Theorem cdleme39a 41089

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 41088 in convention of cdleme32sn1awN 41056. 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 1217	. . . . 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 1218	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝑃 ∈ 𝐴)
4		simp13 1219	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝑄 ∈ 𝐴)
5		simp2 1150	. . . . 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 1215	. . . . 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 40862	. . . . 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 1402	. . . 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 1216	. . . . . . 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 40852	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → ((𝑡 ∨ 𝐸) ∧ 𝑊) = 𝑈)
19	2, 3, 4, 16, 18	syl13anc 1391	. . . . . 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 2809	. . . . 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 7412	. . . 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 2800	. . 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 1298	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝐾 ∈ HL)
24		simp2l 1213	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝑅 ∈ 𝐴)
25		simp3rl 1260	. . . . . 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 39992	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) → (𝑅 ∨ 𝑡) = (𝑡 ∨ 𝑅))
27	23, 24, 25, 26	syl3anc 1390	. . . . 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 7411	. . . 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 7412	. . 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 7414	. 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 2809	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 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142   class class class wbr 5100  ‘cfv 6521  (class class class)co 7396  lecple 17293  joincjn 18343  meetcmee 18344  Atomscatm 39887  HLchlt 39974  LHypclh 40608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-proset 18326  df-poset 18345  df-plt 18360  df-lub 18376  df-glb 18377  df-join 18378  df-meet 18379  df-p0 18455  df-p1 18456  df-lat 18464  df-clat 18531  df-oposet 39800  df-ol 39802  df-oml 39803  df-covers 39890  df-ats 39891  df-atl 39922  df-cvlat 39946  df-hlat 39975  df-psubsp 40127  df-pmap 40128  df-padd 40420  df-lhyp 40612

This theorem is referenced by:  cdleme39n  41090