Theorem recut 28428

Description: The cut involved in defining surreal reals is a genuine cut. (Contributed by Scott Fenton, 15-Apr-2025.)

Assertion

Ref	Expression
recut	⊢ (𝐴 ∈ No → {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 recut

Dummy variables 𝑦 𝑧 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnsex 28323	. . . 4 ⊢ ℕs ∈ V
2	1	abrexex 7987	. . 3 ⊢ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ∈ V
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ∈ V)
4	1	abrexex 7987	. . 3 ⊢ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ∈ V
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ∈ V)
6		1sno 27872	. . . . . . . 8 ⊢ 1s ∈ No
7	6	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
8		nnsno 28329	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
9		nnne0s 28340	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
10	7, 8, 9	divscld 28248	. . . . . 6 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
11		subscl 28092	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
12	10, 11	sylan2 593	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
13		eleq1 2829	. . . . 5 ⊢ (𝑥 = (𝐴 -s ( 1s /su 𝑛)) → (𝑥 ∈ No ↔ (𝐴 -s ( 1s /su 𝑛)) ∈ No ))
14	12, 13	syl5ibrcom 247	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) → 𝑥 ∈ No ))
15	14	rexlimdva 3155	. . 3 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) → 𝑥 ∈ No ))
16	15	abssdv 4068	. 2 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No )
17		addscl 28014	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → (𝐴 +s ( 1s /su 𝑛)) ∈ No )
18	10, 17	sylan2 593	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 +s ( 1s /su 𝑛)) ∈ No )
19		eleq1 2829	. . . . 5 ⊢ (𝑥 = (𝐴 +s ( 1s /su 𝑛)) → (𝑥 ∈ No ↔ (𝐴 +s ( 1s /su 𝑛)) ∈ No ))
20	18, 19	syl5ibrcom 247	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) → 𝑥 ∈ No ))
21	20	rexlimdva 3155	. . 3 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) → 𝑥 ∈ No ))
22	21	abssdv 4068	. 2 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No )
23		vex 3484	. . . . . . 7 ⊢ 𝑦 ∈ V
24		eqeq1 2741	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
25	24	rexbidv 3179	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
26	23, 25	elab 3679	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)))
27		vex 3484	. . . . . . 7 ⊢ 𝑧 ∈ V
28		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
29	28	rexbidv 3179	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
30		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ( 1s /su 𝑛) = ( 1s /su 𝑚))
31	30	oveq2d 7447	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐴 +s ( 1s /su 𝑛)) = (𝐴 +s ( 1s /su 𝑚)))
32	31	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
33	32	cbvrexvw 3238	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚)))
34	29, 33	bitrdi 287	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
35	27, 34	elab 3679	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ↔ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚)))
36	26, 35	anbi12i 628	. . . . 5 ⊢ ((𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ↔ (∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
37		reeanv 3229	. . . . 5 ⊢ (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs (𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝐴 +s ( 1s /su 𝑚))) ↔ (∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
38	36, 37	bitr4i 278	. . . 4 ⊢ ((𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ↔ ∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs (𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
39		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
40	10	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
41	39, 40	subsvald 28091	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
42	41	adantrr 717	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
43	10	negscld 28069	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) ∈ No )
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) ∈ No )
45		0sno 27871	. . . . . . . . . . 11 ⊢ 0s ∈ No
46	45	a1i 11	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → 0s ∈ No )
47	6	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 1s ∈ No )
48		nnsno 28329	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No )
49		nnne0s 28340	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 𝑚 ≠ 0s )
50	47, 48, 49	divscld 28248	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕs → ( 1s /su 𝑚) ∈ No )
51	50	adantl 481	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( 1s /su 𝑚) ∈ No )
52		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ ℕs)
53	52	nnsrecgt0d 28356	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕs → 0s <s ( 1s /su 𝑛))
54	45	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → 0s ∈ No )
55	54, 10	sltnegd 28079	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕs → ( 0s <s ( 1s /su 𝑛) ↔ ( -us ‘( 1s /su 𝑛)) <s ( -us ‘ 0s )))
56	53, 55	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) <s ( -us ‘ 0s ))
57		negs0s 28058	. . . . . . . . . . . 12 ⊢ ( -us ‘ 0s ) = 0s
58	56, 57	breqtrdi 5184	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) <s 0s )
59	58	adantr 480	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) <s 0s )
60		id 22	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ ℕs)
61	60	nnsrecgt0d 28356	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕs → 0s <s ( 1s /su 𝑚))
62	61	adantl 481	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → 0s <s ( 1s /su 𝑚))
63	44, 46, 51, 59, 62	slttrd 27804	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) <s ( 1s /su 𝑚))
64	63	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → ( -us ‘( 1s /su 𝑛)) <s ( 1s /su 𝑚))
65	44	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → ( -us ‘( 1s /su 𝑛)) ∈ No )
66	50	ad2antll 729	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → ( 1s /su 𝑚) ∈ No )
67		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → 𝐴 ∈ No )
68	65, 66, 67	sltadd2d 28030	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (( -us ‘( 1s /su 𝑛)) <s ( 1s /su 𝑚) ↔ (𝐴 +s ( -us ‘( 1s /su 𝑛))) <s (𝐴 +s ( 1s /su 𝑚))))
69	64, 68	mpbid 232	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (𝐴 +s ( -us ‘( 1s /su 𝑛))) <s (𝐴 +s ( 1s /su 𝑚)))
70	42, 69	eqbrtrd 5165	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (𝐴 -s ( 1s /su 𝑛)) <s (𝐴 +s ( 1s /su 𝑚)))
71		breq12 5148	. . . . . 6 ⊢ ((𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝐴 +s ( 1s /su 𝑚))) → (𝑦 <s 𝑧 ↔ (𝐴 -s ( 1s /su 𝑛)) <s (𝐴 +s ( 1s /su 𝑚))))
72	70, 71	syl5ibrcom 247	. . . . 5 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → ((𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝐴 +s ( 1s /su 𝑚))) → 𝑦 <s 𝑧))
73	72	rexlimdvva 3213	. . . 4 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs ∃𝑚 ∈ ℕs (𝑦 = (𝐴 -s ( 1s /su 𝑛)) ∧ 𝑧 = (𝐴 +s ( 1s /su 𝑚))) → 𝑦 <s 𝑧))
74	38, 73	biimtrid 242	. . 3 ⊢ (𝐴 ∈ No → ((𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) → 𝑦 <s 𝑧))
75	74	3impib 1117	. 2 ⊢ ((𝐴 ∈ No ∧ 𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) → 𝑦 <s 𝑧)
76	3, 5, 16, 22, 75	ssltd 27836	1 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  ∃wrex 3070  Vcvv 3480   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431   No csur 27684   <s cslt 27685   <<s csslt 27825   0_s c0s 27867   1_s c1s 27868   +_s cadds 27992   -u_s cnegs 28051   -_s csubs 28052   /_su cdivs 28213  ℕ_scnns 28319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-dc 10486

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-muls 28133  df-divs 28214  df-n0s 28320  df-nns 28321

This theorem is referenced by:  renegscl  28430  readdscl  28431  remulscl  28434