Theorem recut 28881

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 28705	. . . 4 ⊢ ℕs ∈ V
2	1	abrexex 7974	. . 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 7974	. . 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 28197	. . . . . . . 8 ⊢ 1s ∈ No
7	6	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 1s ∈ No)
8		nnno 28711	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No)
9		nnne0s 28724	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
10	7, 8, 9	divscld 28611	. . . . . 6 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No)
11		subscl 28449	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No) → (𝐴 −s ( 1s /su 𝑛)) ∈ No)
12	10, 11	sylan2 605	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 −s ( 1s /su 𝑛)) ∈ No)
13		eleq1 2848	. . . . 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 3163	. . 3 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛)) → 𝑥 ∈ No))
16	15	abssdv 3992	. 2 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))} ⊆ No)
17		addscl 28368	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No) → (𝐴 +s ( 1s /su 𝑛)) ∈ No)
18	10, 17	sylan2 605	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 +s ( 1s /su 𝑛)) ∈ No)
19		eleq1 2848	. . . . 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 3163	. . 3 ⊢ (𝐴 ∈ No → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) → 𝑥 ∈ No))
22	21	abssdv 3992	. 2 ⊢ (𝐴 ∈ No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No)
23		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
24		eqeq1 2764	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = (𝐴 −s ( 1s /su 𝑛)) ↔ 𝑦 = (𝐴 −s ( 1s /su 𝑛))))
25	24	rexbidv 3186	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 −s ( 1s /su 𝑛))))
26	23, 25	elab 3632	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 −s ( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 −s ( 1s /su 𝑛)))
27		vex 3454	. . . . . . 7 ⊢ 𝑧 ∈ V
28		eqeq1 2764	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
29	28	rexbidv 3186	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
30		oveq2 7428	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ( 1s /su 𝑛) = ( 1s /su 𝑚))
31	30	oveq2d 7436	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐴 +s ( 1s /su 𝑛)) = (𝐴 +s ( 1s /su 𝑚)))
32	31	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑚))))
33	32	cbvrexvw 3241	. . . . . . . 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 3632	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ↔ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚)))
36	26, 35	anbi12i 640	. . . . 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 3234	. . . . 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 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → 𝐴 ∈ No)
40	10	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No)
41	39, 40	subsvald 28448	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 −s ( 1s /su 𝑛)) = (𝐴 +s ( -s‘( 1s /su 𝑛))))
42	41	adantrr 730	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (𝐴 −s ( 1s /su 𝑛)) = (𝐴 +s ( -s‘( 1s /su 𝑛))))
43	10	negscld 28424	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ( -s‘( 1s /su 𝑛)) ∈ No)
44	43	adantr 486	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -s‘( 1s /su 𝑛)) ∈ No)
45		0no 28196	. . . . . . . . . . 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 28711	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No)
49		nnne0s 28724	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 𝑚 ≠ 0s )
50	47, 48, 49	divscld 28611	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕs → ( 1s /su 𝑚) ∈ No)
51	50	adantl 487	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( 1s /su 𝑚) ∈ No)
52		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ ℕs)
53	52	nnsrecgt0d 28738	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕs → 0s <s ( 1s /su 𝑛))
54	45	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → 0s ∈ No)
55	54, 10	ltnegsd 28434	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕs → ( 0s <s ( 1s /su 𝑛) ↔ ( -s‘( 1s /su 𝑛)) <s ( -s‘ 0s )))
56	53, 55	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕs → ( -s‘( 1s /su 𝑛)) <s ( -s‘ 0s ))
57		neg0s 28413	. . . . . . . . . . . 12 ⊢ ( -s‘ 0s ) = 0s
58	56, 57	breqtrdi 5145	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ( -s‘( 1s /su 𝑛)) <s 0s )
59	58	adantr 486	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -s‘( 1s /su 𝑛)) <s 0s )
60		id 23	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ ℕs)
61	60	nnsrecgt0d 28738	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕs → 0s <s ( 1s /su 𝑚))
62	61	adantl 487	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → 0s <s ( 1s /su 𝑚))
63	44, 46, 51, 59, 62	ltstrd 28121	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -s‘( 1s /su 𝑛)) <s ( 1s /su 𝑚))
64	63	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → ( -s‘( 1s /su 𝑛)) <s ( 1s /su 𝑚))
65	44	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → ( -s‘( 1s /su 𝑛)) ∈ No)
66	50	ad2antll 742	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → ( 1s /su 𝑚) ∈ No)
67		simpl 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → 𝐴 ∈ No)
68	65, 66, 67	ltadds2d 28384	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (( -s‘( 1s /su 𝑛)) <s ( 1s /su 𝑚) ↔ (𝐴 +s ( -s‘( 1s /su 𝑛))) <s (𝐴 +s ( 1s /su 𝑚))))
69	64, 68	mpbid 235	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (𝐴 +s ( -s‘( 1s /su 𝑛))) <s (𝐴 +s ( 1s /su 𝑚)))
70	42, 69	eqbrtrd 5126	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (𝐴 −s ( 1s /su 𝑛)) <s (𝐴 +s ( 1s /su 𝑚)))
71		breq12 5107	. . . . . 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 3219	. . . 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 1134	. 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 28155	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 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086  Vcvv 3450   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Nocsur 27998   <s clts 27999   <<s cslts 28144   0_s c0s 28192   1_s c1s 28193   +_s cadds 28346   -_scnegs 28406   −_s csubs 28407   /_su cdivs 28574  ℕ_scnns 28700

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9643  ax-dc 10525

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-2o 8478  df-oadd 8481  df-nadd 8676  df-no 28001  df-lts 28002  df-bday 28003  df-les 28103  df-slts 28145  df-cuts 28147  df-0s 28194  df-1s 28195  df-made 28214  df-old 28215  df-left 28217  df-right 28218  df-norec 28325  df-norec2 28336  df-adds 28347  df-negs 28408  df-subs 28409  df-muls 28494  df-divs 28575  df-n0s 28701  df-nns 28702

This theorem is used by:  elreno2  28882  renegscl  28885  readdscl  28886  remulscl  28889