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Theorem cdleme40v 36074
Description: Part of proof of Lemma E in [Crawley] p. 113. Change bound variables in 𝑆 / 𝑢𝑉 (but we use 𝑅 / 𝑢𝑉 for convenience since we have its hypotheses available) . (Contributed by NM, 18-Mar-2013.)
Hypotheses
Ref Expression
cdleme40.b 𝐵 = (Base‘𝐾)
cdleme40.l = (le‘𝐾)
cdleme40.j = (join‘𝐾)
cdleme40.m = (meet‘𝐾)
cdleme40.a 𝐴 = (Atoms‘𝐾)
cdleme40.h 𝐻 = (LHyp‘𝐾)
cdleme40.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme40.e 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
cdleme40.g 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
cdleme40.i 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
cdleme40.n 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
cdleme40.d 𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
cdleme40r.y 𝑌 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))
cdleme40r.t 𝑇 = ((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊)))
cdleme40r.x 𝑋 = ((𝑃 𝑄) (𝑇 ((𝑢 𝑣) 𝑊)))
cdleme40r.o 𝑂 = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋))
cdleme40r.v 𝑉 = if(𝑢 (𝑃 𝑄), 𝑂, 𝑌)
Assertion
Ref Expression
cdleme40v (𝑅𝐴𝑅 / 𝑠𝑁 = 𝑅 / 𝑢𝑉)
Distinct variable groups:   𝑣,𝑢,𝑧,𝐴   𝑢,𝐵,𝑣,𝑧   𝑣,𝐻,𝑧   𝑢, ,𝑣,𝑧   𝑣,𝐾,𝑧   𝑢, ,𝑣,𝑧   𝑢, ,𝑣,𝑧   𝑢,𝑃,𝑣,𝑧   𝑢,𝑄,𝑣,𝑧   𝑣,𝑅,𝑧   𝑢,𝑇   𝑣,𝑈,𝑧   𝑢,𝑊,𝑣,𝑧,𝑠,𝑡,𝑦   𝐴,𝑠   𝑦,𝑡,𝐴   𝐵,𝑠,𝑡,𝑦   𝐸,𝑠   𝑡,𝐻,𝑦   ,𝑠,𝑡,𝑦   𝑡,𝐾,𝑦   ,𝑠,𝑡,𝑦   ,𝑠,𝑡,𝑦   𝑃,𝑠,𝑡,𝑦   𝑄,𝑠,𝑡,𝑦   𝑅,𝑠,𝑡,𝑦   𝑡,𝑈,𝑦   𝑊,𝑠,𝑡,𝑦   𝑦,𝑌   𝑣,𝑡,𝑦   𝑇,𝑠,𝑡,𝑦   𝑣,𝐸,𝑧   𝑢,𝑁,𝑣   𝑢,𝑅   𝑉,𝑠   𝑡,𝑋,𝑦   𝑢,𝑠,𝑧,𝑡,𝑦
Allowed substitution hints:   𝐷(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝑇(𝑧,𝑣)   𝑈(𝑢,𝑠)   𝐸(𝑦,𝑢,𝑡)   𝐺(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝐻(𝑢,𝑠)   𝐼(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝐾(𝑢,𝑠)   𝑁(𝑦,𝑧,𝑡,𝑠)   𝑂(𝑦,𝑧,𝑣,𝑢,𝑡,𝑠)   𝑉(𝑦,𝑧,𝑣,𝑢,𝑡)   𝑋(𝑧,𝑣,𝑢,𝑠)   𝑌(𝑧,𝑣,𝑢,𝑡,𝑠)

Proof of Theorem cdleme40v
StepHypRef Expression
1 breq1 4688 . . . . 5 (𝑠 = 𝑢 → (𝑠 (𝑃 𝑄) ↔ 𝑢 (𝑃 𝑄)))
2 cdleme40.g . . . . . . . . . . . 12 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
3 oveq1 6697 . . . . . . . . . . . . . . 15 (𝑠 = 𝑢 → (𝑠 𝑡) = (𝑢 𝑡))
43oveq1d 6705 . . . . . . . . . . . . . 14 (𝑠 = 𝑢 → ((𝑠 𝑡) 𝑊) = ((𝑢 𝑡) 𝑊))
54oveq2d 6706 . . . . . . . . . . . . 13 (𝑠 = 𝑢 → (𝐸 ((𝑠 𝑡) 𝑊)) = (𝐸 ((𝑢 𝑡) 𝑊)))
65oveq2d 6706 . . . . . . . . . . . 12 (𝑠 = 𝑢 → ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊))) = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))
72, 6syl5eq 2697 . . . . . . . . . . 11 (𝑠 = 𝑢𝐺 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))
87eqeq2d 2661 . . . . . . . . . 10 (𝑠 = 𝑢 → (𝑦 = 𝐺𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))))
98imbi2d 329 . . . . . . . . 9 (𝑠 = 𝑢 → (((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺) ↔ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))))
109ralbidv 3015 . . . . . . . 8 (𝑠 = 𝑢 → (∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺) ↔ ∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))))
1110riotabidv 6653 . . . . . . 7 (𝑠 = 𝑢 → (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺)) = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))))
12 eqeq1 2655 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))) ↔ 𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))))
1312imbi2d 329 . . . . . . . . . 10 (𝑦 = 𝑧 → (((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))) ↔ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))))
1413ralbidv 3015 . . . . . . . . 9 (𝑦 = 𝑧 → (∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))) ↔ ∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))))
15 breq1 4688 . . . . . . . . . . . . 13 (𝑡 = 𝑣 → (𝑡 𝑊𝑣 𝑊))
1615notbid 307 . . . . . . . . . . . 12 (𝑡 = 𝑣 → (¬ 𝑡 𝑊 ↔ ¬ 𝑣 𝑊))
17 breq1 4688 . . . . . . . . . . . . 13 (𝑡 = 𝑣 → (𝑡 (𝑃 𝑄) ↔ 𝑣 (𝑃 𝑄)))
1817notbid 307 . . . . . . . . . . . 12 (𝑡 = 𝑣 → (¬ 𝑡 (𝑃 𝑄) ↔ ¬ 𝑣 (𝑃 𝑄)))
1916, 18anbi12d 747 . . . . . . . . . . 11 (𝑡 = 𝑣 → ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ↔ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄))))
20 oveq1 6697 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑣 → (𝑡 𝑈) = (𝑣 𝑈))
21 oveq2 6698 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑣 → (𝑃 𝑡) = (𝑃 𝑣))
2221oveq1d 6705 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑣 → ((𝑃 𝑡) 𝑊) = ((𝑃 𝑣) 𝑊))
2322oveq2d 6706 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑣 → (𝑄 ((𝑃 𝑡) 𝑊)) = (𝑄 ((𝑃 𝑣) 𝑊)))
2420, 23oveq12d 6708 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑣 → ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊))) = ((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊))))
25 cdleme40.e . . . . . . . . . . . . . . . 16 𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))
26 cdleme40r.t . . . . . . . . . . . . . . . 16 𝑇 = ((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊)))
2724, 25, 263eqtr4g 2710 . . . . . . . . . . . . . . 15 (𝑡 = 𝑣𝐸 = 𝑇)
28 oveq2 6698 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑣 → (𝑢 𝑡) = (𝑢 𝑣))
2928oveq1d 6705 . . . . . . . . . . . . . . 15 (𝑡 = 𝑣 → ((𝑢 𝑡) 𝑊) = ((𝑢 𝑣) 𝑊))
3027, 29oveq12d 6708 . . . . . . . . . . . . . 14 (𝑡 = 𝑣 → (𝐸 ((𝑢 𝑡) 𝑊)) = (𝑇 ((𝑢 𝑣) 𝑊)))
3130oveq2d 6706 . . . . . . . . . . . . 13 (𝑡 = 𝑣 → ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))) = ((𝑃 𝑄) (𝑇 ((𝑢 𝑣) 𝑊))))
32 cdleme40r.x . . . . . . . . . . . . 13 𝑋 = ((𝑃 𝑄) (𝑇 ((𝑢 𝑣) 𝑊)))
3331, 32syl6eqr 2703 . . . . . . . . . . . 12 (𝑡 = 𝑣 → ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))) = 𝑋)
3433eqeq2d 2661 . . . . . . . . . . 11 (𝑡 = 𝑣 → (𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))) ↔ 𝑧 = 𝑋))
3519, 34imbi12d 333 . . . . . . . . . 10 (𝑡 = 𝑣 → (((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))) ↔ ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋)))
3635cbvralv 3201 . . . . . . . . 9 (∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))) ↔ ∀𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋))
3714, 36syl6bb 276 . . . . . . . 8 (𝑦 = 𝑧 → (∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))) ↔ ∀𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋)))
3837cbvriotav 6662 . . . . . . 7 (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))) = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋))
3911, 38syl6eq 2701 . . . . . 6 (𝑠 = 𝑢 → (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺)) = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋)))
40 cdleme40.i . . . . . 6 𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))
41 cdleme40r.o . . . . . 6 𝑂 = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋))
4239, 40, 413eqtr4g 2710 . . . . 5 (𝑠 = 𝑢𝐼 = 𝑂)
43 oveq1 6697 . . . . . . 7 (𝑠 = 𝑢 → (𝑠 𝑈) = (𝑢 𝑈))
44 oveq2 6698 . . . . . . . . 9 (𝑠 = 𝑢 → (𝑃 𝑠) = (𝑃 𝑢))
4544oveq1d 6705 . . . . . . . 8 (𝑠 = 𝑢 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑢) 𝑊))
4645oveq2d 6706 . . . . . . 7 (𝑠 = 𝑢 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑢) 𝑊)))
4743, 46oveq12d 6708 . . . . . 6 (𝑠 = 𝑢 → ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊))) = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊))))
48 cdleme40.d . . . . . 6 𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))
49 cdleme40r.y . . . . . 6 𝑌 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))
5047, 48, 493eqtr4g 2710 . . . . 5 (𝑠 = 𝑢𝐷 = 𝑌)
511, 42, 50ifbieq12d 4146 . . . 4 (𝑠 = 𝑢 → if(𝑠 (𝑃 𝑄), 𝐼, 𝐷) = if(𝑢 (𝑃 𝑄), 𝑂, 𝑌))
52 cdleme40.n . . . 4 𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)
53 cdleme40r.v . . . 4 𝑉 = if(𝑢 (𝑃 𝑄), 𝑂, 𝑌)
5451, 52, 533eqtr4g 2710 . . 3 (𝑠 = 𝑢𝑁 = 𝑉)
5554cbvcsbv 3572 . 2 𝑅 / 𝑠𝑁 = 𝑅 / 𝑢𝑉
5655a1i 11 1 (𝑅𝐴𝑅 / 𝑠𝑁 = 𝑅 / 𝑢𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  csb 3566  ifcif 4119   class class class wbr 4685  cfv 5926  crio 6650  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  Atomscatm 34868  LHypclh 35588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-riota 6651  df-ov 6693
This theorem is referenced by:  cdleme40w  36075
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