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Theorem cdlemg1a 36583
Description: Shorter expression for 𝐺. TODO: fix comment. TODO: shorten using cdleme 36573 or vice-versa? Also, if not shortened with cdleme 36573, 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 36570 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐺𝑇)
13 simpll1 1270 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
14 simplr 786 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → 𝑓𝑇)
1512ad2antrr 718 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → 𝐺𝑇)
16 simpll2 1272 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
17 simpr 478 . . . . . . 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 36511 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐺𝑃) = 𝑄)
1918ad2antrr 718 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → (𝐺𝑃) = 𝑄)
2017, 19eqtr4d 2834 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → (𝑓𝑃) = (𝐺𝑃))
212, 5, 6, 11cdlemd 36220 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑓𝑇𝐺𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑓𝑃) = (𝐺𝑃)) → 𝑓 = 𝐺)
2213, 14, 15, 16, 20, 21syl311anc 1504 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) ∧ (𝑓𝑃) = 𝑄) → 𝑓 = 𝐺)
2322ex 402 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) → ((𝑓𝑃) = 𝑄𝑓 = 𝐺))
2418adantr 473 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) → (𝐺𝑃) = 𝑄)
25 fveq1 6408 . . . . . 6 (𝑓 = 𝐺 → (𝑓𝑃) = (𝐺𝑃))
2625eqeq1d 2799 . . . . 5 (𝑓 = 𝐺 → ((𝑓𝑃) = 𝑄 ↔ (𝐺𝑃) = 𝑄))
2724, 26syl5ibrcom 239 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) → (𝑓 = 𝐺 → (𝑓𝑃) = 𝑄))
2823, 27impbid 204 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑓𝑇) → ((𝑓𝑃) = 𝑄𝑓 = 𝐺))
2912, 28riota5 6863 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑓𝑇 (𝑓𝑃) = 𝑄) = 𝐺)
3029eqcomd 2803 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 385  w3a 1108   = wceq 1653  wcel 2157  wne 2969  wral 3087  csb 3726  ifcif 4275   class class class wbr 4841  cmpt 4920  cfv 6099  crio 6836  (class class class)co 6876  Basecbs 16181  lecple 16271  joincjn 17256  meetcmee 17257  Atomscatm 35276  HLchlt 35363  LHypclh 35997  LTrncltrn 36114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-riotaBAD 34966
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-iin 4711  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-undef 7635  df-map 8095  df-proset 17240  df-poset 17258  df-plt 17270  df-lub 17286  df-glb 17287  df-join 17288  df-meet 17289  df-p0 17351  df-p1 17352  df-lat 17358  df-clat 17420  df-oposet 35189  df-ol 35191  df-oml 35192  df-covers 35279  df-ats 35280  df-atl 35311  df-cvlat 35335  df-hlat 35364  df-llines 35511  df-lplanes 35512  df-lvols 35513  df-lines 35514  df-psubsp 35516  df-pmap 35517  df-padd 35809  df-lhyp 36001  df-laut 36002  df-ldil 36117  df-ltrn 36118  df-trl 36172
This theorem is referenced by:  cdlemg1b2  36584
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