Theorem cdleme27b 41226

Description: Lemma for cdleme27N 41227. (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 5110	. . 3 ⊢ (𝑠 = 𝑡 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑡 ≤ (𝑃 ∨ 𝑄)))
2		oveq1 7423	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑡 → (𝑠 ∨ 𝑧) = (𝑡 ∨ 𝑧))
3	2	oveq1d 7431	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → ((𝑠 ∨ 𝑧) ∧ 𝑊) = ((𝑡 ∨ 𝑧) ∧ 𝑊))
4	3	oveq2d 7432	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → (𝑍 ∨ ((𝑠 ∨ 𝑧) ∧ 𝑊)) = (𝑍 ∨ ((𝑡 ∨ 𝑧) ∧ 𝑊)))
5	4	oveq2d 7432	. . . . . . . . 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 2822	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → 𝑁 = 𝑂)
9	8	eqeq2d 2773	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (𝑢 = 𝑁 ↔ 𝑢 = 𝑂))
10	9	imbi2d 343	. . . . . 6 ⊢ (𝑠 = 𝑡 → (((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁) ↔ ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)))
11	10	ralbidv 3187	. . . . 5 ⊢ (𝑠 = 𝑡 → (∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁) ↔ ∀𝑧 ∈ 𝐴 ((¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑂)))
12	11	riotabidv 7375	. . . 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 2822	. . 3 ⊢ (𝑠 = 𝑡 → 𝐷 = 𝐸)
16		oveq1 7423	. . . . 5 ⊢ (𝑠 = 𝑡 → (𝑠 ∨ 𝑈) = (𝑡 ∨ 𝑈))
17		oveq2 7424	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑡))
18	17	oveq1d 7431	. . . . . 6 ⊢ (𝑠 = 𝑡 → ((𝑃 ∨ 𝑠) ∧ 𝑊) = ((𝑃 ∨ 𝑡) ∧ 𝑊))
19	18	oveq2d 7432	. . . . 5 ⊢ (𝑠 = 𝑡 → (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
20	16, 19	oveq12d 7434	. . . 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 2822	. . 3 ⊢ (𝑠 = 𝑡 → 𝐹 = 𝐺)
24	1, 15, 23	ifbieq12d 4514	. 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 2822	1 ⊢ (𝑠 = 𝑡 → 𝐶 = 𝑌)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570  ∀wral 3078  ifcif 4485   class class class wbr 5107  ‘cfv 6537  ℩crio 7372  (class class class)co 7416  Basecbs 17303  lecple 17351  joincjn 18401  meetcmee 18402  Atomscatm 40121  LHypclh 40842

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-riota 7373  df-ov 7419

This theorem is used by:  cdleme27N  41227  cdleme28c  41230