Theorem cdlemc4 41067

Description: Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)

Hypotheses

Ref	Expression
cdlemc3.l	⊢ ≤ = (le‘𝐾)
cdlemc3.j	⊢ ∨ = (join‘𝐾)
cdlemc3.m	⊢ ∧ = (meet‘𝐾)
cdlemc3.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemc3.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemc3.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemc3.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

Assertion

Ref	Expression
cdlemc4	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) → (𝑄 ∨ (𝑅‘𝐹)) ≠ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))

Proof of Theorem cdlemc4

Step	Hyp	Ref	Expression
1		simpll 779	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝐾 ∈ HL)
2	1	hllatd 40237	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝐾 ∈ Lat)
3		simpl 488	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4		simpr1 1213	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝐹 ∈ 𝑇)
5		simpr2l 1251	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑃 ∈ 𝐴)
6		eqid 2760	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		cdlemc3.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
8	6, 7	atbase 40162	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
9	5, 8	syl 18	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑃 ∈ (Base‘𝐾))
10		cdlemc3.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
11		cdlemc3.t	. . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
12	6, 10, 11	ltrncl 40998	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ (Base‘𝐾)) → (𝐹‘𝑃) ∈ (Base‘𝐾))
13	3, 4, 9, 12	syl3anc 1398	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐹‘𝑃) ∈ (Base‘𝐾))
14		simpr3l 1253	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑄 ∈ 𝐴)
15		cdlemc3.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
16	6, 15, 7	hlatjcl 40240	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
17	1, 5, 14, 16	syl3anc 1398	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
18	6, 10	lhpbase 40871	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
19	18	ad2antlr 740	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑊 ∈ (Base‘𝐾))
20		cdlemc3.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
21	6, 20	latmcl 18528	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾))
22	2, 17, 19, 21	syl3anc 1398	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾))
23		cdlemc3.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
24	6, 23, 15	latlej1 18536	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐹‘𝑃) ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾)) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))
25	2, 13, 22, 24	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))
26		breq2 5107	. . . . 5 ⊢ ((𝑄 ∨ (𝑅‘𝐹)) = ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) → ((𝐹‘𝑃) ≤ (𝑄 ∨ (𝑅‘𝐹)) ↔ (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))))
27	25, 26	syl5ibrcom 250	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝑄 ∨ (𝑅‘𝐹)) = ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) → (𝐹‘𝑃) ≤ (𝑄 ∨ (𝑅‘𝐹))))
28		cdlemc3.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
29	23, 15, 20, 7, 10, 11, 28	cdlemc3 41066	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐹‘𝑃) ≤ (𝑄 ∨ (𝑅‘𝐹)) → 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))))
30	27, 29	syld 48	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝑄 ∨ (𝑅‘𝐹)) = ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) → 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))))
31	30	necon3bd 2969	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃)) → (𝑄 ∨ (𝑅‘𝐹)) ≠ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))))
32	31	3impia 1135	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ¬ 𝑄 ≤ (𝑃 ∨ (𝐹‘𝑃))) → (𝑄 ∨ (𝑅‘𝐹)) ≠ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955   class class class wbr 5103  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  lecple 17349  joincjn 18399  meetcmee 18400  Latclat 18519  Atomscatm 40136  HLchlt 40223  LHypclh 40857  LTrncltrn 40974  trLctrl 41031

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-map 8828  df-proset 18382  df-poset 18401  df-plt 18416  df-lub 18432  df-glb 18433  df-join 18434  df-meet 18435  df-p0 18511  df-p1 18512  df-lat 18520  df-clat 18587  df-oposet 40049  df-ol 40051  df-oml 40052  df-covers 40139  df-ats 40140  df-atl 40171  df-cvlat 40195  df-hlat 40224  df-psubsp 40376  df-pmap 40377  df-padd 40669  df-lhyp 40861  df-laut 40862  df-ldil 40977  df-ltrn 40978  df-trl 41032

This theorem is used by:  cdlemc5  41068