Theorem cdlemc2 39697

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 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐹‘𝑄) ≤ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))

Proof of Theorem cdlemc2

Step	Hyp	Ref	Expression
1		simp1l 1194	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝐾 ∈ HL)
2		simp3ll 1241	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑃 ∈ 𝐴)
3		simp3rl 1243	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑄 ∈ 𝐴)
4		cdlemc2.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
5		cdlemc2.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
6		cdlemc2.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
7	4, 5, 6	hlatlej2 38880	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → 𝑄 ≤ (𝑃 ∨ 𝑄))
8	1, 2, 3, 7	syl3anc 1368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑄 ≤ (𝑃 ∨ 𝑄))
9		simp1 1133	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		eqid 2728	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
11	10, 6	atbase 38793	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
12	3, 11	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑄 ∈ (Base‘𝐾))
13		simp3l 1198	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
14		cdlemc2.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
15		cdlemc2.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
16	10, 4, 5, 14, 6, 15	cdlemc1 39696	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = (𝑃 ∨ 𝑄))
17	9, 12, 13, 16	syl3anc 1368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) = (𝑃 ∨ 𝑄))
18	8, 17	breqtrrd 5180	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑄 ≤ (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))
19		simp2 1134	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝐹 ∈ 𝑇)
20	1	hllatd 38868	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝐾 ∈ Lat)
21	10, 6	atbase 38793	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
22	2, 21	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑃 ∈ (Base‘𝐾))
23	10, 5	latjcl 18438	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
24	20, 22, 12, 23	syl3anc 1368	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
25		simp1r 1195	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑊 ∈ 𝐻)
26	10, 15	lhpbase 39503	. . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
27	25, 26	syl 17	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → 𝑊 ∈ (Base‘𝐾))
28	10, 14	latmcl 18439	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾))
29	20, 24, 27, 28	syl3anc 1368	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾))
30	10, 5	latjcl 18438	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾)) → (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (Base‘𝐾))
31	20, 22, 29, 30	syl3anc 1368	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (Base‘𝐾))
32		cdlemc2.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
33	10, 4, 15, 32	ltrnle 39634	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (Base‘𝐾))) → (𝑄 ≤ (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ↔ (𝐹‘𝑄) ≤ (𝐹‘(𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))))
34	9, 19, 12, 31, 33	syl112anc 1371	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝑄 ≤ (𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)) ↔ (𝐹‘𝑄) ≤ (𝐹‘(𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))))
35	18, 34	mpbid 231	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐹‘𝑄) ≤ (𝐹‘(𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))))
36	10, 5, 15, 32	ltrnj 39637	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾))) → (𝐹‘(𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) = ((𝐹‘𝑃) ∨ (𝐹‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
37	9, 19, 22, 29, 36	syl112anc 1371	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐹‘(𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) = ((𝐹‘𝑃) ∨ (𝐹‘((𝑃 ∨ 𝑄) ∧ 𝑊))))
38	10, 4, 14	latmle2 18464	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
39	20, 24, 27, 38	syl3anc 1368	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)
40	10, 4, 15, 32	ltrnval1 39639	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ 𝑊)) → (𝐹‘((𝑃 ∨ 𝑄) ∧ 𝑊)) = ((𝑃 ∨ 𝑄) ∧ 𝑊))
41	9, 19, 29, 39, 40	syl112anc 1371	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐹‘((𝑃 ∨ 𝑄) ∧ 𝑊)) = ((𝑃 ∨ 𝑄) ∧ 𝑊))
42	41	oveq2d 7442	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → ((𝐹‘𝑃) ∨ (𝐹‘((𝑃 ∨ 𝑄) ∧ 𝑊))) = ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))
43	37, 42	eqtrd 2768	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐹‘(𝑃 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊))) = ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))
44	35, 43	breqtrd 5178	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐹‘𝑄) ≤ ((𝐹‘𝑃) ∨ ((𝑃 ∨ 𝑄) ∧ 𝑊)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   class class class wbr 5152  ‘cfv 6553  (class class class)co 7426  Basecbs 17187  lecple 17247  joincjn 18310  meetcmee 18311  Latclat 18430  Atomscatm 38767  HLchlt 38854  LHypclh 39489  LTrncltrn 39606

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-map 8853  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-p1 18425  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-psubsp 39008  df-pmap 39009  df-padd 39301  df-lhyp 39493  df-laut 39494  df-ldil 39609  df-ltrn 39610

This theorem is referenced by:  cdlemc5  39700