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Theorem cdlemg2cN 37795
 Description: Any translation belongs to the set of functions constructed for cdleme 37766. TODO: Fix comment. (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg2.b 𝐵 = (Base‘𝐾)
cdlemg2.l = (le‘𝐾)
cdlemg2.j = (join‘𝐾)
cdlemg2.m = (meet‘𝐾)
cdlemg2.a 𝐴 = (Atoms‘𝐾)
cdlemg2.h 𝐻 = (LHyp‘𝐾)
cdlemg2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg2.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdlemg2.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdlemg2.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdlemg2.g 𝐺 = (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
cdlemg2cN ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑄) → (𝐹𝑇𝐹 = 𝐺))
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
Allowed substitution hints:   𝐷(𝑡)   𝑇(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐸(𝑡,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐺(𝑥,𝑦,𝑧,𝑡,𝑠)

Proof of Theorem cdlemg2cN
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 cdlemg2.l . . 3 = (le‘𝐾)
2 cdlemg2.a . . 3 𝐴 = (Atoms‘𝐾)
3 cdlemg2.h . . 3 𝐻 = (LHyp‘𝐾)
4 cdlemg2.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
51, 2, 3, 4cdlemg1cN 37793 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑄) → (𝐹𝑇𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄)))
6 cdlemg2.b . . . . 5 𝐵 = (Base‘𝐾)
7 cdlemg2.j . . . . 5 = (join‘𝐾)
8 cdlemg2.m . . . . 5 = (meet‘𝐾)
9 cdlemg2.u . . . . 5 𝑈 = ((𝑃 𝑄) 𝑊)
10 cdlemg2.d . . . . 5 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
11 cdlemg2.e . . . . 5 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
12 cdlemg2.g . . . . 5 𝐺 = (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))
13 eqid 2824 . . . . 5 (𝑓𝑇 (𝑓𝑃) = 𝑄) = (𝑓𝑇 (𝑓𝑃) = 𝑄)
146, 1, 7, 8, 2, 3, 9, 10, 11, 12, 4, 13cdlemg1b2 37777 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑓𝑇 (𝑓𝑃) = 𝑄) = 𝐺)
1514adantr 484 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑄) → (𝑓𝑇 (𝑓𝑃) = 𝑄) = 𝐺)
1615eqeq2d 2835 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑄) → (𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄) ↔ 𝐹 = 𝐺))
175, 16bitrd 282 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑄) → (𝐹𝑇𝐹 = 𝐺))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ 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  LTrncltrn 37307 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-map 8398  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  df-laut 37195  df-ldil 37310  df-ltrn 37311  df-trl 37365 This theorem is referenced by:  cdlemg2dN  37796
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