Theorem cdleme27b 40875

Description: Lemma for cdleme27N 40876. (Contributed by NM, 3-Feb-2013.)

Hypotheses

Ref	Expression
cdleme26.b	⊢ 𝐵 = (Base‘𝐾)
cdleme26.l	⊢ ≤ = (le‘𝐾)
cdleme26.j	⊢ ∨ = (join‘𝐾)
cdleme26.m	⊢ ∧ = (meet‘𝐾)
cdleme26.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme26.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme27.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme27.f	⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme27.z	⊢ 𝑍 = ((𝑧 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑧) ∧ 𝑊)))
cdleme27.n	⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊)))
cdleme27.d	⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))
cdleme27.c	⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹)
cdleme27.g	⊢ 𝐺 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme27.o	⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊)))
cdleme27.e	⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))
cdleme27.y	⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺)

Assertion

Ref	Expression
cdleme27b	⊢ (𝑠 = 𝑡 → 𝐶 = 𝑌)

Distinct variable groups:   𝑡,𝑠,𝑢,𝑧,𝐴   𝐵,𝑠,𝑡,𝑢,𝑧   𝑢,𝐹   𝑢,𝐺   𝐻,𝑠,𝑡,𝑧   ∨ ,𝑠,𝑡,𝑢,𝑧   𝐾,𝑠,𝑡,𝑧   ≤ ,𝑠,𝑡,𝑢,𝑧   ∧ ,𝑠,𝑡,𝑢,𝑧   𝑡,𝑁,𝑢   𝑂,𝑠,𝑢   𝑃,𝑠,𝑡,𝑢,𝑧   𝑄,𝑠,𝑡,𝑢,𝑧   𝑈,𝑠,𝑡,𝑢,𝑧   𝑊,𝑠,𝑡,𝑢,𝑧

Allowed substitution hints:   𝐶(𝑧,𝑢,𝑡,𝑠)   𝐷(𝑧,𝑢,𝑡,𝑠)   𝐸(𝑧,𝑢,𝑡,𝑠)   𝐹(𝑧,𝑡,𝑠)   𝐺(𝑧,𝑡,𝑠)   𝐻(𝑢)   𝐾(𝑢)   𝑁(𝑧,𝑠)   𝑂(𝑧,𝑡)   𝑌(𝑧,𝑢,𝑡,𝑠)   𝑍(𝑧,𝑢,𝑡,𝑠)

Proof of Theorem cdleme27b

Step	Hyp	Ref	Expression
1		breq1 5078	. . 3 ⊢ (𝑠 = 𝑡 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑡 ≤ (𝑃 ∨ 𝑄)))
2		oveq1 7367	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (𝑠 ∨ 𝑧) = (𝑡 ∨ 𝑧))
3	2	oveq1d 7375	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → ((𝑠 ∨ 𝑧) ∧ 𝑊) = ((𝑡 ∨ 𝑧) ∧ 𝑊))
4	3	oveq2d 7376	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊)) = (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊)))
5	4	oveq2d 7376	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊))))
6		cdleme27.n	. . . . . . . . 9 ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊)))
7		cdleme27.o	. . . . . . . . 9 ⊢ 𝑂 = ((𝑃 ∨ 𝑄) ∧ (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊)))
8	5, 6, 7	3eqtr4g 2801	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → 𝑁 = 𝑂)
9	8	eqeq2d 2752	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (𝑢 = 𝑁 ↔ 𝑢 = 𝑂))
10	9	imbi2d 342	. . . . . 6 ⊢ (𝑠 = 𝑡 → (((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁) ↔ ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)))
11	10	ralbidv 3164	. . . . 5 ⊢ (𝑠 = 𝑡 → (∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁) ↔ ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)))
12	11	riotabidv 7319	. . . 4 ⊢ (𝑠 = 𝑡 → (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)) = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)))
13		cdleme27.d	. . . 4 ⊢ 𝐷 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))
14		cdleme27.e	. . . 4 ⊢ 𝐸 = (℩𝑢 ∈ 𝐵 ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂))
15	12, 13, 14	3eqtr4g 2801	. . 3 ⊢ (𝑠 = 𝑡 → 𝐷 = 𝐸)
16		oveq1 7367	. . . . 5 ⊢ (𝑠 = 𝑡 → (𝑠 ∨ 𝑈) = (𝑡 ∨ 𝑈))
17		oveq2 7368	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑡))
18	17	oveq1d 7375	. . . . . 6 ⊢ (𝑠 = 𝑡 → ((𝑃 ∨ 𝑠) ∧ 𝑊) = ((𝑃 ∨ 𝑡) ∧ 𝑊))
19	18	oveq2d 7376	. . . . 5 ⊢ (𝑠 = 𝑡 → (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
20	16, 19	oveq12d 7378	. . . 4 ⊢ (𝑠 = 𝑡 → ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))))
21		cdleme27.f	. . . 4 ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
22		cdleme27.g	. . . 4 ⊢ 𝐺 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
23	20, 21, 22	3eqtr4g 2801	. . 3 ⊢ (𝑠 = 𝑡 → 𝐹 = 𝐺)
24	1, 15, 23	ifbieq12d 4486	. 2 ⊢ (𝑠 = 𝑡 → if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹) = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺))
25		cdleme27.c	. 2 ⊢ 𝐶 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐷, 𝐹)
26		cdleme27.y	. 2 ⊢ 𝑌 = if(𝑡 ≤ (𝑃 ∨ 𝑄), 𝐸, 𝐺)
27	24, 25, 26	3eqtr4g 2801	1 ⊢ (𝑠 = 𝑡 → 𝐶 = 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1548  ∀wral 3055  ifcif 4457   class class class wbr 5075  ‘cfv 6489  ℩crio 7316  (class class class)co 7360  Basecbs 17174  lecple 17222  joincjn 18272  meetcmee 18273  Atomscatm 39770  LHypclh 40491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-iota 6445  df-fv 6497  df-riota 7317  df-ov 7363

This theorem is referenced by:  cdleme27N  40876  cdleme28c  40879