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Theorem cdleme17b 38796
Description: Lemma leading to cdleme17c 38797. (Contributed by NM, 11-Oct-2012.)
Hypotheses
Ref Expression
cdleme17.l ≀ = (leβ€˜πΎ)
cdleme17.j ∨ = (joinβ€˜πΎ)
cdleme17.m ∧ = (meetβ€˜πΎ)
cdleme17.a 𝐴 = (Atomsβ€˜πΎ)
cdleme17.h 𝐻 = (LHypβ€˜πΎ)
cdleme17.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdleme17.f 𝐹 = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))
cdleme17.g 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))
cdleme17.c 𝐢 = ((𝑃 ∨ 𝑆) ∧ π‘Š)
Assertion
Ref Expression
cdleme17b (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝐢 ≀ (𝑃 ∨ 𝑄))

Proof of Theorem cdleme17b
StepHypRef Expression
1 simp33 1212 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))
2 eqid 2733 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3 cdleme17.l . . 3 ≀ = (leβ€˜πΎ)
4 simpl1l 1225 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝐾 ∈ HL)
54hllatd 37872 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝐾 ∈ Lat)
6 simpl32 1256 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑆 ∈ 𝐴)
7 cdleme17.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
82, 7atbase 37797 . . . 4 (𝑆 ∈ 𝐴 β†’ 𝑆 ∈ (Baseβ€˜πΎ))
96, 8syl 17 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑆 ∈ (Baseβ€˜πΎ))
10 simpl2l 1227 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑃 ∈ 𝐴)
11 cdleme17.j . . . . 5 ∨ = (joinβ€˜πΎ)
122, 11, 7hlatjcl 37875 . . . 4 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
134, 10, 6, 12syl3anc 1372 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑃 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
14 simpl31 1255 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑄 ∈ 𝐴)
152, 11, 7hlatjcl 37875 . . . 4 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
164, 10, 14, 15syl3anc 1372 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
173, 11, 7hlatlej2 37884 . . . 4 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) β†’ 𝑆 ≀ (𝑃 ∨ 𝑆))
184, 10, 6, 17syl3anc 1372 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑆 ≀ (𝑃 ∨ 𝑆))
19 simpl1r 1226 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ π‘Š ∈ 𝐻)
20 simpl2r 1228 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ Β¬ 𝑃 ≀ π‘Š)
21 cdleme17.m . . . . . 6 ∧ = (meetβ€˜πΎ)
22 cdleme17.h . . . . . 6 𝐻 = (LHypβ€˜πΎ)
23 cdleme17.c . . . . . 6 𝐢 = ((𝑃 ∨ 𝑆) ∧ π‘Š)
243, 11, 21, 7, 22, 23cdleme8 38759 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ (𝑃 ∨ 𝐢) = (𝑃 ∨ 𝑆))
254, 19, 10, 20, 6, 24syl221anc 1382 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑃 ∨ 𝐢) = (𝑃 ∨ 𝑆))
263, 11, 7hlatlej1 37883 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
274, 10, 14, 26syl3anc 1372 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑃 ≀ (𝑃 ∨ 𝑄))
28 simpr 486 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝐢 ≀ (𝑃 ∨ 𝑄))
292, 7atbase 37797 . . . . . . 7 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
3010, 29syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
312, 11, 21, 7, 22, 23cdleme9b 38761 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ π‘Š ∈ 𝐻)) β†’ 𝐢 ∈ (Baseβ€˜πΎ))
324, 10, 6, 19, 31syl13anc 1373 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝐢 ∈ (Baseβ€˜πΎ))
332, 3, 11latjle12 18344 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝐢 ∈ (Baseβ€˜πΎ) ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))) β†’ ((𝑃 ≀ (𝑃 ∨ 𝑄) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) ↔ (𝑃 ∨ 𝐢) ≀ (𝑃 ∨ 𝑄)))
345, 30, 32, 16, 33syl13anc 1373 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ ((𝑃 ≀ (𝑃 ∨ 𝑄) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) ↔ (𝑃 ∨ 𝐢) ≀ (𝑃 ∨ 𝑄)))
3527, 28, 34mpbi2and 711 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑃 ∨ 𝐢) ≀ (𝑃 ∨ 𝑄))
3625, 35eqbrtrrd 5130 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑃 ∨ 𝑆) ≀ (𝑃 ∨ 𝑄))
372, 3, 5, 9, 13, 16, 18, 36lattrd 18340 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) ∧ 𝐢 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑆 ≀ (𝑃 ∨ 𝑄))
381, 37mtand 815 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝐢 ≀ (𝑃 ∨ 𝑄))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  joincjn 18205  meetcmee 18206  Latclat 18325  Atomscatm 37771  HLchlt 37858  LHypclh 38493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-proset 18189  df-poset 18207  df-plt 18224  df-lub 18240  df-glb 18241  df-join 18242  df-meet 18243  df-p0 18319  df-p1 18320  df-lat 18326  df-clat 18393  df-oposet 37684  df-ol 37686  df-oml 37687  df-covers 37774  df-ats 37775  df-atl 37806  df-cvlat 37830  df-hlat 37859  df-psubsp 38012  df-pmap 38013  df-padd 38305  df-lhyp 38497
This theorem is referenced by:  cdleme17c  38797
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