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Theorem cdleme50rn 37721
 Description: Part of proof of Lemma D in [Crawley] p. 113. TODO: fix comment. (Contributed by NM, 9-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))
Assertion
Ref Expression
cdleme50rn (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ran 𝐹 = 𝐵)
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)   𝐸(𝑡,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)

Proof of Theorem cdleme50rn
Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdlemef50.b . 2 𝐵 = (Base‘𝐾)
2 cdlemef50.l . 2 = (le‘𝐾)
3 cdlemef50.j . 2 = (join‘𝐾)
4 cdlemef50.m . 2 = (meet‘𝐾)
5 cdlemef50.a . 2 𝐴 = (Atoms‘𝐾)
6 cdlemef50.h . 2 𝐻 = (LHyp‘𝐾)
7 cdlemef50.u . 2 𝑈 = ((𝑃 𝑄) 𝑊)
8 cdlemef50.d . 2 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
9 cdlemefs50.e . 2 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
10 cdlemef50.f . 2 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 2821 . 2 ((𝑄 𝑃) 𝑊) = ((𝑄 𝑃) 𝑊)
12 eqid 2821 . 2 ((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊))) = ((𝑣 ((𝑄 𝑃) 𝑊)) (𝑃 ((𝑄 𝑣) 𝑊)))
13 eqid 2821 . 2 ((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 2821 . 2 (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, 14cdleme50rnlem 37720 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ran 𝐹 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3007  ∀wral 3126  ⦋csb 3857  ifcif 4440   class class class wbr 5039   ↦ cmpt 5119  ran crn 5529  ‘cfv 6328  ℩crio 7087  (class class class)co 7130  Basecbs 16461  lecple 16550  joincjn 17532  meetcmee 17533  Atomscatm 36439  HLchlt 36526  LHypclh 37160 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 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-riotaBAD 36129 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 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-undef 7914  df-proset 17516  df-poset 17534  df-plt 17546  df-lub 17562  df-glb 17563  df-join 17564  df-meet 17565  df-p0 17627  df-p1 17628  df-lat 17634  df-clat 17696  df-oposet 36352  df-ol 36354  df-oml 36355  df-covers 36442  df-ats 36443  df-atl 36474  df-cvlat 36498  df-hlat 36527  df-llines 36674  df-lplanes 36675  df-lvols 36676  df-lines 36677  df-psubsp 36679  df-pmap 36680  df-padd 36972  df-lhyp 37164 This theorem is referenced by:  cdleme50f1o  37722
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