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Theorem cdleme51finvtrN 35347
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: fix comment. (Contributed by NM, 14-Apr-2013.) (New usage is discouraged.)
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
cdleme51finvtrN (((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 cdleme51finvtrN
Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑣 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 2621 . . 3 ((𝑄 𝑃) 𝑊) = ((𝑄 𝑃) 𝑊)
12 eqid 2621 . . 3 ((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊))) = ((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊)))
13 eqid 2621 . . 3 ((𝑄 𝑃) (((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊))) = ((𝑄 𝑃) (((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊)))
14 eqid 2621 . . 3 (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = ((𝑄 𝑃) (((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊))))), 𝑢 / 𝑣((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊)))) (𝑎 𝑊)))), 𝑎)) = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = ((𝑄 𝑃) (((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊))))), 𝑢 / 𝑣((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊)))) (𝑎 𝑊)))), 𝑎))
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14cdleme51finvN 35345 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = ((𝑄 𝑃) (((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊))))), 𝑢 / 𝑣((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊)))) (𝑎 𝑊)))), 𝑎)))
16 cdleme50ltrn.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
171, 2, 3, 4, 5, 6, 11, 12, 13, 14, 16cdleme50ltrn 35346 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = ((𝑄 𝑃) (((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊))))), 𝑢 / 𝑣((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊)))) (𝑎 𝑊)))), 𝑎)) ∈ 𝑇)
18173com23 1268 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = ((𝑄 𝑃) (((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊))) ((𝑢 𝑣) 𝑊))))), 𝑢 / 𝑣((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊)))) (𝑎 𝑊)))), 𝑎)) ∈ 𝑇)
1915, 18eqeltrd 2698 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  csb 3515  ifcif 4060   class class class wbr 4615  cmpt 4675  ccnv 5075  cfv 5849  crio 6567  (class class class)co 6607  Basecbs 15784  lecple 15872  joincjn 16868  meetcmee 16869  Atomscatm 34051  HLchlt 34138  LHypclh 34771  LTrncltrn 34888
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 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-riotaBAD 33740
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 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-iin 4490  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-undef 7347  df-map 7807  df-preset 16852  df-poset 16870  df-plt 16882  df-lub 16898  df-glb 16899  df-join 16900  df-meet 16901  df-p0 16963  df-p1 16964  df-lat 16970  df-clat 17032  df-oposet 33964  df-ol 33966  df-oml 33967  df-covers 34054  df-ats 34055  df-atl 34086  df-cvlat 34110  df-hlat 34139  df-llines 34285  df-lplanes 34286  df-lvols 34287  df-lines 34288  df-psubsp 34290  df-pmap 34291  df-padd 34583  df-lhyp 34775  df-laut 34776  df-ldil 34891  df-ltrn 34892
This theorem is referenced by:  cdlemg1finvtrlemN  35364
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