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Theorem cdlemg2klem 40554
Description: cdleme42keg 40445 with simpler hypotheses. TODO: FIX COMMENT. (Contributed by NM, 22-Apr-2013.)
Hypotheses
Ref Expression
cdlemg2.b 𝐵 = (Base‘𝐾)
cdlemg2.l = (le‘𝐾)
cdlemg2.j = (join‘𝐾)
cdlemg2.m = (meet‘𝐾)
cdlemg2.a 𝐴 = (Atoms‘𝐾)
cdlemg2.h 𝐻 = (LHyp‘𝐾)
cdlemg2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemg2ex.u 𝑈 = ((𝑝 𝑞) 𝑊)
cdlemg2ex.d 𝐷 = ((𝑡 𝑈) (𝑞 ((𝑝 𝑡) 𝑊)))
cdlemg2ex.e 𝐸 = ((𝑝 𝑞) (𝐷 ((𝑠 𝑡) 𝑊)))
cdlemg2ex.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))
cdlemg2klem.v 𝑉 = ((𝑃 𝑄) 𝑊)
Assertion
Ref Expression
cdlemg2klem (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → ((𝐹𝑃) (𝐹𝑄)) = ((𝐹𝑃) 𝑉))
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦,𝑧,𝐴   𝐵,𝑠,𝑡,𝑥,𝑦,𝑧   𝐷,𝑠,𝑥,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝐻,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   𝐾,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑠,𝑡,𝑥,𝑦,𝑧   ,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   𝑞,𝑝,𝐴   𝐹,𝑝,𝑞   𝐻,𝑝,𝑞   𝐾,𝑝,𝑞   ,𝑝,𝑞   𝑇,𝑝,𝑞   𝑊,𝑝,𝑞,𝑠,𝑡,𝑥,𝑦,𝑧   ,𝑝,𝑞   𝑃,𝑝,𝑞   𝑄,𝑝,𝑞   𝐵,𝑝,𝑞   ,𝑝,𝑞   𝑉,𝑝,𝑞,𝑠,𝑡,𝑥,𝑧
Allowed substitution hints:   𝐷(𝑡,𝑞,𝑝)   𝑇(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑈(𝑞,𝑝)   𝐸(𝑡,𝑠,𝑞,𝑝)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐺(𝑥,𝑦,𝑧,𝑡,𝑠,𝑞,𝑝)   𝑉(𝑦)

Proof of Theorem cdlemg2klem
StepHypRef Expression
1 cdlemg2.b . . 3 𝐵 = (Base‘𝐾)
2 cdlemg2.l . . 3 = (le‘𝐾)
3 cdlemg2.j . . 3 = (join‘𝐾)
4 cdlemg2.m . . 3 = (meet‘𝐾)
5 cdlemg2.a . . 3 𝐴 = (Atoms‘𝐾)
6 cdlemg2.h . . 3 𝐻 = (LHyp‘𝐾)
7 cdlemg2.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
8 cdlemg2ex.u . . 3 𝑈 = ((𝑝 𝑞) 𝑊)
9 cdlemg2ex.d . . 3 𝐷 = ((𝑡 𝑈) (𝑞 ((𝑝 𝑡) 𝑊)))
10 cdlemg2ex.e . . 3 𝐸 = ((𝑝 𝑞) (𝐷 ((𝑠 𝑡) 𝑊)))
11 cdlemg2ex.g . . 3 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))
12 fveq1 6921 . . . . 5 (𝐹 = 𝐺 → (𝐹𝑃) = (𝐺𝑃))
13 fveq1 6921 . . . . 5 (𝐹 = 𝐺 → (𝐹𝑄) = (𝐺𝑄))
1412, 13oveq12d 7468 . . . 4 (𝐹 = 𝐺 → ((𝐹𝑃) (𝐹𝑄)) = ((𝐺𝑃) (𝐺𝑄)))
1512oveq1d 7465 . . . 4 (𝐹 = 𝐺 → ((𝐹𝑃) 𝑉) = ((𝐺𝑃) 𝑉))
1614, 15eqeq12d 2756 . . 3 (𝐹 = 𝐺 → (((𝐹𝑃) (𝐹𝑄)) = ((𝐹𝑃) 𝑉) ↔ ((𝐺𝑃) (𝐺𝑄)) = ((𝐺𝑃) 𝑉)))
17 vex 3492 . . . . 5 𝑠 ∈ V
18 eqid 2740 . . . . . 6 ((𝑠 𝑈) (𝑞 ((𝑝 𝑠) 𝑊))) = ((𝑠 𝑈) (𝑞 ((𝑝 𝑠) 𝑊)))
199, 18cdleme31sc 40343 . . . . 5 (𝑠 ∈ V → 𝑠 / 𝑡𝐷 = ((𝑠 𝑈) (𝑞 ((𝑝 𝑠) 𝑊))))
2017, 19ax-mp 5 . . . 4 𝑠 / 𝑡𝐷 = ((𝑠 𝑈) (𝑞 ((𝑝 𝑠) 𝑊)))
21 eqid 2740 . . . 4 (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑝 𝑞)) → 𝑦 = 𝐸)) = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑝 𝑞)) → 𝑦 = 𝐸))
22 eqid 2740 . . . 4 if(𝑠 (𝑝 𝑞), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑝 𝑞)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) = if(𝑠 (𝑝 𝑞), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑝 𝑞)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷)
23 eqid 2740 . . . 4 (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑝 𝑞), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑝 𝑞)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (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))))
24 cdlemg2klem.v . . . 4 𝑉 = ((𝑃 𝑄) 𝑊)
251, 2, 3, 4, 5, 6, 8, 20, 9, 10, 21, 22, 23, 11, 24cdleme42keg 40445 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ (𝑞𝐴 ∧ ¬ 𝑞 𝑊)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝐺𝑃) (𝐺𝑄)) = ((𝐺𝑃) 𝑉))
261, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 25cdlemg2ce 40551 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝐹𝑃) (𝐹𝑄)) = ((𝐹𝑃) 𝑉))
27263com23 1126 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → ((𝐹𝑃) (𝐹𝑄)) = ((𝐹𝑃) 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1537  wcel 2108  wne 2946  wral 3067  Vcvv 3488  csb 3921  ifcif 4548   class class class wbr 5166  cmpt 5249  cfv 6575  crio 7405  (class class class)co 7450  Basecbs 17260  lecple 17320  joincjn 18383  meetcmee 18384  Atomscatm 39221  HLchlt 39308  LHypclh 39943  LTrncltrn 40060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772  ax-riotaBAD 38911
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-riota 7406  df-ov 7453  df-oprab 7454  df-mpo 7455  df-1st 8032  df-2nd 8033  df-undef 8316  df-map 8888  df-proset 18367  df-poset 18385  df-plt 18402  df-lub 18418  df-glb 18419  df-join 18420  df-meet 18421  df-p0 18497  df-p1 18498  df-lat 18504  df-clat 18571  df-oposet 39134  df-ol 39136  df-oml 39137  df-covers 39224  df-ats 39225  df-atl 39256  df-cvlat 39280  df-hlat 39309  df-llines 39457  df-lplanes 39458  df-lvols 39459  df-lines 39460  df-psubsp 39462  df-pmap 39463  df-padd 39755  df-lhyp 39947  df-laut 39948  df-ldil 40063  df-ltrn 40064  df-trl 40118
This theorem is referenced by:  cdlemg2k  40560
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