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Theorem cdlemg2cex 37832
Description: Any translation is one of our 𝐹 s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 37804? (Contributed by NM, 22-Apr-2013.)
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)
cdlemg2ex.u 𝑈 = ((𝑝 𝑞) 𝑊)
cdlemg2ex.d 𝐷 = ((𝑡 𝑈) (𝑞 ((𝑝 𝑡) 𝑊)))
cdlemg2ex.e 𝐸 = ((𝑝 𝑞) (𝐷 ((𝑠 𝑡) 𝑊)))
cdlemg2ex.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
cdlemg2cex ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐹𝑇 ↔ ∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 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   𝑈,𝑠,𝑡,𝑥,𝑦,𝑧   𝑊,𝑠,𝑡,𝑥,𝑦,𝑧   𝑞,𝑝,𝐴   𝐹,𝑝,𝑞   𝐻,𝑝,𝑞   𝐾,𝑝,𝑞   ,𝑝,𝑞   𝑇,𝑝,𝑞   𝑊,𝑝,𝑞,𝑠,𝑡,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐵(𝑞,𝑝)   𝐷(𝑡,𝑞,𝑝)   𝑇(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑈(𝑞,𝑝)   𝐸(𝑡,𝑠,𝑞,𝑝)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐺(𝑥,𝑦,𝑧,𝑡,𝑠,𝑞,𝑝)   (𝑞,𝑝)   (𝑞,𝑝)

Proof of Theorem cdlemg2cex
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, 4cdlemg1cex 37829 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐹𝑇 ↔ ∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞))))
6 simplll 774 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → 𝐾 ∈ HL)
7 simpllr 775 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → 𝑊𝐻)
8 simplrl 776 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → 𝑝𝐴)
9 simprl 770 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → ¬ 𝑝 𝑊)
10 simplrr 777 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → 𝑞𝐴)
11 simprr 772 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → ¬ 𝑞 𝑊)
12 cdlemg2.b . . . . . . . 8 𝐵 = (Base‘𝐾)
13 cdlemg2.j . . . . . . . 8 = (join‘𝐾)
14 cdlemg2.m . . . . . . . 8 = (meet‘𝐾)
15 cdlemg2ex.u . . . . . . . 8 𝑈 = ((𝑝 𝑞) 𝑊)
16 cdlemg2ex.d . . . . . . . 8 𝐷 = ((𝑡 𝑈) (𝑞 ((𝑝 𝑡) 𝑊)))
17 cdlemg2ex.e . . . . . . . 8 𝐸 = ((𝑝 𝑞) (𝐷 ((𝑠 𝑡) 𝑊)))
18 cdlemg2ex.g . . . . . . . 8 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))
19 eqid 2824 . . . . . . . 8 (𝑓𝑇 (𝑓𝑝) = 𝑞) = (𝑓𝑇 (𝑓𝑝) = 𝑞)
2012, 1, 13, 14, 2, 3, 15, 16, 17, 18, 4, 19cdlemg1b2 37812 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ (𝑞𝐴 ∧ ¬ 𝑞 𝑊)) → (𝑓𝑇 (𝑓𝑝) = 𝑞) = 𝐺)
216, 7, 8, 9, 10, 11, 20syl222anc 1383 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → (𝑓𝑇 (𝑓𝑝) = 𝑞) = 𝐺)
2221eqeq2d 2835 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → (𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞) ↔ 𝐹 = 𝐺))
2322pm5.32da 582 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) → (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝐹 = 𝐺)))
24 df-3an 1086 . . . 4 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)))
25 df-3an 1086 . . . 4 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝐹 = 𝐺))
2623, 24, 253bitr4g 317 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)) ↔ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺)))
27262rexbidva 3291 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)) ↔ ∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺)))
285, 27bitrd 282 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐹𝑇 ↔ ∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 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  wrex 3134  csb 3866  ifcif 4450   class class class wbr 5052  cmpt 5132  cfv 6343  crio 7106  (class class class)co 7149  Basecbs 16483  lecple 16572  joincjn 17554  meetcmee 17555  Atomscatm 36504  HLchlt 36591  LHypclh 37225  LTrncltrn 37342
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 7455  ax-riotaBAD 36194
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 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  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 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-undef 7935  df-map 8404  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-p1 17650  df-lat 17656  df-clat 17718  df-oposet 36417  df-ol 36419  df-oml 36420  df-covers 36507  df-ats 36508  df-atl 36539  df-cvlat 36563  df-hlat 36592  df-llines 36739  df-lplanes 36740  df-lvols 36741  df-lines 36742  df-psubsp 36744  df-pmap 36745  df-padd 37037  df-lhyp 37229  df-laut 37230  df-ldil 37345  df-ltrn 37346  df-trl 37400
This theorem is referenced by:  cdlemg2ce  37833
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