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Theorem cdleme40v 36074
 Description: Part of proof of Lemma E in [Crawley] p. 113. Change bound variables in ⦋𝑆 / 𝑢⦌𝑉 (but we use ⦋𝑅 / 𝑢⦌𝑉 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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