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Theorem cdleme40v 40452
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 5151 . . . . 5 (𝑠 = 𝑢 → (𝑠 (𝑃 𝑄) ↔ 𝑢 (𝑃 𝑄)))
2 cdleme40.g . . . . . . . . . . . 12 𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))
3 oveq1 7438 . . . . . . . . . . . . . . 15 (𝑠 = 𝑢 → (𝑠 𝑡) = (𝑢 𝑡))
43oveq1d 7446 . . . . . . . . . . . . . 14 (𝑠 = 𝑢 → ((𝑠 𝑡) 𝑊) = ((𝑢 𝑡) 𝑊))
54oveq2d 7447 . . . . . . . . . . . . 13 (𝑠 = 𝑢 → (𝐸 ((𝑠 𝑡) 𝑊)) = (𝐸 ((𝑢 𝑡) 𝑊)))
65oveq2d 7447 . . . . . . . . . . . 12 (𝑠 = 𝑢 → ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊))) = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))
72, 6eqtrid 2787 . . . . . . . . . . 11 (𝑠 = 𝑢𝐺 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))
87eqeq2d 2746 . . . . . . . . . 10 (𝑠 = 𝑢 → (𝑦 = 𝐺𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))))
98imbi2d 340 . . . . . . . . 9 (𝑠 = 𝑢 → (((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺) ↔ ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))))
109ralbidv 3176 . . . . . . . 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 7390 . . . . . . 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 2739 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))) ↔ 𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))))
1312imbi2d 340 . . . . . . . . . 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 3176 . . . . . . . . 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 5151 . . . . . . . . . . . . 13 (𝑡 = 𝑣 → (𝑡 𝑊𝑣 𝑊))
1615notbid 318 . . . . . . . . . . . 12 (𝑡 = 𝑣 → (¬ 𝑡 𝑊 ↔ ¬ 𝑣 𝑊))
17 breq1 5151 . . . . . . . . . . . . 13 (𝑡 = 𝑣 → (𝑡 (𝑃 𝑄) ↔ 𝑣 (𝑃 𝑄)))
1817notbid 318 . . . . . . . . . . . 12 (𝑡 = 𝑣 → (¬ 𝑡 (𝑃 𝑄) ↔ ¬ 𝑣 (𝑃 𝑄)))
1916, 18anbi12d 632 . . . . . . . . . . 11 (𝑡 = 𝑣 → ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) ↔ (¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄))))
20 oveq1 7438 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑣 → (𝑡 𝑈) = (𝑣 𝑈))
21 oveq2 7439 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑣 → (𝑃 𝑡) = (𝑃 𝑣))
2221oveq1d 7446 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑣 → ((𝑃 𝑡) 𝑊) = ((𝑃 𝑣) 𝑊))
2322oveq2d 7447 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑣 → (𝑄 ((𝑃 𝑡) 𝑊)) = (𝑄 ((𝑃 𝑣) 𝑊)))
2420, 23oveq12d 7449 . . . . . . . . . . . . . . . 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 2800 . . . . . . . . . . . . . . 15 (𝑡 = 𝑣𝐸 = 𝑇)
28 oveq2 7439 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑣 → (𝑢 𝑡) = (𝑢 𝑣))
2928oveq1d 7446 . . . . . . . . . . . . . . 15 (𝑡 = 𝑣 → ((𝑢 𝑡) 𝑊) = ((𝑢 𝑣) 𝑊))
3027, 29oveq12d 7449 . . . . . . . . . . . . . 14 (𝑡 = 𝑣 → (𝐸 ((𝑢 𝑡) 𝑊)) = (𝑇 ((𝑢 𝑣) 𝑊)))
3130oveq2d 7447 . . . . . . . . . . . . 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, 32eqtr4di 2793 . . . . . . . . . . . 12 (𝑡 = 𝑣 → ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))) = 𝑋)
3433eqeq2d 2746 . . . . . . . . . . 11 (𝑡 = 𝑣 → (𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))) ↔ 𝑧 = 𝑋))
3519, 34imbi12d 344 . . . . . . . . . 10 (𝑡 = 𝑣 → (((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))) ↔ ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋)))
3635cbvralvw 3235 . . . . . . . . 9 (∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑧 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))) ↔ ∀𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋))
3714, 36bitrdi 287 . . . . . . . 8 (𝑦 = 𝑧 → (∀𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊)))) ↔ ∀𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋)))
3837cbvriotavw 7398 . . . . . . 7 (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))) = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋))
3911, 38eqtrdi 2791 . . . . . 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 2800 . . . . 5 (𝑠 = 𝑢𝐼 = 𝑂)
43 oveq1 7438 . . . . . . 7 (𝑠 = 𝑢 → (𝑠 𝑈) = (𝑢 𝑈))
44 oveq2 7439 . . . . . . . . 9 (𝑠 = 𝑢 → (𝑃 𝑠) = (𝑃 𝑢))
4544oveq1d 7446 . . . . . . . 8 (𝑠 = 𝑢 → ((𝑃 𝑠) 𝑊) = ((𝑃 𝑢) 𝑊))
4645oveq2d 7447 . . . . . . 7 (𝑠 = 𝑢 → (𝑄 ((𝑃 𝑠) 𝑊)) = (𝑄 ((𝑃 𝑢) 𝑊)))
4743, 46oveq12d 7449 . . . . . 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 2800 . . . . 5 (𝑠 = 𝑢𝐷 = 𝑌)
511, 42, 50ifbieq12d 4559 . . . 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 2800 . . 3 (𝑠 = 𝑢𝑁 = 𝑉)
5554cbvcsbv 3920 . 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 395   = wceq 1537  wcel 2106  wral 3059  csb 3908  ifcif 4531   class class class wbr 5148  cfv 6563  crio 7387  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Atomscatm 39245  LHypclh 39967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-riota 7388  df-ov 7434
This theorem is referenced by:  cdleme40w  40453
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