Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg2cex Structured version   Visualization version   GIF version

Theorem cdlemg2cex 34897
 Description: Any translation is one of our 𝐹 s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 34869? (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 34894 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐹𝑇 ↔ ∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞))))
6 simplll 794 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → 𝐾 ∈ HL)
7 simpllr 795 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → 𝑊𝐻)
8 simplrl 796 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → 𝑝𝐴)
9 simprl 790 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → ¬ 𝑝 𝑊)
10 simplrr 797 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → 𝑞𝐴)
11 simprr 792 . . . . . . 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 2610 . . . . . . . 8 (𝑓𝑇 (𝑓𝑝) = 𝑞) = (𝑓𝑇 (𝑓𝑝) = 𝑞)
2012, 1, 13, 14, 2, 3, 15, 16, 17, 18, 4, 19cdlemg1b2 34877 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ (𝑞𝐴 ∧ ¬ 𝑞 𝑊)) → (𝑓𝑇 (𝑓𝑝) = 𝑞) = 𝐺)
216, 7, 8, 9, 10, 11, 20syl222anc 1334 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → (𝑓𝑇 (𝑓𝑝) = 𝑞) = 𝐺)
2221eqeq2d 2620 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → (𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞) ↔ 𝐹 = 𝐺))
2322pm5.32da 671 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) → (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝐹 = 𝐺)))
24 df-3an 1033 . . . 4 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)))
25 df-3an 1033 . . . 4 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝐹 = 𝐺))
2623, 24, 253bitr4g 302 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)) ↔ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺)))
27262rexbidva 3038 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)) ↔ ∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺)))
285, 27bitrd 267 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐹𝑇 ↔ ∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ⦋csb 3499  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Atomscatm 33568  HLchlt 33655  LHypclh 34288  LTrncltrn 34405 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-undef 7286  df-map 7746  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292  df-laut 34293  df-ldil 34408  df-ltrn 34409  df-trl 34464 This theorem is referenced by:  cdlemg2ce  34898
 Copyright terms: Public domain W3C validator