Theorem cdleme4 41432

Description: Part of proof of Lemma E in [Crawley] p. 113. 𝐹 and 𝐺 represent f(s) and f_s(r). Here show p ∨ q = r ∨ u at the top of p. 114. (Contributed by NM, 7-Jun-2012.)

Hypotheses

Ref	Expression
cdleme4.l	⊢ ≤ = (le‘𝐾)
cdleme4.j	⊢ ∨ = (join‘𝐾)
cdleme4.m	⊢ ∧ = (meet‘𝐾)
cdleme4.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme4.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme4.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)

Assertion

Ref	Expression
cdleme4	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑈))

Proof of Theorem cdleme4

Step	Hyp	Ref	Expression
1		cdleme4.u	. . 3 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
2	1	oveq2i 7435	. 2 ⊢ (𝑅 ∨ 𝑈) = (𝑅 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))
3		simp1l 1216	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ HL)
4		simp23l 1313	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑅 ∈ 𝐴)
5		simp21 1225	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ∈ 𝐴)
6		simp22 1226	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑄 ∈ 𝐴)
7		eqid 2760	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
8		cdleme4.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
9		cdleme4.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
10	7, 8, 9	hlatjcl 40561	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
11	3, 5, 6, 10	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
12		simp1r 1217	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑊 ∈ 𝐻)
13		cdleme4.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
14	7, 13	lhpbase 41192	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
15	12, 14	syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑊 ∈ (Base‘𝐾))
16		simp3 1156	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑅 ≤ (𝑃 ∨ 𝑄))
17		cdleme4.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
18		cdleme4.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
19	7, 17, 8, 18, 9	atmod3i1 41058	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑅 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑊)))
20	3, 4, 11, 15, 16, 19	syl131anc 1410	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑅 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑊)))
21		simp1 1154	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
22		simp23 1227	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
23		eqid 2760	. . . . . 6 ⊢ (1.‘𝐾) = (1.‘𝐾)
24	17, 8, 23, 9, 13	lhpjat2 41215	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → (𝑅 ∨ 𝑊) = (1.‘𝐾))
25	21, 22, 24	syl2anc 596	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑅 ∨ 𝑊) = (1.‘𝐾))
26	25	oveq2d 7440	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((𝑃 ∨ 𝑄) ∧ (𝑅 ∨ 𝑊)) = ((𝑃 ∨ 𝑄) ∧ (1.‘𝐾)))
27		hlol 40555	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
28	3, 27	syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝐾 ∈ OL)
29	7, 18, 23	olm11 40421	. . . 4 ⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ (1.‘𝐾)) = (𝑃 ∨ 𝑄))
30	28, 11, 29	syl2anc 596	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((𝑃 ∨ 𝑄) ∧ (1.‘𝐾)) = (𝑃 ∨ 𝑄))
31	20, 26, 30	3eqtrd 2799	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑅 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = (𝑃 ∨ 𝑄))
32	2, 31	eqtr2id 2808	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑈))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  lecple 17444  joincjn 18494  meetcmee 18495  1.cp1 18605  OLcol 40368  Atomscatm 40457  HLchlt 40544  LHypclh 41178

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-proset 18477  df-poset 18496  df-plt 18511  df-lub 18527  df-glb 18528  df-join 18529  df-meet 18530  df-p0 18606  df-p1 18607  df-lat 18615  df-clat 18682  df-oposet 40370  df-ol 40372  df-oml 40373  df-covers 40460  df-ats 40461  df-atl 40492  df-cvlat 40516  df-hlat 40545  df-psubsp 40697  df-pmap 40698  df-padd 40990  df-lhyp 41182

This theorem is used by:  cdleme5  41434  cdleme7aa  41436  cdleme7c  41439  cdleme7e  41441  cdleme20i  41511  cdleme36a  41654  cdleme37m  41656  cdleme39a  41659