Theorem cdleme39a 40928

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 40927 in convention of cdleme32sn1awN 40895. 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 1205	. . . . 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 1206	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝑃 ∈ 𝐴)
4		simp13 1207	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝑄 ∈ 𝐴)
5		simp2 1138	. . . . 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 1203	. . . . 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 40701	. . . . 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 1386	. . . 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 1204	. . . . . . 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 40691	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → ((𝑡 ∨ 𝐸) ∧ 𝑊) = 𝑈)
19	2, 3, 4, 16, 18	syl13anc 1375	. . . . . 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 7377	. . . 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 1286	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝐾 ∈ HL)
24		simp2l 1201	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ≤ (𝑃 ∨ 𝑄) ∧ (𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊))) → 𝑅 ∈ 𝐴)
25		simp3rl 1248	. . . . . 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 39831	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) → (𝑅 ∨ 𝑡) = (𝑡 ∨ 𝑅))
27	23, 24, 25, 26	syl3anc 1374	. . . . 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 7376	. . . 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 7377	. . 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 7379	. 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 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361  lecple 17221  joincjn 18271  meetcmee 18272  Atomscatm 39726  HLchlt 39813  LHypclh 40447

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-proset 18254  df-poset 18273  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-p1 18384  df-lat 18392  df-clat 18459  df-oposet 39639  df-ol 39641  df-oml 39642  df-covers 39729  df-ats 39730  df-atl 39761  df-cvlat 39785  df-hlat 39814  df-psubsp 39966  df-pmap 39967  df-padd 40259  df-lhyp 40451

This theorem is referenced by:  cdleme39n  40929