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Theorem cdleme48gfv 39000
Description: TODO: fix comment. (Contributed by NM, 9-Apr-2013.)
Hypotheses
Ref Expression
cdlemef46g.b 𝐵 = (Base‘𝐾)
cdlemef46g.l = (le‘𝐾)
cdlemef46g.j = (join‘𝐾)
cdlemef46g.m = (meet‘𝐾)
cdlemef46g.a 𝐴 = (Atoms‘𝐾)
cdlemef46g.h 𝐻 = (LHyp‘𝐾)
cdlemef46g.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdlemef46g.d 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdlemefs46g.e 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
cdlemef46g.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))
cdlemef46.v 𝑉 = ((𝑄 𝑃) 𝑊)
cdlemef46.n 𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))
cdlemefs46.o 𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))
cdlemef46.g 𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))
Assertion
Ref Expression
cdleme48gfv ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) → (𝐺‘(𝐹𝑋)) = 𝑋)
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   𝑎,𝑏,𝑐,𝑢,𝑣,𝐴   𝐵,𝑎,𝑏,𝑐,𝑢,𝑣   𝑣,𝐷   𝐺,𝑠,𝑡,𝑥,𝑦,𝑧   𝐻,𝑎,𝑏,𝑐,𝑢,𝑣   ,𝑎,𝑏,𝑐,𝑢,𝑣   𝐾,𝑎,𝑏,𝑐,𝑢,𝑣   ,𝑎,𝑏,𝑐,𝑢,𝑣   ,𝑎,𝑏,𝑐,𝑢,𝑣   𝑁,𝑎,𝑏,𝑐   𝑂,𝑎,𝑏,𝑐   𝑃,𝑎,𝑏,𝑐,𝑢,𝑣   𝑄,𝑎,𝑏,𝑐,𝑢,𝑣   𝑉,𝑎,𝑏,𝑐   𝑊,𝑎,𝑏,𝑐,𝑢,𝑣,𝑥,𝑦,𝑧   𝑢,𝑁,𝑥,𝑦,𝑧   𝑥,𝑂,𝑦,𝑧   𝑣,𝑡   𝑢,𝑉   𝑥,𝑣,𝑦,𝑧,𝑉   𝐷,𝑎,𝑏,𝑐   𝐸,𝑎,𝑏,𝑐   𝐹,𝑎,𝑏,𝑐,𝑢,𝑣   𝑡,𝑁   𝑈,𝑎,𝑏,𝑐,𝑣   𝑡,𝑉   𝑠,𝑎,𝑡,𝑏,𝑐,𝑥,𝑦,𝑧,𝑢,𝑣   𝑋,𝑎,𝑐,𝑠,𝑡,𝑢,𝑣,𝑥,𝑧
Allowed substitution hints:   𝐷(𝑢,𝑡)   𝑈(𝑢)   𝐸(𝑣,𝑢,𝑡,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐺(𝑣,𝑢,𝑎,𝑏,𝑐)   𝑁(𝑣,𝑠)   𝑂(𝑣,𝑢,𝑡,𝑠)   𝑉(𝑠)   𝑋(𝑦,𝑏)

Proof of Theorem cdleme48gfv
StepHypRef Expression
1 simpll 765 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
2 simprl 769 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑃𝑄)
3 simplr 767 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑋𝐵)
4 simprr 771 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → ¬ 𝑋 𝑊)
53, 4jca 512 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝑋𝐵 ∧ ¬ 𝑋 𝑊))
6 cdlemef46g.b . . . 4 𝐵 = (Base‘𝐾)
7 cdlemef46g.l . . . 4 = (le‘𝐾)
8 cdlemef46g.j . . . 4 = (join‘𝐾)
9 cdlemef46g.m . . . 4 = (meet‘𝐾)
10 cdlemef46g.a . . . 4 𝐴 = (Atoms‘𝐾)
11 cdlemef46g.h . . . 4 𝐻 = (LHyp‘𝐾)
12 cdlemef46g.u . . . 4 𝑈 = ((𝑃 𝑄) 𝑊)
13 cdlemef46g.d . . . 4 𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
14 cdlemefs46g.e . . . 4 𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))
15 cdlemef46g.f . . . 4 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))
16 cdlemef46.v . . . 4 𝑉 = ((𝑄 𝑃) 𝑊)
17 cdlemef46.n . . . 4 𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))
18 cdlemefs46.o . . . 4 𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))
19 cdlemef46.g . . . 4 𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))
206, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19cdleme48gfv1 38999 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊))) → (𝐺‘(𝐹𝑋)) = 𝑋)
211, 2, 5, 20syl12anc 835 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐺‘(𝐹𝑋)) = 𝑋)
2215cdleme31fv2 38856 . . . . . 6 ((𝑋𝐵 ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = 𝑋)
2322adantll 712 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) = 𝑋)
24 simplr 767 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → 𝑋𝐵)
2523, 24eqeltrd 2838 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐹𝑋) ∈ 𝐵)
26 simpr 485 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊))
27 necom 2997 . . . . . . 7 (𝑄𝑃𝑃𝑄)
2827a1i 11 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝑄𝑃𝑃𝑄))
2923breq1d 5115 . . . . . . 7 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → ((𝐹𝑋) 𝑊𝑋 𝑊))
3029notbid 317 . . . . . 6 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (¬ (𝐹𝑋) 𝑊 ↔ ¬ 𝑋 𝑊))
3128, 30anbi12d 631 . . . . 5 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → ((𝑄𝑃 ∧ ¬ (𝐹𝑋) 𝑊) ↔ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)))
3226, 31mtbird 324 . . . 4 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → ¬ (𝑄𝑃 ∧ ¬ (𝐹𝑋) 𝑊))
3319cdleme31fv2 38856 . . . 4 (((𝐹𝑋) ∈ 𝐵 ∧ ¬ (𝑄𝑃 ∧ ¬ (𝐹𝑋) 𝑊)) → (𝐺‘(𝐹𝑋)) = (𝐹𝑋))
3425, 32, 33syl2anc 584 . . 3 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐺‘(𝐹𝑋)) = (𝐹𝑋))
3534, 23eqtrd 2776 . 2 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) → (𝐺‘(𝐹𝑋)) = 𝑋)
3621, 35pm2.61dan 811 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) → (𝐺‘(𝐹𝑋)) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  csb 3855  ifcif 4486   class class class wbr 5105  cmpt 5188  cfv 6496  crio 7312  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  Atomscatm 37725  HLchlt 37812  LHypclh 38447
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-riotaBAD 37415
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-undef 8204  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963  df-lines 37964  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451
This theorem is referenced by:  cdleme48fgv  39001  cdlemeg49lebilem  39002
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