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Theorem cdleme50ltrn 36359
Description: Part of proof of Lemma E in [Crawley] p. 113. 𝐹 is a lattice translation. TODO: fix comment. (Contributed by NM, 10-Apr-2013.)
Hypotheses
Ref Expression
cdlemef50.b 𝐵 = (Base‘𝐾)
cdlemef50.l = (le‘𝐾)
cdlemef50.j = (join‘𝐾)
cdlemef50.m = (meet‘𝐾)
cdlemef50.a 𝐴 = (Atoms‘𝐾)
cdlemef50.h 𝐻 = (LHyp‘𝐾)
cdlemef50.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdlemef50.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdlemefs50.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdlemef50.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))
cdleme50ltrn.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
cdleme50ltrn (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹𝑇)
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,   ,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐴,𝑠,𝑡,𝑥,𝑦,𝑧   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝐷,𝑠,𝑥,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝐻,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,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)

Proof of Theorem cdleme50ltrn
Dummy variables 𝑒 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdlemef50.b . . 3 𝐵 = (Base‘𝐾)
2 cdlemef50.l . . 3 = (le‘𝐾)
3 cdlemef50.j . . 3 = (join‘𝐾)
4 cdlemef50.m . . 3 = (meet‘𝐾)
5 cdlemef50.a . . 3 𝐴 = (Atoms‘𝐾)
6 cdlemef50.h . . 3 𝐻 = (LHyp‘𝐾)
7 cdlemef50.u . . 3 𝑈 = ((𝑃 𝑄) 𝑊)
8 cdlemef50.d . . 3 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
9 cdlemefs50.e . . 3 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
10 cdlemef50.f . . 3 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))
11 eqid 2770 . . 3 ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cdleme50ldil 36350 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
13 simp1 1129 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑑𝐴𝑒𝐴) ∧ (¬ 𝑑 𝑊 ∧ ¬ 𝑒 𝑊)) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
14 simp2l 1240 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑑𝐴𝑒𝐴) ∧ (¬ 𝑑 𝑊 ∧ ¬ 𝑒 𝑊)) → 𝑑𝐴)
15 simp3l 1242 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑑𝐴𝑒𝐴) ∧ (¬ 𝑑 𝑊 ∧ ¬ 𝑒 𝑊)) → ¬ 𝑑 𝑊)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10cdleme50trn123 36356 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑑𝐴 ∧ ¬ 𝑑 𝑊)) → ((𝑑 (𝐹𝑑)) 𝑊) = 𝑈)
1713, 14, 15, 16syl12anc 1473 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑑𝐴𝑒𝐴) ∧ (¬ 𝑑 𝑊 ∧ ¬ 𝑒 𝑊)) → ((𝑑 (𝐹𝑑)) 𝑊) = 𝑈)
18 simp2r 1241 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑑𝐴𝑒𝐴) ∧ (¬ 𝑑 𝑊 ∧ ¬ 𝑒 𝑊)) → 𝑒𝐴)
19 simp3r 1243 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑑𝐴𝑒𝐴) ∧ (¬ 𝑑 𝑊 ∧ ¬ 𝑒 𝑊)) → ¬ 𝑒 𝑊)
201, 2, 3, 4, 5, 6, 7, 8, 9, 10cdleme50trn123 36356 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑒𝐴 ∧ ¬ 𝑒 𝑊)) → ((𝑒 (𝐹𝑒)) 𝑊) = 𝑈)
2113, 18, 19, 20syl12anc 1473 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑑𝐴𝑒𝐴) ∧ (¬ 𝑑 𝑊 ∧ ¬ 𝑒 𝑊)) → ((𝑒 (𝐹𝑒)) 𝑊) = 𝑈)
2217, 21eqtr4d 2807 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑑𝐴𝑒𝐴) ∧ (¬ 𝑑 𝑊 ∧ ¬ 𝑒 𝑊)) → ((𝑑 (𝐹𝑑)) 𝑊) = ((𝑒 (𝐹𝑒)) 𝑊))
23223exp 1111 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑑𝐴𝑒𝐴) → ((¬ 𝑑 𝑊 ∧ ¬ 𝑒 𝑊) → ((𝑑 (𝐹𝑑)) 𝑊) = ((𝑒 (𝐹𝑒)) 𝑊))))
2423ralrimivv 3118 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ∀𝑑𝐴𝑒𝐴 ((¬ 𝑑 𝑊 ∧ ¬ 𝑒 𝑊) → ((𝑑 (𝐹𝑑)) 𝑊) = ((𝑒 (𝐹𝑒)) 𝑊)))
25 cdleme50ltrn.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
262, 3, 4, 5, 6, 11, 25isltrn 35920 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐹𝑇 ↔ (𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑑𝐴𝑒𝐴 ((¬ 𝑑 𝑊 ∧ ¬ 𝑒 𝑊) → ((𝑑 (𝐹𝑑)) 𝑊) = ((𝑒 (𝐹𝑒)) 𝑊)))))
27263ad2ant1 1126 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐹𝑇 ↔ (𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑑𝐴𝑒𝐴 ((¬ 𝑑 𝑊 ∧ ¬ 𝑒 𝑊) → ((𝑑 (𝐹𝑑)) 𝑊) = ((𝑒 (𝐹𝑒)) 𝑊)))))
2812, 24, 27mpbir2and 684 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  csb 3680  ifcif 4223   class class class wbr 4784  cmpt 4861  cfv 6031  crio 6752  (class class class)co 6792  Basecbs 16063  lecple 16155  joincjn 17151  meetcmee 17152  Atomscatm 35065  HLchlt 35152  LHypclh 35785  LDilcldil 35901  LTrncltrn 35902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-riotaBAD 34754
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-undef 7550  df-map 8010  df-preset 17135  df-poset 17153  df-plt 17165  df-lub 17181  df-glb 17182  df-join 17183  df-meet 17184  df-p0 17246  df-p1 17247  df-lat 17253  df-clat 17315  df-oposet 34978  df-ol 34980  df-oml 34981  df-covers 35068  df-ats 35069  df-atl 35100  df-cvlat 35124  df-hlat 35153  df-llines 35299  df-lplanes 35300  df-lvols 35301  df-lines 35302  df-psubsp 35304  df-pmap 35305  df-padd 35597  df-lhyp 35789  df-laut 35790  df-ldil 35905  df-ltrn 35906
This theorem is referenced by:  cdleme51finvtrN  36360  cdleme50ex  36361  cdlemg1a  36372  cdlemg1ltrnlem  36376
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