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Theorem cdlemg1a 37865
 Description: Shorter expression for 𝐺. TODO: fix comment. TODO: shorten using cdleme 37855 or vice-versa? Also, if not shortened with cdleme 37855, then it can be moved up to save repeating hypotheses. (Contributed by NM, 15-Apr-2013.)
Hypotheses
Ref Expression
cdlemg1.b 𝐵 = (Base‘𝐾)
cdlemg1.l = (le‘𝐾)
cdlemg1.j = (join‘𝐾)
cdlemg1.m = (meet‘𝐾)
cdlemg1.a 𝐴 = (Atoms‘𝐾)
cdlemg1.h 𝐻 = (LHyp‘𝐾)
cdlemg1.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdlemg1.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdlemg1.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdlemg1.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))
cdlemg1.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemg1a (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐺 = (𝑓𝑇 (𝑓𝑃) = 𝑄))
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,𝐴,𝑓   𝐵,𝑓,𝑠,𝑡,𝑥,𝑦,𝑧   𝐷,𝑓,𝑠,𝑥,𝑦,𝑧   𝑓,𝐸,𝑥,𝑦,𝑧   𝐻,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑓,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑓,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   𝐴,𝑓   𝑓,𝐻   𝑓,𝐾   ,𝑓   𝑃,𝑓   𝑄,𝑓   𝑇,𝑓   𝑓,𝑊   𝑓,𝐺
Allowed substitution hints:   𝐷(𝑡)   𝑇(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑈(𝑓)   𝐸(𝑡,𝑠)   𝐺(𝑥,𝑦,𝑧,𝑡,𝑠)

Proof of Theorem cdlemg1a
StepHypRef Expression
1 cdlemg1.b . . . 4 𝐵 = (Base‘𝐾)
2 cdlemg1.l . . . 4 = (le‘𝐾)
3 cdlemg1.j . . . 4 = (join‘𝐾)
4 cdlemg1.m . . . 4 = (meet‘𝐾)
5 cdlemg1.a . . . 4 𝐴 = (Atoms‘𝐾)
6 cdlemg1.h . . . 4 𝐻 = (LHyp‘𝐾)
7 cdlemg1.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
8 cdlemg1.d . . . 4 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
9 cdlemg1.e . . . 4 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
10 cdlemg1.g . . . 4 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))
11 cdlemg1.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11cdleme50ltrn 37852 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐺𝑇)
13 simpll1 1209 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
14 simplr 768 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → 𝑓𝑇)
1512ad2antrr 725 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → 𝐺𝑇)
16 simpll2 1210 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
17 simpr 488 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → (𝑓𝑃) = 𝑄)
181, 2, 3, 4, 5, 6, 7, 8, 9, 10cdleme17d 37793 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐺𝑃) = 𝑄)
1918ad2antrr 725 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → (𝐺𝑃) = 𝑄)
2017, 19eqtr4d 2839 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → (𝑓𝑃) = (𝐺𝑃))
212, 5, 6, 11cdlemd 37502 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓𝑇𝐺𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑓𝑃) = (𝐺𝑃)) → 𝑓 = 𝐺)
2213, 14, 15, 16, 20, 21syl311anc 1381 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → 𝑓 = 𝐺)
2322ex 416 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) → ((𝑓𝑃) = 𝑄𝑓 = 𝐺))
2418adantr 484 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) → (𝐺𝑃) = 𝑄)
25 fveq1 6648 . . . . . 6 (𝑓 = 𝐺 → (𝑓𝑃) = (𝐺𝑃))
2625eqeq1d 2803 . . . . 5 (𝑓 = 𝐺 → ((𝑓𝑃) = 𝑄 ↔ (𝐺𝑃) = 𝑄))
2724, 26syl5ibrcom 250 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) → (𝑓 = 𝐺 → (𝑓𝑃) = 𝑄))
2823, 27impbid 215 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) → ((𝑓𝑃) = 𝑄𝑓 = 𝐺))
2912, 28riota5 7126 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑓𝑇 (𝑓𝑃) = 𝑄) = 𝐺)
3029eqcomd 2807 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐺 = (𝑓𝑇 (𝑓𝑃) = 𝑄))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  ⦋csb 3831  ifcif 4428   class class class wbr 5033   ↦ cmpt 5113  ‘cfv 6328  ℩crio 7096  (class class class)co 7139  Basecbs 16479  lecple 16568  joincjn 17550  meetcmee 17551  Atomscatm 36558  HLchlt 36645  LHypclh 37279  LTrncltrn 37396 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-riotaBAD 36248 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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  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 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-undef 7926  df-map 8395  df-proset 17534  df-poset 17552  df-plt 17564  df-lub 17580  df-glb 17581  df-join 17582  df-meet 17583  df-p0 17645  df-p1 17646  df-lat 17652  df-clat 17714  df-oposet 36471  df-ol 36473  df-oml 36474  df-covers 36561  df-ats 36562  df-atl 36593  df-cvlat 36617  df-hlat 36646  df-llines 36793  df-lplanes 36794  df-lvols 36795  df-lines 36796  df-psubsp 36798  df-pmap 36799  df-padd 37091  df-lhyp 37283  df-laut 37284  df-ldil 37399  df-ltrn 37400  df-trl 37454 This theorem is referenced by:  cdlemg1b2  37866
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