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Theorem cdleme40v 34574
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 4576 . . . . 5 (𝑠 = 𝑢 → (𝑠 (𝑃 𝑄) ↔ 𝑢 (𝑃 𝑄)))
2 cdleme40.g . . . . . . . . . . . 12 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
3 oveq1 6530 . . . . . . . . . . . . . . 15 (𝑠 = 𝑢 → (𝑠 𝑡) = (𝑢 𝑡))
43oveq1d 6538 . . . . . . . . . . . . . 14 (𝑠 = 𝑢 → ((𝑠 𝑡) 𝑊) = ((𝑢 𝑡) 𝑊))
54oveq2d 6539 . . . . . . . . . . . . 13 (𝑠 = 𝑢 → (𝐸 ((𝑠 𝑡) 𝑊)) = (𝐸 ((𝑢 𝑡) 𝑊)))
65oveq2d 6539 . . . . . . . . . . . 12 (𝑠 = 𝑢 → ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊))) = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))
72, 6syl5eq 2651 . . . . . . . . . . 11 (𝑠 = 𝑢𝐺 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))
87eqeq2d 2615 . . . . . . . . . 10 (𝑠 = 𝑢 → (𝑦 = 𝐺𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))))
98imbi2d 328 . . . . . . . . 9 (𝑠 = 𝑢 → (((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺) ↔ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))))
109ralbidv 2964 . . . . . . . 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 6487 . . . . . . 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 2609 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))) ↔ 𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))))
1312imbi2d 328 . . . . . . . . . 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 2964 . . . . . . . . 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 4576 . . . . . . . . . . . . 13 (𝑡 = 𝑣 → (𝑡 𝑊𝑣 𝑊))
1615notbid 306 . . . . . . . . . . . 12 (𝑡 = 𝑣 → (¬ 𝑡 𝑊 ↔ ¬ 𝑣 𝑊))
17 breq1 4576 . . . . . . . . . . . . 13 (𝑡 = 𝑣 → (𝑡 (𝑃 𝑄) ↔ 𝑣 (𝑃 𝑄)))
1817notbid 306 . . . . . . . . . . . 12 (𝑡 = 𝑣 → (¬ 𝑡 (𝑃 𝑄) ↔ ¬ 𝑣 (𝑃 𝑄)))
1916, 18anbi12d 742 . . . . . . . . . . 11 (𝑡 = 𝑣 → ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ↔ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄))))
20 oveq1 6530 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑣 → (𝑡 𝑈) = (𝑣 𝑈))
21 oveq2 6531 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑣 → (𝑃 𝑡) = (𝑃 𝑣))
2221oveq1d 6538 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑣 → ((𝑃 𝑡) 𝑊) = ((𝑃 𝑣) 𝑊))
2322oveq2d 6539 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑣 → (𝑄 ((𝑃 𝑡) 𝑊)) = (𝑄 ((𝑃 𝑣) 𝑊)))
2420, 23oveq12d 6541 . . . . . . . . . . . . . . . 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 2664 . . . . . . . . . . . . . . 15 (𝑡 = 𝑣𝐸 = 𝑇)
28 oveq2 6531 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑣 → (𝑢 𝑡) = (𝑢 𝑣))
2928oveq1d 6538 . . . . . . . . . . . . . . 15 (𝑡 = 𝑣 → ((𝑢 𝑡) 𝑊) = ((𝑢 𝑣) 𝑊))
3027, 29oveq12d 6541 . . . . . . . . . . . . . 14 (𝑡 = 𝑣 → (𝐸 ((𝑢 𝑡) 𝑊)) = (𝑇 ((𝑢 𝑣) 𝑊)))
3130oveq2d 6539 . . . . . . . . . . . . 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 2657 . . . . . . . . . . . 12 (𝑡 = 𝑣 → ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))) = 𝑋)
3433eqeq2d 2615 . . . . . . . . . . 11 (𝑡 = 𝑣 → (𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))) ↔ 𝑧 = 𝑋))
3519, 34imbi12d 332 . . . . . . . . . 10 (𝑡 = 𝑣 → (((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))) ↔ ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋)))
3635cbvralv 3142 . . . . . . . . 9 (∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))) ↔ ∀𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋))
3714, 36syl6bb 274 . . . . . . . 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 6496 . . . . . . 7 (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))) = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋))
3911, 38syl6eq 2655 . . . . . 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 2664 . . . . 5 (𝑠 = 𝑢𝐼 = 𝑂)
43 oveq1 6530 . . . . . . 7 (𝑠 = 𝑢 → (𝑠 𝑈) = (𝑢 𝑈))
44 oveq2 6531 . . . . . . . . 9 (𝑠 = 𝑢 → (𝑃 𝑠) = (𝑃 𝑢))
4544oveq1d 6538 . . . . . . . 8 (𝑠 = 𝑢 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑢) 𝑊))
4645oveq2d 6539 . . . . . . 7 (𝑠 = 𝑢 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑢) 𝑊)))
4743, 46oveq12d 6541 . . . . . 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 2664 . . . . 5 (𝑠 = 𝑢𝐷 = 𝑌)
511, 42, 50ifbieq12d 4058 . . . 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 2664 . . 3 (𝑠 = 𝑢𝑁 = 𝑉)
5554cbvcsbv 3500 . 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 382   = wceq 1474  wcel 1975  wral 2891  csb 3494  ifcif 4031   class class class wbr 4573  cfv 5786  crio 6484  (class class class)co 6523  Basecbs 15637  lecple 15717  joincjn 16709  meetcmee 16710  Atomscatm 33367  LHypclh 34087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-iota 5750  df-fv 5794  df-riota 6485  df-ov 6526
This theorem is referenced by:  cdleme40w  34575
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