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Theorem cdleme48fvg 37706
Description: Remove 𝑃𝑄 condition in cdleme48fv 37705. TODO: Can this replace uses of cdleme32a 37647? TODO: Can this be used to help prove the 𝑅 or 𝑆 case where 𝑋 is an atom? TODO: Can this be proved more directly by eliminating 𝑃𝑄 in earlier theorems? Should this replace uses of cdleme48fv 37705? (Contributed by NM, 23-Apr-2013.)
Hypotheses
Ref Expression
cdlemef46.b 𝐵 = (Base‘𝐾)
cdlemef46.l = (le‘𝐾)
cdlemef46.j = (join‘𝐾)
cdlemef46.m = (meet‘𝐾)
cdlemef46.a 𝐴 = (Atoms‘𝐾)
cdlemef46.h 𝐻 = (LHyp‘𝐾)
cdlemef46.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdlemef46.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdlemefs46.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdlemef46.f 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))
Assertion
Ref Expression
cdleme48fvg ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) = ((𝐹𝑆) (𝑋 𝑊)))
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,𝐴   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝐷,𝑠,𝑥,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝐻,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝑃,𝑠,𝑡,𝑥,𝑦,𝑧   𝑄,𝑠,𝑡,𝑥,𝑦,𝑧   𝑈,𝑠,𝑡,𝑥,𝑦,𝑧   𝑊,𝑠,𝑡,𝑥,𝑦,𝑧   𝑆,𝑠,𝑡,𝑥,𝑦,𝑧   𝑋,𝑠,𝑡,𝑥,𝑧
Allowed substitution hints:   𝐷(𝑡)   𝐸(𝑡,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑋(𝑦)

Proof of Theorem cdleme48fvg
StepHypRef Expression
1 simpl3r 1226 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) ∧ 𝑃 = 𝑄) → (𝑆 (𝑋 𝑊)) = 𝑋)
2 simp3ll 1241 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) → 𝑆𝐴)
32adantr 484 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) ∧ 𝑃 = 𝑄) → 𝑆𝐴)
4 cdlemef46.b . . . . . . 7 𝐵 = (Base‘𝐾)
5 cdlemef46.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
64, 5atbase 36495 . . . . . 6 (𝑆𝐴𝑆𝐵)
73, 6syl 17 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) ∧ 𝑃 = 𝑄) → 𝑆𝐵)
8 cdlemef46.f . . . . . 6 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))
98cdleme31id 37600 . . . . 5 ((𝑆𝐵𝑃 = 𝑄) → (𝐹𝑆) = 𝑆)
107, 9sylancom 591 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) ∧ 𝑃 = 𝑄) → (𝐹𝑆) = 𝑆)
1110oveq1d 7160 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) ∧ 𝑃 = 𝑄) → ((𝐹𝑆) (𝑋 𝑊)) = (𝑆 (𝑋 𝑊)))
12 simp2l 1196 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) → 𝑋𝐵)
138cdleme31id 37600 . . . 4 ((𝑋𝐵𝑃 = 𝑄) → (𝐹𝑋) = 𝑋)
1412, 13sylan 583 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) ∧ 𝑃 = 𝑄) → (𝐹𝑋) = 𝑋)
151, 11, 143eqtr4rd 2870 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) ∧ 𝑃 = 𝑄) → (𝐹𝑋) = ((𝐹𝑆) (𝑋 𝑊)))
16 simpl1 1188 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) ∧ 𝑃𝑄) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
17 simpr 488 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) ∧ 𝑃𝑄) → 𝑃𝑄)
18 simpl2 1189 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) ∧ 𝑃𝑄) → (𝑋𝐵 ∧ ¬ 𝑋 𝑊))
19 simpl3 1190 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) ∧ 𝑃𝑄) → ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋))
20 cdlemef46.l . . . 4 = (le‘𝐾)
21 cdlemef46.j . . . 4 = (join‘𝐾)
22 cdlemef46.m . . . 4 = (meet‘𝐾)
23 cdlemef46.h . . . 4 𝐻 = (LHyp‘𝐾)
24 cdlemef46.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
25 cdlemef46.d . . . 4 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
26 cdlemefs46.e . . . 4 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
274, 20, 21, 22, 5, 23, 24, 25, 26, 8cdleme48fv 37705 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) = ((𝐹𝑆) (𝑋 𝑊)))
2816, 17, 18, 19, 27syl121anc 1372 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) ∧ 𝑃𝑄) → (𝐹𝑋) = ((𝐹𝑆) (𝑋 𝑊)))
2915, 28pm2.61dane 3101 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) = ((𝐹𝑆) (𝑋 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wral 3133  csb 3866  ifcif 4449   class class class wbr 5052  cmpt 5132  cfv 6343  crio 7102  (class class class)co 7145  Basecbs 16479  lecple 16568  joincjn 17550  meetcmee 17551  Atomscatm 36469  HLchlt 36556  LHypclh 37190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-riotaBAD 36159
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7679  df-2nd 7680  df-undef 7929  df-proset 17534  df-poset 17552  df-plt 17564  df-lub 17580  df-glb 17581  df-join 17582  df-meet 17583  df-p0 17645  df-p1 17646  df-lat 17652  df-clat 17714  df-oposet 36382  df-ol 36384  df-oml 36385  df-covers 36472  df-ats 36473  df-atl 36504  df-cvlat 36528  df-hlat 36557  df-llines 36704  df-lplanes 36705  df-lvols 36706  df-lines 36707  df-psubsp 36709  df-pmap 36710  df-padd 37002  df-lhyp 37194
This theorem is referenced by:  cdlemg2fvlem  37800
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