Theorem cdleme31sn 37510

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

Hypotheses

Ref	Expression
cdleme31sn.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
cdleme31sn.c	⊢ 𝐶 = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷)

Assertion

Ref	Expression
cdleme31sn	⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)

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

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

Proof of Theorem cdleme31sn

Step	Hyp	Ref	Expression
1		nfv 1911	. . . . 5 ⊢ Ⅎ𝑠 𝑅 ≤ (𝑃 ∨ 𝑄)
2		nfcsb1v 3906	. . . . 5 ⊢ Ⅎ𝑠⦋𝑅 / 𝑠⦌𝐼
3		nfcsb1v 3906	. . . . 5 ⊢ Ⅎ𝑠⦋𝑅 / 𝑠⦌𝐷
4	1, 2, 3	nfif 4495	. . . 4 ⊢ Ⅎ𝑠if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷)
5	4	a1i 11	. . 3 ⊢ (𝑅 ∈ 𝐴 → Ⅎ𝑠if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷))
6		breq1 5061	. . . 4 ⊢ (𝑠 = 𝑅 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑅 ≤ (𝑃 ∨ 𝑄)))
7		csbeq1a 3896	. . . 4 ⊢ (𝑠 = 𝑅 → 𝐼 = ⦋𝑅 / 𝑠⦌𝐼)
8		csbeq1a 3896	. . . 4 ⊢ (𝑠 = 𝑅 → 𝐷 = ⦋𝑅 / 𝑠⦌𝐷)
9	6, 7, 8	ifbieq12d 4493	. . 3 ⊢ (𝑠 = 𝑅 → if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷))
10	5, 9	csbiegf 3915	. 2 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷))
11		cdleme31sn.n	. . 3 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
12	11	csbeq2i 3890	. 2 ⊢ ⦋𝑅 / 𝑠⦌𝑁 = ⦋𝑅 / 𝑠⦌if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
13		cdleme31sn.c	. 2 ⊢ 𝐶 = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷)
14	10, 12, 13	3eqtr4g 2881	1 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  Ⅎwnfc 2961  ⦋csb 3882  ifcif 4466   class class class wbr 5058  (class class class)co 7150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059

This theorem is referenced by:  cdleme31sn1  37511  cdleme31sn2  37519