Theorem recut 28347

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 28211	. . . 4 ⊢ ℕs ∈ V
2	1	abrexex 7941	. . 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 7941	. . 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 27739	. . . . . . . 8 ⊢ 1s ∈ No
7	6	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
8		nnsno 28217	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
9		nnne0s 28229	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
10	7, 8, 9	divscld 28126	. . . . . 6 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
11		subscl 27966	. . . . . 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 2816	. . . . 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 3134	. . 3 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) → 𝑥 ∈ No ))
16	15	abssdv 4031	. 2 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No )
17		addscl 27888	. . . . . 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 2816	. . . . 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 3134	. . 3 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) → 𝑥 ∈ No ))
22	21	abssdv 4031	. 2 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No )
23		vex 3451	. . . . . . 7 ⊢ 𝑦 ∈ V
24		eqeq1 2733	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
25	24	rexbidv 3157	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
26	23, 25	elab 3646	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)))
27		vex 3451	. . . . . . 7 ⊢ 𝑧 ∈ V
28		eqeq1 2733	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
29	28	rexbidv 3157	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
30		oveq2 7395	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ( 1s /su 𝑛) = ( 1s /su 𝑚))
31	30	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐴 +s ( 1s /su 𝑛)) = (𝐴 +s ( 1s /su 𝑚)))
32	31	eqeq2d 2740	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
33	32	cbvrexvw 3216	. . . . . . . 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 3646	. . . . . 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 3209	. . . . 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 27965	. . . . . . . 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 27943	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) ∈ No )
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) ∈ No )
45		0sno 27738	. . . . . . . . . . 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 28217	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No )
49		nnne0s 28229	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 𝑚 ≠ 0s )
50	47, 48, 49	divscld 28126	. . . . . . . . . . 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 28243	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕs → 0s <s ( 1s /su 𝑛))
54	45	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → 0s ∈ No )
55	54, 10	sltnegd 27953	. . . . . . . . . . . . 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 27932	. . . . . . . . . . . 12 ⊢ ( -us ‘ 0s ) = 0s
58	56, 57	breqtrdi 5148	. . . . . . . . . . 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 28243	. . . . . . . . . . 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 27671	. . . . . . . . 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 27904	. . . . . . . 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 5129	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (𝐴 -s ( 1s /su 𝑛)) <s (𝐴 +s ( 1s /su 𝑚)))
71		breq12 5112	. . . . . 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 3194	. . . 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 1116	. 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 27703	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 2109  {cab 2707  ∃wrex 3053  Vcvv 3447   class class class wbr 5107  ‘cfv 6511  (class class class)co 7387   No csur 27551   <s cslt 27552   <<s csslt 27692   0_s c0s 27734   1_s c1s 27735   +_s cadds 27866   -u_s cnegs 27925   -_s csubs 27926   /_su cdivs 28090  ℕ_scnns 28207

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-dc 10399

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-nadd 8630  df-no 27554  df-slt 27555  df-bday 27556  df-sle 27657  df-sslt 27693  df-scut 27695  df-0s 27736  df-1s 27737  df-made 27755  df-old 27756  df-left 27758  df-right 27759  df-norec 27845  df-norec2 27856  df-adds 27867  df-negs 27927  df-subs 27928  df-muls 28010  df-divs 28091  df-n0s 28208  df-nns 28209

This theorem is referenced by:  renegscl  28349  readdscl  28350  remulscl  28353