Theorem cdleme39a 39422

Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 ∨ 𝑄 line. TODO: FIX COMMENT. 𝐸, 𝑌, 𝐺, 𝑍 serve as f(t), f(u), f_t(𝑅), f_t(𝑆). Put hypotheses of cdleme38n 39421 in convention of cdleme32sn1awN 39389. TODO see if this hypothesis conversion would be better if done earlier. (Contributed by NM, 15-Mar-2013.)

Hypotheses

Ref	Expression
cdleme39.l	⊢ ≤ = (le‘𝐾)
cdleme39.j	⊢ ∨ = (join‘𝐾)
cdleme39.m	⊢ ∧ = (meet‘𝐾)
cdleme39.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme39.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme39.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme39.e	⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme39.g	⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
cdleme39a.v	⊢ 𝑉 = ((𝑡 ∨ 𝐸) ∧ 𝑊)

Assertion

Ref	Expression
cdleme39a	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝐺 = ((𝑅 ∨ 𝑉) ∧ (𝐸 ∨ ((𝑡 ∨ 𝑅) ∧ 𝑊))))

Proof of Theorem cdleme39a

Step	Hyp	Ref	Expression
1		cdleme39.g	. 2 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
2		simp11 1203	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		simp12 1204	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝑃 ∈ 𝐴)
4		simp13 1205	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝑄 ∈ 𝐴)
5		simp2 1137	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
6		simp3l 1201	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝑅 ≤ (𝑃 ∨ 𝑄))
7		cdleme39.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
8		cdleme39.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
9		cdleme39.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
10		cdleme39.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
11		cdleme39.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
12		cdleme39.u	. . . . . 6 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
13	7, 8, 9, 10, 11, 12	cdleme4 39195	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑈))
14	2, 3, 4, 5, 6, 13	syl131anc 1383	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑈))
15		cdleme39a.v	. . . . . 6 ⊢ 𝑉 = ((𝑡 ∨ 𝐸) ∧ 𝑊)
16		simp3r 1202	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))
17		cdleme39.e	. . . . . . . 8 ⊢ 𝐸 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
18	7, 8, 9, 10, 11, 12, 17	cdleme2 39185	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → ((𝑡 ∨ 𝐸) ∧ 𝑊) = 𝑈)
19	2, 3, 4, 16, 18	syl13anc 1372	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → ((𝑡 ∨ 𝐸) ∧ 𝑊) = 𝑈)
20	15, 19	eqtrid 2784	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝑉 = 𝑈)
21	20	oveq2d 7427	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → (𝑅 ∨ 𝑉) = (𝑅 ∨ 𝑈))
22	14, 21	eqtr4d 2775	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑉))
23		simp11l 1284	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝐾 ∈ HL)
24		simp2l 1199	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝑅 ∈ 𝐴)
25		simp3rl 1246	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝑡 ∈ 𝐴)
26	8, 10	hlatjcom 38324	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) → (𝑅 ∨ 𝑡) = (𝑡 ∨ 𝑅))
27	23, 24, 25, 26	syl3anc 1371	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → (𝑅 ∨ 𝑡) = (𝑡 ∨ 𝑅))
28	27	oveq1d 7426	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → ((𝑅 ∨ 𝑡) ∧ 𝑊) = ((𝑡 ∨ 𝑅) ∧ 𝑊))
29	28	oveq2d 7427	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)) = (𝐸 ∨ ((𝑡 ∨ 𝑅) ∧ 𝑊)))
30	22, 29	oveq12d 7429	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) = ((𝑅 ∨ 𝑉) ∧ (𝐸 ∨ ((𝑡 ∨ 𝑅) ∧ 𝑊))))
31	1, 30	eqtrid 2784	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝐺 = ((𝑅 ∨ 𝑉) ∧ (𝐸 ∨ ((𝑡 ∨ 𝑅) ∧ 𝑊))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  ‘cfv 6543  (class class class)co 7411  lecple 17206  joincjn 18266  meetcmee 18267  Atomscatm 38219  HLchlt 38306  LHypclh 38941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-proset 18250  df-poset 18268  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18387  df-clat 18454  df-oposet 38132  df-ol 38134  df-oml 38135  df-covers 38222  df-ats 38223  df-atl 38254  df-cvlat 38278  df-hlat 38307  df-psubsp 38460  df-pmap 38461  df-padd 38753  df-lhyp 38945

This theorem is referenced by:  cdleme39n  39423