Theorem cdleme17c 41169

Description: Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph. 𝐶 represents s₁. We show, in their notation, (p ∨ q) ∧ (q ∨ s₁)=q. (Contributed by NM, 11-Oct-2012.)

Hypotheses

Ref	Expression
cdleme17.l	⊢ ≤ = (le‘𝐾)
cdleme17.j	⊢ ∨ = (join‘𝐾)
cdleme17.m	⊢ ∧ = (meet‘𝐾)
cdleme17.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme17.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme17.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme17.f	⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
cdleme17.g	⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
cdleme17.c	⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊)

Assertion

Ref	Expression
cdleme17c	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ 𝐶)) = 𝑄)

Proof of Theorem cdleme17c

Step	Hyp	Ref	Expression
1		simp1l 1216	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ HL)
2		simp2l 1218	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ 𝐴)
3		simp31 1228	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ 𝐴)
4		cdleme17.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		cdleme17.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	hlatjcom 40249	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
7	1, 2, 3, 6	syl3anc 1398	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
8	7	oveq1d 7432	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ 𝐶)) = ((𝑄 ∨ 𝑃) ∧ (𝑄 ∨ 𝐶)))
9		simp1r 1217	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑊 ∈ 𝐻)
10		simp2r 1219	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑃 ≤ 𝑊)
11		simp32 1229	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑆 ∈ 𝐴)
12	1	hllatd 40245	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐾 ∈ Lat)
13		eqid 2762	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
14	13, 5	atbase 40170	. . . . . 6 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
15	11, 14	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑆 ∈ (Base‘𝐾))
16	13, 5	atbase 40170	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
17	2, 16	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ∈ (Base‘𝐾))
18	13, 5	atbase 40170	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
19	3, 18	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑄 ∈ (Base‘𝐾))
20		simp33 1230	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
21		cdleme17.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
22	13, 21, 4	latnlej1l 18551	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑆 ≠ 𝑃)
23	22	necomd 3012	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑆)
24	12, 15, 17, 19, 20, 23	syl131anc 1410	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑃 ≠ 𝑆)
25		cdleme17.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
26		cdleme17.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
27		cdleme17.c	. . . . 5 ⊢ 𝐶 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
28	21, 4, 25, 5, 26, 27	cdleme9a 41132	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆)) → 𝐶 ∈ 𝐴)
29	1, 9, 2, 10, 11, 24, 28	syl222anc 1413	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝐶 ∈ 𝐴)
30		cdleme17.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
31		cdleme17.f	. . . 4 ⊢ 𝐹 = ((𝑆 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
32		cdleme17.g	. . . 4 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑃 ∨ 𝑆) ∧ 𝑊)))
33	21, 4, 25, 5, 26, 30, 31, 32, 27	cdleme17b 41168	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
34	21, 4, 25, 5	2llnma1 40668	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) → ((𝑄 ∨ 𝑃) ∧ (𝑄 ∨ 𝐶)) = 𝑄)
35	1, 2, 3, 29, 33, 34	syl131anc 1410	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ((𝑄 ∨ 𝑃) ∧ (𝑄 ∨ 𝐶)) = 𝑄)
36	8, 35	eqtrd 2797	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑄) ∧ (𝑄 ∨ 𝐶)) = 𝑄)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957   class class class wbr 5107  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  lecple 17355  joincjn 18405  meetcmee 18406  Latclat 18525  Atomscatm 40144  HLchlt 40231  LHypclh 40865

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-proset 18388  df-poset 18407  df-plt 18422  df-lub 18438  df-glb 18439  df-join 18440  df-meet 18441  df-p0 18517  df-p1 18518  df-lat 18526  df-clat 18593  df-oposet 40057  df-ol 40059  df-oml 40060  df-covers 40147  df-ats 40148  df-atl 40179  df-cvlat 40203  df-hlat 40232  df-psubsp 40384  df-pmap 40385  df-padd 40677  df-lhyp 40869

This theorem is used by:  cdleme17d1  41170