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Theorem cdlemg2cex 35356
Description: Any translation is one of our 𝐹 s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 35328? (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 35353 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐹𝑇 ↔ ∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞))))
6 simplll 797 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → 𝐾 ∈ HL)
7 simpllr 798 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → 𝑊𝐻)
8 simplrl 799 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → 𝑝𝐴)
9 simprl 793 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → ¬ 𝑝 𝑊)
10 simplrr 800 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → 𝑞𝐴)
11 simprr 795 . . . . . . 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 2621 . . . . . . . 8 (𝑓𝑇 (𝑓𝑝) = 𝑞) = (𝑓𝑇 (𝑓𝑝) = 𝑞)
2012, 1, 13, 14, 2, 3, 15, 16, 17, 18, 4, 19cdlemg1b2 35336 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ (𝑞𝐴 ∧ ¬ 𝑞 𝑊)) → (𝑓𝑇 (𝑓𝑝) = 𝑞) = 𝐺)
216, 7, 8, 9, 10, 11, 20syl222anc 1339 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → (𝑓𝑇 (𝑓𝑝) = 𝑞) = 𝐺)
2221eqeq2d 2631 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) ∧ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊)) → (𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞) ↔ 𝐹 = 𝐺))
2322pm5.32da 672 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) → (((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝐹 = 𝐺)))
24 df-3an 1038 . . . 4 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)))
25 df-3an 1038 . . . 4 ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺) ↔ ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊) ∧ 𝐹 = 𝐺))
2623, 24, 253bitr4g 303 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴𝑞𝐴)) → ((¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)) ↔ (¬ 𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺)))
27262rexbidva 3049 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞)) ↔ ∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺)))
285, 27bitrd 268 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐹𝑇 ↔ ∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  csb 3514  ifcif 4058   class class class wbr 4613  cmpt 4673  cfv 5847  crio 6564  (class class class)co 6604  Basecbs 15781  lecple 15869  joincjn 16865  meetcmee 16866  Atomscatm 34027  HLchlt 34114  LHypclh 34747  LTrncltrn 34864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-riotaBAD 33716
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-undef 7344  df-map 7804  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-p1 16961  df-lat 16967  df-clat 17029  df-oposet 33940  df-ol 33942  df-oml 33943  df-covers 34030  df-ats 34031  df-atl 34062  df-cvlat 34086  df-hlat 34115  df-llines 34261  df-lplanes 34262  df-lvols 34263  df-lines 34264  df-psubsp 34266  df-pmap 34267  df-padd 34559  df-lhyp 34751  df-laut 34752  df-ldil 34867  df-ltrn 34868  df-trl 34923
This theorem is referenced by:  cdlemg2ce  35357
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