Theorem elreno 28586

Description: Membership in the set of surreal reals. (Contributed by Scott Fenton, 15-Apr-2025.)

Assertion

Ref	Expression
elreno	⊢ (𝐴 ∈ ℝs ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))))

Distinct variable group:   𝑥,𝐴,𝑛

Proof of Theorem elreno

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5106	. . . . 5 ⊢ (𝑦 = 𝐴 → (( -us ‘𝑛) <s 𝑦 ↔ ( -us ‘𝑛) <s 𝐴))
2		breq1 5105	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 <s 𝑛 ↔ 𝐴 <s 𝑛))
3	1, 2	anbi12d 641	. . . 4 ⊢ (𝑦 = 𝐴 → ((( -us ‘𝑛) <s 𝑦 ∧ 𝑦 <s 𝑛) ↔ (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛)))
4	3	rexbidv 3188	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝑦 ∧ 𝑦 <s 𝑛) ↔ ∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛)))
5		id 22	. . . 4 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
6		oveq1 7405	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 -s ( 1s /su 𝑛)) = (𝐴 -s ( 1s /su 𝑛)))
7	6	eqeq2d 2775	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝑦 -s ( 1s /su 𝑛)) ↔ 𝑥 = (𝐴 -s ( 1s /su 𝑛))))
8	7	rexbidv 3188	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))))
9	8	abbidv 2830	. . . . 5 ⊢ (𝑦 = 𝐴 → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))})
10		oveq1 7405	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑦 +s ( 1s /su 𝑛)) = (𝐴 +s ( 1s /su 𝑛)))
11	10	eqeq2d 2775	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝑦 +s ( 1s /su 𝑛)) ↔ 𝑥 = (𝐴 +s ( 1s /su 𝑛))))
12	11	rexbidv 3188	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))))
13	12	abbidv 2830	. . . . 5 ⊢ (𝑦 = 𝐴 → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
14	9, 13	oveq12d 7416	. . . 4 ⊢ (𝑦 = 𝐴 → ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛))}) = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))
15	5, 14	eqeq12d 2780	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛))}) ↔ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})))
16	4, 15	anbi12d 641	. 2 ⊢ (𝑦 = 𝐴 → ((∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝑦 ∧ 𝑦 <s 𝑛) ∧ 𝑦 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛))})) ↔ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))))
17		df-reno 28585	. 2 ⊢ ℝs = {𝑦 ∈ No ∣ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝑦 ∧ 𝑦 <s 𝑛) ∧ 𝑦 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝑦 +s ( 1s /su 𝑛))}))}
18	16, 17	elrab2 3656	1 ⊢ (𝐴 ∈ ℝs ↔ (𝐴 ∈ No ∧ (∃𝑛 ∈ ℕs (( -us ‘𝑛) <s 𝐴 ∧ 𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  {cab 2742  ∃wrex 3088   class class class wbr 5102  ‘cfv 6523  (class class class)co 7398   No csur 27706   <s clts 27707   |s ccuts 27854   1_s c1s 27901   +_s cadds 28054   -u_s cnegs 28114   -_s csubs 28115   /_su cdivs 28282  ℕ_scnns 28408  ℝ_screno 28584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531  df-ov 7401  df-reno 28585

This theorem is referenced by:  reno  28587  elreno2  28590  renegscl  28593  readdscl  28594  remulscl  28597