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Theorem cdlemefr32sn2aw 36213
Description: Show that 𝑅 / 𝑠𝑁 is an atom not under 𝑊 when ¬ 𝑅 (𝑃 𝑄). (Contributed by NM, 28-Mar-2013.)
Hypotheses
Ref Expression
cdlemefr27.b 𝐵 = (Base‘𝐾)
cdlemefr27.l = (le‘𝐾)
cdlemefr27.j = (join‘𝐾)
cdlemefr27.m = (meet‘𝐾)
cdlemefr27.a 𝐴 = (Atoms‘𝐾)
cdlemefr27.h 𝐻 = (LHyp‘𝐾)
cdlemefr27.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdlemefr27.c 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdlemefr27.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
Assertion
Ref Expression
cdlemefr32sn2aw ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑅 / 𝑠𝑁𝐴 ∧ ¬ 𝑅 / 𝑠𝑁 𝑊))
Distinct variable groups:   𝐴,𝑠   ,𝑠   ,𝑠   ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑈,𝑠   𝑊,𝑠
Allowed substitution hints:   𝐵(𝑠)   𝐶(𝑠)   𝐻(𝑠)   𝐼(𝑠)   𝐾(𝑠)   𝑁(𝑠)

Proof of Theorem cdlemefr32sn2aw
StepHypRef Expression
1 simp11 1245 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 simp12 1246 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
3 simp13 1247 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4 simp2r 1242 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑅𝐴 ∧ ¬ 𝑅 𝑊))
5 simp2l 1241 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑃𝑄)
6 simp3 1132 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ¬ 𝑅 (𝑃 𝑄))
7 cdlemefr27.l . . . . 5 = (le‘𝐾)
8 cdlemefr27.j . . . . 5 = (join‘𝐾)
9 cdlemefr27.m . . . . 5 = (meet‘𝐾)
10 cdlemefr27.a . . . . 5 𝐴 = (Atoms‘𝐾)
11 cdlemefr27.h . . . . 5 𝐻 = (LHyp‘𝐾)
12 cdlemefr27.u . . . . 5 𝑈 = ((𝑃 𝑄) 𝑊)
13 eqid 2771 . . . . 5 ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
147, 8, 9, 10, 11, 12, 13cdleme3fa 36045 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))) ∈ 𝐴)
157, 8, 9, 10, 11, 12, 13cdleme3 36046 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ¬ ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))) 𝑊)
1614, 15jca 501 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → (((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))) ∈ 𝐴 ∧ ¬ ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))) 𝑊))
171, 2, 3, 4, 5, 6, 16syl132anc 1494 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))) ∈ 𝐴 ∧ ¬ ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))) 𝑊))
18 simp2rl 1308 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅𝐴)
19 cdlemefr27.c . . . . . 6 𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
20 cdlemefr27.n . . . . . 6 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)
2119, 20, 13cdleme31sn2 36198 . . . . 5 ((𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
2218, 6, 21syl2anc 573 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
2322eleq1d 2835 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑅 / 𝑠𝑁𝐴 ↔ ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))) ∈ 𝐴))
2422breq1d 4796 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑅 / 𝑠𝑁 𝑊 ↔ ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))) 𝑊))
2524notbid 307 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (¬ 𝑅 / 𝑠𝑁 𝑊 ↔ ¬ ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))) 𝑊))
2623, 25anbi12d 616 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑅 / 𝑠𝑁𝐴 ∧ ¬ 𝑅 / 𝑠𝑁 𝑊) ↔ (((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))) ∈ 𝐴 ∧ ¬ ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))) 𝑊)))
2717, 26mpbird 247 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑅 / 𝑠𝑁𝐴 ∧ ¬ 𝑅 / 𝑠𝑁 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  csb 3682  ifcif 4225   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Atomscatm 35072  HLchlt 35159  LHypclh 35792
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 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  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-ral 3066  df-rex 3067  df-reu 3068  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 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  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-lines 35309  df-psubsp 35311  df-pmap 35312  df-padd 35604  df-lhyp 35796
This theorem is referenced by:  cdlemefr32snb  36214  cdleme32sn2awN  36243  cdleme32snaw  36244
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