Theorem cdleme31sn2 41053

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.)

Hypotheses

Ref	Expression
cdleme32sn2.d	⊢ 𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme31sn2.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
cdleme31sn2.c	⊢ 𝐶 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme31sn2	⊢ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)

Distinct variable groups:   𝐴,𝑠   ∨ ,𝑠   ≤ ,𝑠   ∧ ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠   𝑈,𝑠   𝑊,𝑠

Allowed substitution hints:   𝐶(𝑠)   𝐷(𝑠)   𝐼(𝑠)   𝑁(𝑠)

Proof of Theorem cdleme31sn2

Step	Hyp	Ref	Expression
1		cdleme31sn2.n	. . . . 5 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
2		eqid 2769	. . . . 5 ⊢ if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷)
3	1, 2	cdleme31sn 41044	. . . 4 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷))
4	3	adantr 485	. . 3 ⊢ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷))
5		iffalse 4501	. . . . 5 ⊢ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) → if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = ⦋𝑅 / 𝑠⦌𝐷)
6		cdleme32sn2.d	. . . . . 6 ⊢ 𝐷 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
7	6	csbeq2i 3869	. . . . 5 ⊢ ⦋𝑅 / 𝑠⦌𝐷 = ⦋𝑅 / 𝑠⦌((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
8	5, 7	eqtrdi 2820	. . . 4 ⊢ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) → if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = ⦋𝑅 / 𝑠⦌((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))))
9		nfcvd 2932	. . . . 5 ⊢ (𝑅 ∈ 𝐴 → Ⅎ𝑠((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
10		oveq1 7418	. . . . . 6 ⊢ (𝑠 = 𝑅 → (𝑠 ∨ 𝑈) = (𝑅 ∨ 𝑈))
11		oveq2 7419	. . . . . . . 8 ⊢ (𝑠 = 𝑅 → (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑅))
12	11	oveq1d 7426	. . . . . . 7 ⊢ (𝑠 = 𝑅 → ((𝑃 ∨ 𝑠) ∧ 𝑊) = ((𝑃 ∨ 𝑅) ∧ 𝑊))
13	12	oveq2d 7427	. . . . . 6 ⊢ (𝑠 = 𝑅 → (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
14	10, 13	oveq12d 7429	. . . . 5 ⊢ (𝑠 = 𝑅 → ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
15	9, 14	csbiegf 3894	. . . 4 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
16	8, 15	sylan9eqr 2826	. . 3 ⊢ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
17	4, 16	eqtrd 2804	. 2 ⊢ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊))))
18		cdleme31sn2.c	. 2 ⊢ 𝐶 = ((𝑅 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ 𝑊)))
19	17, 18	eqtr4di 2822	1 ⊢ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ⦋csb 3861  ifcif 4492   class class class wbr 5113  (class class class)co 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414

This theorem is referenced by:  cdlemefr32sn2aw  41068  cdleme43frv1snN  41072  cdlemefr31fv1  41075  cdleme35sn2aw  41122  cdleme35sn3a  41123