ∧ π‘Š)))))
9 cdleme32.l . . . . . . 7 ≀ = (leβ€˜πΎ)
10 cdleme32.j . . . . . . 7 ∨ = (joinβ€˜πΎ)
11 cdleme32.m . . . . . . 7 ∧ = (meetβ€˜πΎ)
12 cdleme32.a . . . . . . 7 𝐴 = (Atomsβ€˜πΎ)
13 cdleme32.h . . . . . . 7 𝐻 = (LHypβ€˜πΎ)
14 cdleme32.u . . . . . . 7 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
15 cdleme32.c . . . . . . 7 𝐢 = ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)))
16 cdleme32.d . . . . . . 7 𝐷 = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
17 cdleme32.e . . . . . . 7 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑑) ∧ π‘Š)))
18 cdleme32.i . . . . . . 7 𝐼 = (℩𝑦 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑦 = 𝐸))
19 cdleme32.n . . . . . . 7 𝑁 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐼, 𝐢)
201, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 4, 5cdleme32fvcl 39824 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝑋 ∈ 𝐡) β†’ (πΉβ€˜π‘‹) ∈ 𝐡)
2120adantrr 714 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑋 ∈ 𝐡 ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑋 ≀ π‘Š))) β†’ (πΉβ€˜π‘‹) ∈ 𝐡)
228, 21riotasvd 38339 . . . 4 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑋 ∈ 𝐡 ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑋 ≀ π‘Š))) ∧ 𝐡 ∈ V) β†’ ((𝑠 ∈ 𝐴 ∧ (Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋)) β†’ (πΉβ€˜π‘‹) = (𝑁 ∨ (𝑋 ∧ π‘Š))))
233, 22biimtrid 241 . . 3 (((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑋 ∈ 𝐡 ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑋 ≀ π‘Š))) ∧ 𝐡 ∈ V) β†’ (((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ (πΉβ€˜π‘‹) = (𝑁 ∨ (𝑋 ∧ π‘Š))))
242, 23mpan2 688 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑋 ∈ 𝐡 ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑋 ≀ π‘Š))) β†’ (((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋) β†’ (πΉβ€˜π‘‹) = (𝑁 ∨ (𝑋 ∧ π‘Š))))
25243impia 1114 1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑋 ∈ 𝐡 ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑋 ≀ π‘Š)) ∧ ((𝑠 ∈ 𝐴 ∧ Β¬ 𝑠 ≀ π‘Š) ∧ (𝑠 ∨ (𝑋 ∧ π‘Š)) = 𝑋)) β†’ (πΉβ€˜π‘‹) = (𝑁 ∨ (𝑋 ∧ π‘Š)))
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  βˆ€wral 3055  Vcvv 3468  ifcif 4523   class class class wbr 5141   ↦ cmpt 5224  β€˜cfv 6537  β„©crio 7360  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  meetcmee 18277  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  ax-riotaBAD 38336
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  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-iin 4993  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-mpo 7410  df-1st 7974  df-2nd 7975  df-undef 8259  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-p1 18391  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  df-llines 38882  df-lplanes 38883  df-lvols 38884  df-lines 38885  df-psubsp 38887  df-pmap 38888  df-padd 39180  df-lhyp 39372
This theorem is referenced by:  cdleme32b  39826  cdleme32c  39827  cdleme32e  39829
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