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Theorem cdlemc2 37320
Description: Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
Hypotheses
Ref Expression
cdlemc2.l = (le‘𝐾)
cdlemc2.j = (join‘𝐾)
cdlemc2.m = (meet‘𝐾)
cdlemc2.a 𝐴 = (Atoms‘𝐾)
cdlemc2.h 𝐻 = (LHyp‘𝐾)
cdlemc2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemc2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝐹𝑄) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))

Proof of Theorem cdlemc2
StepHypRef Expression
1 simp1l 1191 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝐾 ∈ HL)
2 simp3ll 1238 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑃𝐴)
3 simp3rl 1240 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑄𝐴)
4 cdlemc2.l . . . . . 6 = (le‘𝐾)
5 cdlemc2.j . . . . . 6 = (join‘𝐾)
6 cdlemc2.a . . . . . 6 𝐴 = (Atoms‘𝐾)
74, 5, 6hlatlej2 36504 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))
81, 2, 3, 7syl3anc 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑄 (𝑃 𝑄))
9 simp1 1130 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
10 eqid 2819 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
1110, 6atbase 36417 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
123, 11syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑄 ∈ (Base‘𝐾))
13 simp3l 1195 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
14 cdlemc2.m . . . . . 6 = (meet‘𝐾)
15 cdlemc2.h . . . . . 6 𝐻 = (LHyp‘𝐾)
1610, 4, 5, 14, 6, 15cdlemc1 37319 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 ((𝑃 𝑄) 𝑊)) = (𝑃 𝑄))
179, 12, 13, 16syl3anc 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑃 ((𝑃 𝑄) 𝑊)) = (𝑃 𝑄))
188, 17breqtrrd 5085 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑄 (𝑃 ((𝑃 𝑄) 𝑊)))
19 simp2 1131 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝐹𝑇)
201hllatd 36492 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝐾 ∈ Lat)
2110, 6atbase 36417 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
222, 21syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑃 ∈ (Base‘𝐾))
2310, 5latjcl 17653 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
2420, 22, 12, 23syl3anc 1365 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑃 𝑄) ∈ (Base‘𝐾))
25 simp1r 1192 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑊𝐻)
2610, 15lhpbase 37126 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑊 ∈ (Base‘𝐾))
2810, 14latmcl 17654 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
2920, 24, 27, 28syl3anc 1365 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
3010, 5latjcl 17653 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾)) → (𝑃 ((𝑃 𝑄) 𝑊)) ∈ (Base‘𝐾))
3120, 22, 29, 30syl3anc 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑃 ((𝑃 𝑄) 𝑊)) ∈ (Base‘𝐾))
32 cdlemc2.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
3310, 4, 15, 32ltrnle 37257 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ((𝑃 𝑄) 𝑊)) ∈ (Base‘𝐾))) → (𝑄 (𝑃 ((𝑃 𝑄) 𝑊)) ↔ (𝐹𝑄) (𝐹‘(𝑃 ((𝑃 𝑄) 𝑊)))))
349, 19, 12, 31, 33syl112anc 1368 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 (𝑃 ((𝑃 𝑄) 𝑊)) ↔ (𝐹𝑄) (𝐹‘(𝑃 ((𝑃 𝑄) 𝑊)))))
3518, 34mpbid 234 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝐹𝑄) (𝐹‘(𝑃 ((𝑃 𝑄) 𝑊))))
3610, 5, 15, 32ltrnj 37260 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))) → (𝐹‘(𝑃 ((𝑃 𝑄) 𝑊))) = ((𝐹𝑃) (𝐹‘((𝑃 𝑄) 𝑊))))
379, 19, 22, 29, 36syl112anc 1368 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝐹‘(𝑃 ((𝑃 𝑄) 𝑊))) = ((𝐹𝑃) (𝐹‘((𝑃 𝑄) 𝑊))))
3810, 4, 14latmle2 17679 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
3920, 24, 27, 38syl3anc 1365 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝑃 𝑄) 𝑊) 𝑊)
4010, 4, 15, 32ltrnval1 37262 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾) ∧ ((𝑃 𝑄) 𝑊) 𝑊)) → (𝐹‘((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) 𝑊))
419, 19, 29, 39, 40syl112anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝐹‘((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) 𝑊))
4241oveq2d 7164 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝐹𝑃) (𝐹‘((𝑃 𝑄) 𝑊))) = ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
4337, 42eqtrd 2854 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝐹‘(𝑃 ((𝑃 𝑄) 𝑊))) = ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
4435, 43breqtrd 5083 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝐹𝑄) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  w3a 1081   = wceq 1530  wcel 2107   class class class wbr 5057  cfv 6348  (class class class)co 7148  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  Latclat 17647  Atomscatm 36391  HLchlt 36478  LHypclh 37112  LTrncltrn 37229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-map 8400  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36304  df-ol 36306  df-oml 36307  df-covers 36394  df-ats 36395  df-atl 36426  df-cvlat 36450  df-hlat 36479  df-psubsp 36631  df-pmap 36632  df-padd 36924  df-lhyp 37116  df-laut 37117  df-ldil 37232  df-ltrn 37233
This theorem is referenced by:  cdlemc5  37323
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