Theorem recut 28698

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 28522	. . . 4 ⊢ ℕs ∈ V
2	1	abrexex 7957	. . 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 7957	. . 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		1no 28014	. . . . . . . 8 ⊢ 1s ∈ No
7	6	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
8		nnno 28528	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
9		nnne0s 28541	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
10	7, 8, 9	divscld 28428	. . . . . 6 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
11		subscl 28266	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
12	10, 11	sylan2 604	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
13		eleq1 2850	. . . . 5 ⊢ (𝑥 = (𝐴 -s ( 1s /su 𝑛)) → (𝑥 ∈ No ↔ (𝐴 -s ( 1s /su 𝑛)) ∈ No ))
14	12, 13	syl5ibrcom 250	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) → 𝑥 ∈ No ))
15	14	rexlimdva 3165	. . 3 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) → 𝑥 ∈ No ))
16	15	abssdv 4020	. 2 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No )
17		addscl 28185	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → (𝐴 +s ( 1s /su 𝑛)) ∈ No )
18	10, 17	sylan2 604	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 +s ( 1s /su 𝑛)) ∈ No )
19		eleq1 2850	. . . . 5 ⊢ (𝑥 = (𝐴 +s ( 1s /su 𝑛)) → (𝑥 ∈ No ↔ (𝐴 +s ( 1s /su 𝑛)) ∈ No ))
20	18, 19	syl5ibrcom 250	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) → 𝑥 ∈ No ))
21	20	rexlimdva 3165	. . 3 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) → 𝑥 ∈ No ))
22	21	abssdv 4020	. 2 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No )
23		vex 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
24		eqeq1 2766	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
25	24	rexbidv 3188	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
26	23, 25	elab 3637	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)))
27		vex 3458	. . . . . . 7 ⊢ 𝑧 ∈ V
28		eqeq1 2766	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
29	28	rexbidv 3188	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
30		oveq2 7420	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ( 1s /su 𝑛) = ( 1s /su 𝑚))
31	30	oveq2d 7428	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐴 +s ( 1s /su 𝑛)) = (𝐴 +s ( 1s /su 𝑚)))
32	31	eqeq2d 2773	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
33	32	cbvrexvw 3243	. . . . . . . 8 ⊢ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚)))
34	29, 33	bitrdi 290	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
35	27, 34	elab 3637	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ↔ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚)))
36	26, 35	anbi12i 639	. . . . 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 3236	. . . . 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 281	. . . 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 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No )
40	10	adantl 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
41	39, 40	subsvald 28265	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
42	41	adantrr 729	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
43	10	negscld 28241	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) ∈ No )
44	43	adantr 485	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) ∈ No )
45		0no 28013	. . . . . . . . . . 11 ⊢ 0s ∈ No
46	45	a1i 11	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → 0s ∈ No )
47	6	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 1s ∈ No )
48		nnno 28528	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No )
49		nnne0s 28541	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 𝑚 ≠ 0s )
50	47, 48, 49	divscld 28428	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕs → ( 1s /su 𝑚) ∈ No )
51	50	adantl 486	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( 1s /su 𝑚) ∈ No )
52		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ ℕs)
53	52	nnsrecgt0d 28555	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕs → 0s <s ( 1s /su 𝑛))
54	45	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → 0s ∈ No )
55	54, 10	ltnegsd 28251	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕs → ( 0s <s ( 1s /su 𝑛) ↔ ( -us ‘( 1s /su 𝑛)) <s ( -us ‘ 0s )))
56	53, 55	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) <s ( -us ‘ 0s ))
57		neg0s 28230	. . . . . . . . . . . 12 ⊢ ( -us ‘ 0s ) = 0s
58	56, 57	breqtrdi 5151	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) <s 0s )
59	58	adantr 485	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) <s 0s )
60		id 23	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ ℕs)
61	60	nnsrecgt0d 28555	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕs → 0s <s ( 1s /su 𝑚))
62	61	adantl 486	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → 0s <s ( 1s /su 𝑚))
63	44, 46, 51, 59, 62	ltstrd 27938	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) <s ( 1s /su 𝑚))
64	63	adantl 486	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → ( -us ‘( 1s /su 𝑛)) <s ( 1s /su 𝑚))
65	44	adantl 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → ( -us ‘( 1s /su 𝑛)) ∈ No )
66	50	ad2antll 741	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → ( 1s /su 𝑚) ∈ No )
67		simpl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → 𝐴 ∈ No )
68	65, 66, 67	ltadds2d 28201	. . . . . . . 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 235	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (𝐴 +s ( -us ‘( 1s /su 𝑛))) <s (𝐴 +s ( 1s /su 𝑚)))
70	42, 69	eqbrtrd 5132	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (𝐴 -s ( 1s /su 𝑛)) <s (𝐴 +s ( 1s /su 𝑚)))
71		breq12 5113	. . . . . 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 250	. . . . 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 3221	. . . 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 245	. . 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 1133	. 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	sltsd 27972	1 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {cab 2740  ∃wrex 3088  Vcvv 3454   class class class wbr 5108  ‘cfv 6536  (class class class)co 7412   No csur 27815   <s clts 27816   <<s cslts 27961   0_s c0s 28009   1_s c1s 28010   +_s cadds 28163   -u_s cnegs 28223   -_s csubs 28224   /_su cdivs 28391  ℕ_scnns 28517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-inf2 9608  ax-dc 10436

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-oadd 8455  df-nadd 8650  df-no 27818  df-lts 27819  df-bday 27820  df-les 27920  df-slts 27962  df-cuts 27964  df-0s 28011  df-1s 28012  df-made 28031  df-old 28032  df-left 28034  df-right 28035  df-norec 28142  df-norec2 28153  df-adds 28164  df-negs 28225  df-subs 28226  df-muls 28311  df-divs 28392  df-n0s 28518  df-nns 28519

This theorem is used by:  elreno2  28699  renegscl  28702  readdscl  28703  remulscl  28706