Theorem cdleme31sn1 39791

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

Hypotheses

Ref	Expression
cdleme31sn1.i	⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
cdleme31sn1.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
cdleme31sn1.c	⊢ 𝐶 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺))

Assertion

Ref	Expression
cdleme31sn1	⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)

Distinct variable groups:   𝑡,𝑠,𝑦,𝐴   𝐵,𝑠   ∨ ,𝑠   ≤ ,𝑠   𝑃,𝑠   𝑄,𝑠   𝑅,𝑠,𝑡,𝑦   𝑊,𝑠

Allowed substitution hints:   𝐵(𝑦,𝑡)   𝐶(𝑦,𝑡,𝑠)   𝐷(𝑦,𝑡,𝑠)   𝑃(𝑦,𝑡)   𝑄(𝑦,𝑡)   𝐺(𝑦,𝑡,𝑠)   𝐼(𝑦,𝑡,𝑠)   ∨ (𝑦,𝑡)   ≤ (𝑦,𝑡)   𝑁(𝑦,𝑡,𝑠)   𝑊(𝑦,𝑡)

Proof of Theorem cdleme31sn1

Step	Hyp	Ref	Expression
1		cdleme31sn1.n	. . . 4 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷)
2		eqid 2727	. . . 4 ⊢ if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷)
3	1, 2	cdleme31sn 39790	. . 3 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷))
4	3	adantr 480	. 2 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷))
5		iftrue 4530	. . . . 5 ⊢ (𝑅 ≤ (𝑃 ∨ 𝑄) → if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = ⦋𝑅 / 𝑠⦌𝐼)
6		cdleme31sn1.i	. . . . . 6 ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
7	6	csbeq2i 3897	. . . . 5 ⊢ ⦋𝑅 / 𝑠⦌𝐼 = ⦋𝑅 / 𝑠⦌(℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺))
8	5, 7	eqtrdi 2783	. . . 4 ⊢ (𝑅 ≤ (𝑃 ∨ 𝑄) → if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = ⦋𝑅 / 𝑠⦌(℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺)))
9		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑠𝐴
10		nfv 1910	. . . . . . . . 9 ⊢ Ⅎ𝑠(¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))
11		nfcsb1v 3914	. . . . . . . . . 10 ⊢ Ⅎ𝑠⦋𝑅 / 𝑠⦌𝐺
12	11	nfeq2 2915	. . . . . . . . 9 ⊢ Ⅎ𝑠 𝑦 = ⦋𝑅 / 𝑠⦌𝐺
13	10, 12	nfim 1892	. . . . . . . 8 ⊢ Ⅎ𝑠((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)
14	9, 13	nfralw 3303	. . . . . . 7 ⊢ Ⅎ𝑠∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)
15		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑠𝐵
16	14, 15	nfriota 7383	. . . . . 6 ⊢ Ⅎ𝑠(℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺))
17	16	a1i 11	. . . . 5 ⊢ (𝑅 ∈ 𝐴 → Ⅎ𝑠(℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)))
18		csbeq1a 3903	. . . . . . . . 9 ⊢ (𝑠 = 𝑅 → 𝐺 = ⦋𝑅 / 𝑠⦌𝐺)
19	18	eqeq2d 2738	. . . . . . . 8 ⊢ (𝑠 = 𝑅 → (𝑦 = 𝐺 ↔ 𝑦 = ⦋𝑅 / 𝑠⦌𝐺))
20	19	imbi2d 340	. . . . . . 7 ⊢ (𝑠 = 𝑅 → (((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺) ↔ ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)))
21	20	ralbidv 3172	. . . . . 6 ⊢ (𝑠 = 𝑅 → (∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺) ↔ ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)))
22	21	riotabidv 7372	. . . . 5 ⊢ (𝑠 = 𝑅 → (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺)) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)))
23	17, 22	csbiegf 3923	. . . 4 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌(℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺)) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)))
24	8, 23	sylan9eqr 2789	. . 3 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)))
25		cdleme31sn1.c	. . 3 ⊢ 𝐶 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺))
26	24, 25	eqtr4di 2785	. 2 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) = 𝐶)
27	4, 26	eqtrd 2767	1 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Ⅎwnfc 2878  ∀wral 3056  ⦋csb 3889  ifcif 4524   class class class wbr 5142  ℩crio 7369  (class class class)co 7414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-riota 7370

This theorem is referenced by:  cdleme31sn1c  39798