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Theorem cdleme27b 38860
Description: Lemma for cdleme27N 38861. (Contributed by NM, 3-Feb-2013.)
Hypotheses
Ref Expression
cdleme26.b 𝐡 = (Baseβ€˜πΎ)
cdleme26.l ≀ = (leβ€˜πΎ)
cdleme26.j ∨ = (joinβ€˜πΎ)
cdleme26.m ∧ = (meetβ€˜πΎ)
cdleme26.a 𝐴 = (Atomsβ€˜πΎ)
cdleme26.h 𝐻 = (LHypβ€˜πΎ)
cdleme27.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdleme27.f 𝐹 = ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)))
cdleme27.z 𝑍 = ((𝑧 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ π‘Š)))
cdleme27.n 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ π‘Š)))
cdleme27.d 𝐷 = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁))
cdleme27.c 𝐢 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐷, 𝐹)
cdleme27.g 𝐺 = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
cdleme27.o 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š)))
cdleme27.e 𝐸 = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑂))
cdleme27.y π‘Œ = if(𝑑 ≀ (𝑃 ∨ 𝑄), 𝐸, 𝐺)
Assertion
Ref Expression
cdleme27b (𝑠 = 𝑑 β†’ 𝐢 = π‘Œ)
Distinct variable groups:   𝑑,𝑠,𝑒,𝑧,𝐴   𝐡,𝑠,𝑑,𝑒,𝑧   𝑒,𝐹   𝑒,𝐺   𝐻,𝑠,𝑑,𝑧   ∨ ,𝑠,𝑑,𝑒,𝑧   𝐾,𝑠,𝑑,𝑧   ≀ ,𝑠,𝑑,𝑒,𝑧   ∧ ,𝑠,𝑑,𝑒,𝑧   𝑑,𝑁,𝑒   𝑂,𝑠,𝑒   𝑃,𝑠,𝑑,𝑒,𝑧   𝑄,𝑠,𝑑,𝑒,𝑧   π‘ˆ,𝑠,𝑑,𝑒,𝑧   π‘Š,𝑠,𝑑,𝑒,𝑧
Allowed substitution hints:   𝐢(𝑧,𝑒,𝑑,𝑠)   𝐷(𝑧,𝑒,𝑑,𝑠)   𝐸(𝑧,𝑒,𝑑,𝑠)   𝐹(𝑧,𝑑,𝑠)   𝐺(𝑧,𝑑,𝑠)   𝐻(𝑒)   𝐾(𝑒)   𝑁(𝑧,𝑠)   𝑂(𝑧,𝑑)   π‘Œ(𝑧,𝑒,𝑑,𝑠)   𝑍(𝑧,𝑒,𝑑,𝑠)

Proof of Theorem cdleme27b
StepHypRef Expression
1 breq1 5113 . . 3 (𝑠 = 𝑑 β†’ (𝑠 ≀ (𝑃 ∨ 𝑄) ↔ 𝑑 ≀ (𝑃 ∨ 𝑄)))
2 oveq1 7369 . . . . . . . . . . . 12 (𝑠 = 𝑑 β†’ (𝑠 ∨ 𝑧) = (𝑑 ∨ 𝑧))
32oveq1d 7377 . . . . . . . . . . 11 (𝑠 = 𝑑 β†’ ((𝑠 ∨ 𝑧) ∧ π‘Š) = ((𝑑 ∨ 𝑧) ∧ π‘Š))
43oveq2d 7378 . . . . . . . . . 10 (𝑠 = 𝑑 β†’ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ π‘Š)) = (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š)))
54oveq2d 7378 . . . . . . . . 9 (𝑠 = 𝑑 β†’ ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ π‘Š))) = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š))))
6 cdleme27.n . . . . . . . . 9 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ π‘Š)))
7 cdleme27.o . . . . . . . . 9 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑑 ∨ 𝑧) ∧ π‘Š)))
85, 6, 73eqtr4g 2802 . . . . . . . 8 (𝑠 = 𝑑 β†’ 𝑁 = 𝑂)
98eqeq2d 2748 . . . . . . 7 (𝑠 = 𝑑 β†’ (𝑒 = 𝑁 ↔ 𝑒 = 𝑂))
109imbi2d 341 . . . . . 6 (𝑠 = 𝑑 β†’ (((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁) ↔ ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑂)))
1110ralbidv 3175 . . . . 5 (𝑠 = 𝑑 β†’ (βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁) ↔ βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑂)))
1211riotabidv 7320 . . . 4 (𝑠 = 𝑑 β†’ (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁)) = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑂)))
13 cdleme27.d . . . 4 𝐷 = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑁))
14 cdleme27.e . . . 4 𝐸 = (℩𝑒 ∈ 𝐡 βˆ€π‘§ ∈ 𝐴 ((Β¬ 𝑧 ≀ π‘Š ∧ Β¬ 𝑧 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑒 = 𝑂))
1512, 13, 143eqtr4g 2802 . . 3 (𝑠 = 𝑑 β†’ 𝐷 = 𝐸)
16 oveq1 7369 . . . . 5 (𝑠 = 𝑑 β†’ (𝑠 ∨ π‘ˆ) = (𝑑 ∨ π‘ˆ))
17 oveq2 7370 . . . . . . 7 (𝑠 = 𝑑 β†’ (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑑))
1817oveq1d 7377 . . . . . 6 (𝑠 = 𝑑 β†’ ((𝑃 ∨ 𝑠) ∧ π‘Š) = ((𝑃 ∨ 𝑑) ∧ π‘Š))
1918oveq2d 7378 . . . . 5 (𝑠 = 𝑑 β†’ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)) = (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
2016, 19oveq12d 7380 . . . 4 (𝑠 = 𝑑 β†’ ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š))) = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š))))
21 cdleme27.f . . . 4 𝐹 = ((𝑠 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ π‘Š)))
22 cdleme27.g . . . 4 𝐺 = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
2320, 21, 223eqtr4g 2802 . . 3 (𝑠 = 𝑑 β†’ 𝐹 = 𝐺)
241, 15, 23ifbieq12d 4519 . 2 (𝑠 = 𝑑 β†’ if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐷, 𝐹) = if(𝑑 ≀ (𝑃 ∨ 𝑄), 𝐸, 𝐺))
25 cdleme27.c . 2 𝐢 = if(𝑠 ≀ (𝑃 ∨ 𝑄), 𝐷, 𝐹)
26 cdleme27.y . 2 π‘Œ = if(𝑑 ≀ (𝑃 ∨ 𝑄), 𝐸, 𝐺)
2724, 25, 263eqtr4g 2802 1 (𝑠 = 𝑑 β†’ 𝐢 = π‘Œ)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542  βˆ€wral 3065  ifcif 4491   class class class wbr 5110  β€˜cfv 6501  β„©crio 7317  (class class class)co 7362  Basecbs 17090  lecple 17147  joincjn 18207  meetcmee 18208  Atomscatm 37754  LHypclh 38476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-riota 7318  df-ov 7365
This theorem is referenced by:  cdleme27N  38861  cdleme28c  38864
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