Theorem addhalfcut 28845

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 28711	. . 3 ⊢ (𝜑 → 𝐴 ∈ No)
3		1no 28196	. . . . 5 ⊢ 1s ∈ No
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 1s ∈ No)
5	2, 4	addscld 28366	. . 3 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ No)
6	2	ltsp1d 28401	. . 3 ⊢ (𝜑 → 𝐴 <s (𝐴 +s 1s ))
7		no2times 28803	. . . . . . 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 7435	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ((𝐴 +s 𝐴) +s 1s ))
10	2, 2, 4	addsassd 28392	. . . . 5 ⊢ (𝜑 → ((𝐴 +s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
11	9, 10	eqtr2d 2796	. . . 4 ⊢ (𝜑 → (𝐴 +s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s 1s ))
12		2nns 28804	. . . . . . . . . 10 ⊢ 2s ∈ ℕs
13		nnn0s 28713	. . . . . . . . . 10 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
14	12, 13	ax-mp 5	. . . . . . . . 9 ⊢ 2s ∈ ℕ0s
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2s ∈ ℕ0s)
16		n0mulscl 28731	. . . . . . . 8 ⊢ ((2s ∈ ℕ0s ∧ 𝐴 ∈ ℕ0s) → (2s ·s 𝐴) ∈ ℕ0s)
17	15, 1, 16	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) ∈ ℕ0s)
18		1n0s 28734	. . . . . . . 8 ⊢ 1s ∈ ℕ0s
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1s ∈ ℕ0s)
20		n0addscl 28730	. . . . . . 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 28720	. . . . . 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 28805	. . . . . . . . . 10 ⊢ 2s ∈ No
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2s ∈ No)
26	25, 2	mulscld 28521	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐴) ∈ No)
27		pncans 28458	. . . . . . . 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 4595	. . . . . 6 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) −s 1s )} = {(2s ·s 𝐴)})
30	29	oveq1d 7435	. . . . 5 ⊢ (𝜑 → ({(((2s ·s 𝐴) +s 1s ) −s 1s )} |s ∅) = ({(2s ·s 𝐴)} |s ∅))
31	23, 30	eqtrd 2795	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ({(2s ·s 𝐴)} |s ∅))
32		snelpwi 5411	. . . . . 6 ⊢ ((2s ·s 𝐴) ∈ No → {(2s ·s 𝐴)} ∈ 𝒫 No)
33		nulsgts 28162	. . . . . 6 ⊢ ({(2s ·s 𝐴)} ∈ 𝒫 No → {(2s ·s 𝐴)} <<s ∅)
34	26, 32, 33	3syl 19	. . . . 5 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s ∅)
35		lesid 28124	. . . . . . 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 7453	. . . . . . . . 9 ⊢ (2s ·s 𝐴) ∈ V
38		breq2 5106	. . . . . . . . 9 ⊢ (𝑦 = (2s ·s 𝐴) → (𝑥 ≤s 𝑦 ↔ 𝑥 ≤s (2s ·s 𝐴)))
39	37, 38	rexsn 4642	. . . . . . . 8 ⊢ (∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦 ↔ 𝑥 ≤s (2s ·s 𝐴))
40	39	ralbii 3108	. . . . . . 7 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦 ↔ ∀𝑥 ∈ {(2s ·s 𝐴)}𝑥 ≤s (2s ·s 𝐴))
41		breq1 5105	. . . . . . . 8 ⊢ (𝑥 = (2s ·s 𝐴) → (𝑥 ≤s (2s ·s 𝐴) ↔ (2s ·s 𝐴) ≤s (2s ·s 𝐴)))
42	37, 41	ralsn 4641	. . . . . . 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 4453	. . . . . 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 28366	. . . . . . 7 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) ∈ No)
48	26	ltsp1d 28401	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) <s ((2s ·s 𝐴) +s 1s ))
49	26, 47, 48	sltssn 28156	. . . . . 6 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s {((2s ·s 𝐴) +s 1s )})
50	31	sneqd 4595	. . . . . 6 ⊢ (𝜑 → {((2s ·s 𝐴) +s 1s )} = {({(2s ·s 𝐴)} |s ∅)})
51	49, 50	breqtrd 5130	. . . . 5 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s {({(2s ·s 𝐴)} |s ∅)})
52	25, 5	mulscld 28521	. . . . . . 7 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) ∈ No)
53	4	ltsp1d 28401	. . . . . . . . . 10 ⊢ (𝜑 → 1s <s ( 1s +s 1s ))
54		1p1e2s 28802	. . . . . . . . . 10 ⊢ ( 1s +s 1s ) = 2s
55	53, 54	breqtrdi 5145	. . . . . . . . 9 ⊢ (𝜑 → 1s <s 2s)
56	4, 25, 26	ltadds2d 28383	. . . . . . . . 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 28542	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s )))
59		mulsrid 28499	. . . . . . . . . . 11 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
60	24, 59	ax-mp 5	. . . . . . . . . 10 ⊢ (2s ·s 1s ) = 2s
61	60	oveq2i 7431	. . . . . . . . 9 ⊢ ((2s ·s 𝐴) +s (2s ·s 1s )) = ((2s ·s 𝐴) +s 2s)
62	58, 61	eqtrdi 2811	. . . . . . . 8 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s 2s))
63	57, 62	breqtrrd 5132	. . . . . . 7 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) <s (2s ·s (𝐴 +s 1s )))
64	47, 52, 63	sltssn 28156	. . . . . 6 ⊢ (𝜑 → {((2s ·s 𝐴) +s 1s )} <<s {(2s ·s (𝐴 +s 1s ))})
65	50, 64	eqbrtrrd 5128	. . . . 5 ⊢ (𝜑 → {({(2s ·s 𝐴)} |s ∅)} <<s {(2s ·s (𝐴 +s 1s ))})
66	34, 44, 46, 51, 65	cofcut1d 28307	. . . 4 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s ∅) = ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}))
67	11, 31, 66	3eqtrrd 2800	. . 3 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}) = (𝐴 +s (𝐴 +s 1s )))
68		eqid 2760	. . 3 ⊢ ({𝐴} |s {(𝐴 +s 1s )}) = ({𝐴} |s {(𝐴 +s 1s )})
69	2, 5, 6, 67, 68	halfcut 28844	. 2 ⊢ (𝜑 → ({𝐴} |s {(𝐴 +s 1s )}) = ((𝐴 +s (𝐴 +s 1s )) /su 2s))
70	11	oveq1d 7435	. 2 ⊢ (𝜑 → ((𝐴 +s (𝐴 +s 1s )) /su 2s) = (((2s ·s 𝐴) +s 1s ) /su 2s))
71		2ne0s 28806	. . . . 5 ⊢ 2s ≠ 0s
72	71	a1i 11	. . . 4 ⊢ (𝜑 → 2s ≠ 0s )
73	26, 4, 25, 72	divsdird 28621	. . 3 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) /su 2s) = (((2s ·s 𝐴) /su 2s) +s ( 1s /su 2s)))
74	2, 25, 72	divscan3d 28622	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) /su 2s) = 𝐴)
75	74	oveq1d 7435	. . 3 ⊢ (𝜑 → (((2s ·s 𝐴) /su 2s) +s ( 1s /su 2s)) = (𝐴 +s ( 1s /su 2s)))
76	73, 75	eqtrd 2795	. 2 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) /su 2s) = (𝐴 +s ( 1s /su 2s)))
77	69, 70, 76	3eqtrd 2799	1 ⊢ (𝜑 → ({𝐴} |s {(𝐴 +s 1s )}) = (𝐴 +s ( 1s /su 2s)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  ∅c0 4278  𝒫 cpw 4556  {csn 4583   class class class wbr 5102  (class class class)co 7420  Nocsur 27997   <s clts 27998   ≤s cles 28101   <<s cslts 28143   |s ccuts 28145   0_s c0s 28191   1_s c1s 28192   +_s cadds 28345   −_s csubs 28406   ·_s cmuls 28492   /_su cdivs 28573  ℕ_0scn0s 28698  ℕ_scnns 28699  2_sc2s 28796

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-dc 10524

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 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-nadd 8675  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-1s 28194  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-norec2 28335  df-adds 28346  df-negs 28407  df-subs 28408  df-muls 28493  df-divs 28574  df-n0s 28700  df-nns 28701  df-2s 28797

This theorem is used by: (None)