Theorem 1reno 28892

Description: Surreal one is a surreal real. (Contributed by Scott Fenton, 18-Feb-2026.)

Assertion

Ref	Expression
1reno	⊢ 1s ∈ ℝs

Proof of Theorem 1reno

Dummy variables 𝑛 𝑥_𝑂 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1no 28205	. 2 ⊢ 1s ∈ No
2		2nns 28813	. . . 4 ⊢ 2s ∈ ℕs
3		2no 28814	. . . . . . . . 9 ⊢ 2s ∈ No
4	3	a1i 11	. . . . . . . 8 ⊢ (⊤ → 2s ∈ No)
5	4	negscld 28432	. . . . . . 7 ⊢ (⊤ → ( -s‘2s) ∈ No)
6		0no 28204	. . . . . . . 8 ⊢ 0s ∈ No
7	6	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s ∈ No)
8	1	a1i 11	. . . . . . 7 ⊢ (⊤ → 1s ∈ No)
9		nnsgt0 28734	. . . . . . . . 9 ⊢ (2s ∈ ℕs → 0s <s 2s)
10	2, 9	ax-mp 5	. . . . . . . 8 ⊢ 0s <s 2s
11	4	lt0negs2d 28446	. . . . . . . 8 ⊢ (⊤ → ( 0s <s 2s ↔ ( -s‘2s) <s 0s ))
12	10, 11	mpbii 236	. . . . . . 7 ⊢ (⊤ → ( -s‘2s) <s 0s )
13		0lt1s 28207	. . . . . . . 8 ⊢ 0s <s 1s
14	13	a1i 11	. . . . . . 7 ⊢ (⊤ → 0s <s 1s )
15	5, 7, 8, 12, 14	ltstrd 28129	. . . . . 6 ⊢ (⊤ → ( -s‘2s) <s 1s )
16	15	mptru 1576	. . . . 5 ⊢ ( -s‘2s) <s 1s
17	8	ltsp1d 28410	. . . . . . 7 ⊢ (⊤ → 1s <s ( 1s +s 1s ))
18	17	mptru 1576	. . . . . 6 ⊢ 1s <s ( 1s +s 1s )
19		1p1e2s 28811	. . . . . 6 ⊢ ( 1s +s 1s ) = 2s
20	18, 19	breqtri 5129	. . . . 5 ⊢ 1s <s 2s
21	16, 20	pm3.2i 476	. . . 4 ⊢ (( -s‘2s) <s 1s ∧ 1s <s 2s)
22		fveq2 6890	. . . . . . 7 ⊢ (𝑛 = 2s → ( -s‘𝑛) = ( -s‘2s))
23	22	breq1d 5112	. . . . . 6 ⊢ (𝑛 = 2s → (( -s‘𝑛) <s 1s ↔ ( -s‘2s) <s 1s ))
24		breq2 5106	. . . . . 6 ⊢ (𝑛 = 2s → ( 1s <s 𝑛 ↔ 1s <s 2s))
25	23, 24	anbi12d 644	. . . . 5 ⊢ (𝑛 = 2s → ((( -s‘𝑛) <s 1s ∧ 1s <s 𝑛) ↔ (( -s‘2s) <s 1s ∧ 1s <s 2s)))
26	25	rspcev 3576	. . . 4 ⊢ ((2s ∈ ℕs ∧ (( -s‘2s) <s 1s ∧ 1s <s 2s)) → ∃𝑛 ∈ ℕs (( -s‘𝑛) <s 1s ∧ 1s <s 𝑛))
27	2, 21, 26	mp2an 705	. . 3 ⊢ ∃𝑛 ∈ ℕs (( -s‘𝑛) <s 1s ∧ 1s <s 𝑛)
28		1nns 28744	. . . . 5 ⊢ 1s ∈ ℕs
29		lesid 28133	. . . . . 6 ⊢ ( 1s ∈ No → 1s ≤s 1s )
30	1, 29	ax-mp 5	. . . . 5 ⊢ 1s ≤s 1s
31		oveq2 7432	. . . . . . . 8 ⊢ (𝑛 = 1s → ( 1s /su 𝑛) = ( 1s /su 1s ))
32		divs1 28599	. . . . . . . . 9 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
33	1, 32	ax-mp 5	. . . . . . . 8 ⊢ ( 1s /su 1s ) = 1s
34	31, 33	eqtrdi 2811	. . . . . . 7 ⊢ (𝑛 = 1s → ( 1s /su 𝑛) = 1s )
35	34	breq1d 5112	. . . . . 6 ⊢ (𝑛 = 1s → (( 1s /su 𝑛) ≤s 1s ↔ 1s ≤s 1s ))
36	35	rspcev 3576	. . . . 5 ⊢ (( 1s ∈ ℕs ∧ 1s ≤s 1s ) → ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s )
37	28, 30, 36	mp2an 705	. . . 4 ⊢ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s
38		left1s 28290	. . . . . . . 8 ⊢ (L‘ 1s ) = { 0s }
39		right1s 28291	. . . . . . . 8 ⊢ (R‘ 1s ) = ∅
40	38, 39	uneq12i 4164	. . . . . . 7 ⊢ ((L‘ 1s ) ∪ (R‘ 1s )) = ({ 0s } ∪ ∅)
41		un0 4343	. . . . . . 7 ⊢ ({ 0s } ∪ ∅) = { 0s }
42	40, 41	eqtri 2783	. . . . . 6 ⊢ ((L‘ 1s ) ∪ (R‘ 1s )) = { 0s }
43	42	raleqi 3317	. . . . 5 ⊢ (∀𝑥𝑂 ∈ ((L‘ 1s ) ∪ (R‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)) ↔ ∀𝑥𝑂 ∈ { 0s }∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)))
44	6	elexi 3472	. . . . . 6 ⊢ 0s ∈ V
45		oveq2 7432	. . . . . . . . . . 11 ⊢ (𝑥𝑂 = 0s → ( 1s −s 𝑥𝑂) = ( 1s −s 0s ))
46		subsid1 28463	. . . . . . . . . . . 12 ⊢ ( 1s ∈ No → ( 1s −s 0s ) = 1s )
47	1, 46	ax-mp 5	. . . . . . . . . . 11 ⊢ ( 1s −s 0s ) = 1s
48	45, 47	eqtrdi 2811	. . . . . . . . . 10 ⊢ (𝑥𝑂 = 0s → ( 1s −s 𝑥𝑂) = 1s )
49	48	fveq2d 6894	. . . . . . . . 9 ⊢ (𝑥𝑂 = 0s → (abss‘( 1s −s 𝑥𝑂)) = (abss‘ 1s ))
50	7, 8, 14	ltlesd 28139	. . . . . . . . . . 11 ⊢ (⊤ → 0s ≤s 1s )
51	50	mptru 1576	. . . . . . . . . 10 ⊢ 0s ≤s 1s
52		abssid 28636	. . . . . . . . . 10 ⊢ (( 1s ∈ No ∧ 0s ≤s 1s ) → (abss‘ 1s ) = 1s )
53	1, 51, 52	mp2an 705	. . . . . . . . 9 ⊢ (abss‘ 1s ) = 1s
54	49, 53	eqtrdi 2811	. . . . . . . 8 ⊢ (𝑥𝑂 = 0s → (abss‘( 1s −s 𝑥𝑂)) = 1s )
55	54	breq2d 5114	. . . . . . 7 ⊢ (𝑥𝑂 = 0s → (( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)) ↔ ( 1s /su 𝑛) ≤s 1s ))
56	55	rexbidv 3186	. . . . . 6 ⊢ (𝑥𝑂 = 0s → (∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s ))
57	44, 56	ralsn 4638	. . . . 5 ⊢ (∀𝑥𝑂 ∈ { 0s }∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s )
58	43, 57	bitri 278	. . . 4 ⊢ (∀𝑥𝑂 ∈ ((L‘ 1s ) ∪ (R‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)) ↔ ∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s 1s )
59	37, 58	mpbir 234	. . 3 ⊢ ∀𝑥𝑂 ∈ ((L‘ 1s ) ∪ (R‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂))
60	27, 59	pm3.2i 476	. 2 ⊢ (∃𝑛 ∈ ℕs (( -s‘𝑛) <s 1s ∧ 1s <s 𝑛) ∧ ∀𝑥𝑂 ∈ ((L‘ 1s ) ∪ (R‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)))
61		elreno2 28890	. 2 ⊢ ( 1s ∈ ℝs ↔ ( 1s ∈ No ∧ (∃𝑛 ∈ ℕs (( -s‘𝑛) <s 1s ∧ 1s <s 𝑛) ∧ ∀𝑥𝑂 ∈ ((L‘ 1s ) ∪ (R‘ 1s ))∃𝑛 ∈ ℕs ( 1s /su 𝑛) ≤s (abss‘( 1s −s 𝑥𝑂)))))
62	1, 60, 61	mpbir2an 724	1 ⊢ 1s ∈ ℝs

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  ⊤wtru 1570   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ∪ cun 4148  ∅c0 4277  {csn 4584   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  Nocsur 28006   <s clts 28007   ≤s cles 28110   0_s c0s 28200   1_s c1s 28201  Lcleft 28220  Rcright 28221   +_s cadds 28354   -_scnegs 28414   −_s csubs 28415   /_su cdivs 28582  abs_scabss 28632  ℕ_scnns 28708  2_sc2s 28805  ℝ_screno 28884

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 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649  ax-dc 10531

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 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-2o 8484  df-oadd 8487  df-nadd 8682  df-no 28009  df-lts 28010  df-bday 28011  df-les 28111  df-slts 28153  df-cuts 28155  df-0s 28202  df-1s 28203  df-made 28222  df-old 28223  df-left 28225  df-right 28226  df-norec 28333  df-norec2 28344  df-adds 28355  df-negs 28416  df-subs 28417  df-muls 28502  df-divs 28583  df-abss 28633  df-n0s 28709  df-nns 28710  df-2s 28806  df-reno 28885

This theorem is used by: (None)