Theorem recut 28473

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 28299	. . . 4 ⊢ ℕs ∈ V
2	1	abrexex 7908	. . 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 7908	. . 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 27808	. . . . . . . 8 ⊢ 1s ∈ No
7	6	a1i 11	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 1s ∈ No )
8		nnsno 28305	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ∈ No )
9		nnne0s 28317	. . . . . . 7 ⊢ (𝑛 ∈ ℕs → 𝑛 ≠ 0s )
10	7, 8, 9	divscld 28205	. . . . . 6 ⊢ (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
11		subscl 28044	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
12	10, 11	sylan2 594	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) ∈ No )
13		eleq1 2825	. . . . 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 3138	. . 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 27963	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( 1s /su 𝑛) ∈ No ) → (𝐴 +s ( 1s /su 𝑛)) ∈ No )
18	10, 17	sylan2 594	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 +s ( 1s /su 𝑛)) ∈ No )
19		eleq1 2825	. . . . 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 3138	. . 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 3445	. . . . . . 7 ⊢ 𝑦 ∈ V
24		eqeq1 2741	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
25	24	rexbidv 3161	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛))))
26	23, 25	elab 3635	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑦 = (𝐴 -s ( 1s /su 𝑛)))
27		vex 3445	. . . . . . 7 ⊢ 𝑧 ∈ V
28		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
29	28	rexbidv 3161	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
30		oveq2 7368	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ( 1s /su 𝑛) = ( 1s /su 𝑚))
31	30	oveq2d 7376	. . . . . . . . . 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 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 3635	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ↔ ∃𝑚 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑚)))
36	26, 35	anbi12i 629	. . . . 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 28043	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
42	41	adantrr 718	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
43	10	negscld 28019	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) ∈ No )
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) ∈ No )
45		0sno 27807	. . . . . . . . . . 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 28305	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 𝑚 ∈ No )
49		nnne0s 28317	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕs → 𝑚 ≠ 0s )
50	47, 48, 49	divscld 28205	. . . . . . . . . . 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 28331	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕs → 0s <s ( 1s /su 𝑛))
54	45	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕs → 0s ∈ No )
55	54, 10	sltnegd 28029	. . . . . . . . . . . . 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 28008	. . . . . . . . . . . 12 ⊢ ( -us ‘ 0s ) = 0s
58	56, 57	breqtrdi 5140	. . . . . . . . . . 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 28331	. . . . . . . . . . 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 27735	. . . . . . . . 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 730	. . . . . . . . 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 27979	. . . . . . . 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 5121	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ (𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs)) → (𝐴 -s ( 1s /su 𝑛)) <s (𝐴 +s ( 1s /su 𝑚)))
71		breq12 5104	. . . . . 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 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 27768	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 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3061  Vcvv 3441   class class class wbr 5099  ‘cfv 6493  (class class class)co 7360   No csur 27611   <s cslt 27612   <<s csslt 27757   0_s c0s 27803   1_s c1s 27804   +_s cadds 27941   -u_s cnegs 28001   -_s csubs 28002   /_su cdivs 28169  ℕ_scnns 28294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554  ax-dc 10360

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-nadd 8596  df-no 27614  df-slt 27615  df-bday 27616  df-sle 27717  df-sslt 27758  df-scut 27760  df-0s 27805  df-1s 27806  df-made 27825  df-old 27826  df-left 27828  df-right 27829  df-norec 27920  df-norec2 27931  df-adds 27942  df-negs 28003  df-subs 28004  df-muls 28089  df-divs 28170  df-n0s 28295  df-nns 28296

This theorem is referenced by:  elreno2  28474  renegscl  28477  readdscl  28478  remulscl  28481