Theorem addhalfcut 28875

Description: The cut of a surreal non-negative integer and its successor is the original number plus one half. Part of theorem 4.2 of [Gonshor] p. 30. (Contributed by Scott Fenton, 13-Aug-2025.)

Hypothesis

Ref	Expression
addhalfcut.1	⊢ (𝜑 → 𝐴 ∈ ℕ0s)

Assertion

Ref	Expression
addhalfcut	⊢ (𝜑 → ({𝐴} |s {(𝐴 +s 1s )}) = (𝐴 +s ( 1s /su 2s)))

Proof of Theorem addhalfcut

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addhalfcut.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℕ0s)
2	1	n0nod 28741	. . 3 ⊢ (𝜑 → 𝐴 ∈ No)
3		1no 28226	. . . . 5 ⊢ 1s ∈ No
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 1s ∈ No)
5	2, 4	addscld 28396	. . 3 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ No)
6	2	ltsp1d 28431	. . 3 ⊢ (𝜑 → 𝐴 <s (𝐴 +s 1s ))
7		no2times 28833	. . . . . . 7 ⊢ (𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴))
8	2, 7	syl 18	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝐴) = (𝐴 +s 𝐴))
9	8	oveq1d 7458	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ((𝐴 +s 𝐴) +s 1s ))
10	2, 2, 4	addsassd 28422	. . . . 5 ⊢ (𝜑 → ((𝐴 +s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
11	9, 10	eqtr2d 2799	. . . 4 ⊢ (𝜑 → (𝐴 +s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s 1s ))
12		2nns 28834	. . . . . . . . . 10 ⊢ 2s ∈ ℕs
13		nnn0s 28743	. . . . . . . . . 10 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
14	12, 13	ax-mp 5	. . . . . . . . 9 ⊢ 2s ∈ ℕ0s
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2s ∈ ℕ0s)
16		n0mulscl 28761	. . . . . . . 8 ⊢ ((2s ∈ ℕ0s ∧ 𝐴 ∈ ℕ0s) → (2s ·s 𝐴) ∈ ℕ0s)
17	15, 1, 16	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) ∈ ℕ0s)
18		1n0s 28764	. . . . . . . 8 ⊢ 1s ∈ ℕ0s
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1s ∈ ℕ0s)
20		n0addscl 28760	. . . . . . 7 ⊢ (((2s ·s 𝐴) ∈ ℕ0s ∧ 1s ∈ ℕ0s) → ((2s ·s 𝐴) +s 1s ) ∈ ℕ0s)
21	17, 19, 20	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) ∈ ℕ0s)
22		n0cut 28750	. . . . . 6 ⊢ (((2s ·s 𝐴) +s 1s ) ∈ ℕ0s → ((2s ·s 𝐴) +s 1s ) = ({(((2s ·s 𝐴) +s 1s ) −s 1s )} |s ∅))
23	21, 22	syl 18	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ({(((2s ·s 𝐴) +s 1s ) −s 1s )} |s ∅))
24		2no 28835	. . . . . . . . . 10 ⊢ 2s ∈ No
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2s ∈ No)
26	25, 2	mulscld 28551	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐴) ∈ No)
27		pncans 28488	. . . . . . . 8 ⊢ (((2s ·s 𝐴) ∈ No ∧ 1s ∈ No) → (((2s ·s 𝐴) +s 1s ) −s 1s ) = (2s ·s 𝐴))
28	26, 4, 27	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) −s 1s ) = (2s ·s 𝐴))
29	28	sneqd 4596	. . . . . 6 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) −s 1s )} = {(2s ·s 𝐴)})
30	29	oveq1d 7458	. . . . 5 ⊢ (𝜑 → ({(((2s ·s 𝐴) +s 1s ) −s 1s )} |s ∅) = ({(2s ·s 𝐴)} |s ∅))
31	23, 30	eqtrd 2798	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ({(2s ·s 𝐴)} |s ∅))
32		snelpwi 5416	. . . . . 6 ⊢ ((2s ·s 𝐴) ∈ No → {(2s ·s 𝐴)} ∈ 𝒫 No)
33		nulsgts 28192	. . . . . 6 ⊢ ({(2s ·s 𝐴)} ∈ 𝒫 No → {(2s ·s 𝐴)} <<s ∅)
34	26, 32, 33	3syl 19	. . . . 5 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s ∅)
35		lesid 28154	. . . . . . 7 ⊢ ((2s ·s 𝐴) ∈ No → (2s ·s 𝐴) ≤s (2s ·s 𝐴))
36	26, 35	syl 18	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝐴) ≤s (2s ·s 𝐴))
37		ovex 7476	. . . . . . . . 9 ⊢ (2s ·s 𝐴) ∈ V
38		breq2 5110	. . . . . . . . 9 ⊢ (𝑦 = (2s ·s 𝐴) → (𝑥 ≤s 𝑦 ↔ 𝑥 ≤s (2s ·s 𝐴)))
39	37, 38	rexsn 4643	. . . . . . . 8 ⊢ (∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦 ↔ 𝑥 ≤s (2s ·s 𝐴))
40	39	ralbii 3112	. . . . . . 7 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦 ↔ ∀𝑥 ∈ {(2s ·s 𝐴)}𝑥 ≤s (2s ·s 𝐴))
41		breq1 5109	. . . . . . . 8 ⊢ (𝑥 = (2s ·s 𝐴) → (𝑥 ≤s (2s ·s 𝐴) ↔ (2s ·s 𝐴) ≤s (2s ·s 𝐴)))
42	37, 41	ralsn 4642	. . . . . . 7 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}𝑥 ≤s (2s ·s 𝐴) ↔ (2s ·s 𝐴) ≤s (2s ·s 𝐴))
43	40, 42	bitri 278	. . . . . 6 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦 ↔ (2s ·s 𝐴) ≤s (2s ·s 𝐴))
44	36, 43	sylibr 237	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦)
45		ral0 4457	. . . . . 6 ⊢ ∀𝑥 ∈ ∅ ∃𝑦 ∈ {(2s ·s (𝐴 +s 1s ))}𝑦 ≤s 𝑥
46	45	a1i 11	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ∅ ∃𝑦 ∈ {(2s ·s (𝐴 +s 1s ))}𝑦 ≤s 𝑥)
47	26, 4	addscld 28396	. . . . . . 7 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) ∈ No)
48	26	ltsp1d 28431	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) <s ((2s ·s 𝐴) +s 1s ))
49	26, 47, 48	sltssn 28186	. . . . . 6 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s {((2s ·s 𝐴) +s 1s )})
50	31	sneqd 4596	. . . . . 6 ⊢ (𝜑 → {((2s ·s 𝐴) +s 1s )} = {({(2s ·s 𝐴)} |s ∅)})
51	49, 50	breqtrd 5134	. . . . 5 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s {({(2s ·s 𝐴)} |s ∅)})
52	25, 5	mulscld 28551	. . . . . . 7 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) ∈ No)
53	4	ltsp1d 28431	. . . . . . . . . 10 ⊢ (𝜑 → 1s <s ( 1s +s 1s ))
54		1p1e2s 28832	. . . . . . . . . 10 ⊢ ( 1s +s 1s ) = 2s
55	53, 54	breqtrdi 5149	. . . . . . . . 9 ⊢ (𝜑 → 1s <s 2s)
56	4, 25, 26	ltadds2d 28413	. . . . . . . . 9 ⊢ (𝜑 → ( 1s <s 2s ↔ ((2s ·s 𝐴) +s 1s ) <s ((2s ·s 𝐴) +s 2s)))
57	55, 56	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) <s ((2s ·s 𝐴) +s 2s))
58	25, 2, 4	addsdid 28572	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s )))
59		mulsrid 28529	. . . . . . . . . . 11 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
60	24, 59	ax-mp 5	. . . . . . . . . 10 ⊢ (2s ·s 1s ) = 2s
61	60	oveq2i 7454	. . . . . . . . 9 ⊢ ((2s ·s 𝐴) +s (2s ·s 1s )) = ((2s ·s 𝐴) +s 2s)
62	58, 61	eqtrdi 2814	. . . . . . . 8 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s 2s))
63	57, 62	breqtrrd 5136	. . . . . . 7 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) <s (2s ·s (𝐴 +s 1s )))
64	47, 52, 63	sltssn 28186	. . . . . 6 ⊢ (𝜑 → {((2s ·s 𝐴) +s 1s )} <<s {(2s ·s (𝐴 +s 1s ))})
65	50, 64	eqbrtrrd 5132	. . . . 5 ⊢ (𝜑 → {({(2s ·s 𝐴)} |s ∅)} <<s {(2s ·s (𝐴 +s 1s ))})
66	34, 44, 46, 51, 65	cofcut1d 28337	. . . 4 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s ∅) = ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}))
67	11, 31, 66	3eqtrrd 2803	. . 3 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}) = (𝐴 +s (𝐴 +s 1s )))
68		eqid 2763	. . 3 ⊢ ({𝐴} |s {(𝐴 +s 1s )}) = ({𝐴} |s {(𝐴 +s 1s )})
69	2, 5, 6, 67, 68	halfcut 28874	. 2 ⊢ (𝜑 → ({𝐴} |s {(𝐴 +s 1s )}) = ((𝐴 +s (𝐴 +s 1s )) /su 2s))
70	11	oveq1d 7458	. 2 ⊢ (𝜑 → ((𝐴 +s (𝐴 +s 1s )) /su 2s) = (((2s ·s 𝐴) +s 1s ) /su 2s))
71		2ne0s 28836	. . . . 5 ⊢ 2s ≠ 0s
72	71	a1i 11	. . . 4 ⊢ (𝜑 → 2s ≠ 0s )
73	26, 4, 25, 72	divsdird 28651	. . 3 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) /su 2s) = (((2s ·s 𝐴) /su 2s) +s ( 1s /su 2s)))
74	2, 25, 72	divscan3d 28652	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) /su 2s) = 𝐴)
75	74	oveq1d 7458	. . 3 ⊢ (𝜑 → (((2s ·s 𝐴) /su 2s) +s ( 1s /su 2s)) = (𝐴 +s ( 1s /su 2s)))
76	73, 75	eqtrd 2798	. 2 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) /su 2s) = (𝐴 +s ( 1s /su 2s)))
77	69, 70, 76	3eqtrd 2802	1 ⊢ (𝜑 → ({𝐴} |s {(𝐴 +s 1s )}) = (𝐴 +s ( 1s /su 2s)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  ∅c0 4281  𝒫 cpw 4560  {csn 4588   class class class wbr 5106  (class class class)co 7443  Nocsur 28027   <s clts 28028   ≤s cles 28131   <<s cslts 28173   |s ccuts 28175   0_s c0s 28221   1_s c1s 28222   +_s cadds 28375   −_s csubs 28436   ·_s cmuls 28522   /_su cdivs 28603  ℕ_0scn0s 28728  ℕ_scnns 28729  2_sc2s 28826

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-dc 10551

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-1s 28224  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-norec2 28365  df-adds 28376  df-negs 28437  df-subs 28438  df-muls 28523  df-divs 28604  df-n0s 28730  df-nns 28731  df-2s 28827

This theorem is used by: (None)