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Theorem cdleme40v 40471
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:   ,𝑠,𝑡,𝑦   𝑧,𝑈   𝑧,𝑅   𝑇,𝑠,𝑡,𝑦   𝑅,𝑠,𝑡,𝑣,𝑦   𝑄,𝑠,𝑡,𝑦   𝑧,𝐾   𝑢,𝑃,𝑧   𝑢,𝑄,𝑣,𝑧   𝑧,𝐻   𝑃,𝑠,𝑡,𝑣,𝑦   𝐸,𝑠   𝑢,𝑊,𝑧,𝑠,𝑡,𝑣,𝑦   𝐵,𝑠,𝑡,𝑦,𝑢,𝑣,𝑧   𝑦,𝑌   𝑢, ,𝑧,𝑠,𝑡,𝑣,𝑦   𝑢, ,𝑧,𝑠,𝑡,𝑣,𝑦   𝐴,𝑠,𝑡,𝑣,𝑦   𝑢, ,𝑣,𝑧   𝑡,𝑈,𝑣,𝑦   𝑡,𝐾,𝑣,𝑦   𝑡,𝐻,𝑣,𝑦   𝑢,𝐴,𝑧   𝑢,𝑇   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 5146 . . . . 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 2789 . . . . . . . . . . 11 (𝑠 = 𝑢𝐺 = ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))))
87eqeq2d 2748 . . . . . . . . . 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 3178 . . . . . . . 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 2741 . . . . . . . . . . 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 3178 . . . . . . . . 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 5146 . . . . . . . . . . . . 13 (𝑡 = 𝑣 → (𝑡 𝑊𝑣 𝑊))
1615notbid 318 . . . . . . . . . . . 12 (𝑡 = 𝑣 → (¬ 𝑡 𝑊 ↔ ¬ 𝑣 𝑊))
17 breq1 5146 . . . . . . . . . . . . 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 2802 . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . 12 (𝑡 = 𝑣 → ((𝑃 𝑄) (𝐸 ((𝑢 𝑡) 𝑊))) = 𝑋)
3433eqeq2d 2748 . . . . . . . . . . 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 3237 . . . . . . . . 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 2793 . . . . . 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 2802 . . . . 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 2802 . . . . 5 (𝑠 = 𝑢𝐷 = 𝑌)
511, 42, 50ifbieq12d 4554 . . . 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 2802 . . 3 (𝑠 = 𝑢𝑁 = 𝑉)
5554cbvcsbv 3911 . 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 1540  wcel 2108  wral 3061  csb 3899  ifcif 4525   class class class wbr 5143  cfv 6561  crio 7387  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  Atomscatm 39264  LHypclh 39986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-riota 7388  df-ov 7434
This theorem is referenced by:  cdleme40w  40472
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