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Theorem cdleme41fva11 36287
 Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(r) is one-to-one for r in W (r an atom not under w). TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013.)
Hypotheses
Ref Expression
cdleme41.b 𝐵 = (Base‘𝐾)
cdleme41.l = (le‘𝐾)
cdleme41.j = (join‘𝐾)
cdleme41.m = (meet‘𝐾)
cdleme41.a 𝐴 = (Atoms‘𝐾)
cdleme41.h 𝐻 = (LHyp‘𝐾)
cdleme41.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme41.d 𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme41.e 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme41.g 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
cdleme41.i 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
cdleme41.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
cdleme41.o 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
cdleme41.f 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
Assertion
Ref Expression
cdleme41fva11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑅𝑆) → (𝐹𝑅) ≠ (𝐹𝑆))
Distinct variable groups:   𝐴,𝑠   ,𝑠   ,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑆,𝑠   𝑈,𝑠   𝑊,𝑠   𝑦,𝑡,𝐴,𝑠   𝐵,𝑠,𝑡,𝑦   𝑦,𝐷   𝑦,𝐺   𝐸,𝑠,𝑦   𝐻,𝑠,𝑡,𝑦   𝑡, ,𝑦   𝐾,𝑠,𝑡,𝑦   𝑡, ,𝑦   𝑡, ,𝑦   𝑡,𝑃,𝑦   𝑡,𝑄,𝑦   𝑡,𝑅,𝑦   𝑡,𝑆,𝑦   𝑡,𝑈,𝑦   𝑡,𝑊,𝑦   𝑥,𝑧,𝐴   𝑥,𝐵,𝑧   𝑧,𝐸,𝑠   𝑧,𝐻   𝑥, ,𝑧   𝑧,𝐾   𝑥, ,𝑧   𝑥, ,𝑧   𝑥,𝑁,𝑧   𝑥,𝑃,𝑧   𝑥,𝑄,𝑧   𝑥,𝑅,𝑧   𝑥,𝑆,𝑧   𝑥,𝑈,𝑧   𝑥,𝑊,𝑧,𝑠,𝑡,𝑦
Allowed substitution hints:   𝐷(𝑥,𝑧,𝑡,𝑠)   𝐸(𝑥,𝑡)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐺(𝑥,𝑧,𝑡,𝑠)   𝐻(𝑥)   𝐼(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐾(𝑥)   𝑁(𝑦,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡,𝑠)

Proof of Theorem cdleme41fva11
StepHypRef Expression
1 cdleme41.b . . 3 𝐵 = (Base‘𝐾)
2 cdleme41.l . . 3 = (le‘𝐾)
3 cdleme41.j . . 3 = (join‘𝐾)
4 cdleme41.m . . 3 = (meet‘𝐾)
5 cdleme41.a . . 3 𝐴 = (Atoms‘𝐾)
6 cdleme41.h . . 3 𝐻 = (LHyp‘𝐾)
7 cdleme41.u . . 3 𝑈 = ((𝑃 𝑄) 𝑊)
8 cdleme41.d . . 3 𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
9 cdleme41.e . . 3 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
10 cdleme41.g . . 3 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
11 cdleme41.i . . 3 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
12 cdleme41.n . . 3 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cdleme41snaw 36286 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑅𝑆) → 𝑅 / 𝑠𝑁𝑆 / 𝑠𝑁)
14 simp1 1130 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑅𝑆) → ((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
15 simp22 1249 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑅𝑆) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
16 simp21 1248 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑅𝑆) → 𝑃𝑄)
17 cdleme41.o . . . 4 𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))
18 cdleme41.f . . . 4 𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 18cdleme32fva1 36248 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (𝐹𝑅) = 𝑅 / 𝑠𝑁)
2014, 15, 16, 19syl3anc 1476 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑅𝑆) → (𝐹𝑅) = 𝑅 / 𝑠𝑁)
21 simp23 1250 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑅𝑆) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
221, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 17, 18cdleme32fva1 36248 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑃𝑄) → (𝐹𝑆) = 𝑆 / 𝑠𝑁)
2314, 21, 16, 22syl3anc 1476 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑅𝑆) → (𝐹𝑆) = 𝑆 / 𝑠𝑁)
2413, 20, 233netr4d 3020 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑅𝑆) → (𝐹𝑅) ≠ (𝐹𝑆))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061  ⦋csb 3682  ifcif 4226   class class class wbr 4787   ↦ cmpt 4864  ‘cfv 6030  ℩crio 6756  (class class class)co 6796  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Atomscatm 35072  HLchlt 35159  LHypclh 35793 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-riotaBAD 34761 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-undef 7555  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-p1 17248  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35307  df-lplanes 35308  df-lvols 35309  df-lines 35310  df-psubsp 35312  df-pmap 35313  df-padd 35605  df-lhyp 35797 This theorem is referenced by:  cdleme42k  36294
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