Theorem addhalfcut 28374

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	n0snod 28285	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
3		1sno 27827	. . . . 5 ⊢ 1s ∈ No
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 1s ∈ No )
5	2, 4	addscld 27968	. . 3 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ No )
6	2	sltp1d 28003	. . 3 ⊢ (𝜑 → 𝐴 <s (𝐴 +s 1s ))
7		no2times 28356	. . . . . . 7 ⊢ (𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴))
8	2, 7	syl 17	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝐴) = (𝐴 +s 𝐴))
9	8	oveq1d 7429	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ((𝐴 +s 𝐴) +s 1s ))
10	2, 2, 4	addsassd 27994	. . . . 5 ⊢ (𝜑 → ((𝐴 +s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
11	9, 10	eqtr2d 2770	. . . 4 ⊢ (𝜑 → (𝐴 +s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s 1s ))
12		2nns 28357	. . . . . . . . . 10 ⊢ 2s ∈ ℕs
13		nnn0s 28287	. . . . . . . . . 10 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
14	12, 13	ax-mp 5	. . . . . . . . 9 ⊢ 2s ∈ ℕ0s
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2s ∈ ℕ0s)
16		n0mulscl 28303	. . . . . . . 8 ⊢ ((2s ∈ ℕ0s ∧ 𝐴 ∈ ℕ0s) → (2s ·s 𝐴) ∈ ℕ0s)
17	15, 1, 16	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) ∈ ℕ0s)
18		1n0s 28306	. . . . . . . 8 ⊢ 1s ∈ ℕ0s
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1s ∈ ℕ0s)
20		n0addscl 28302	. . . . . . 7 ⊢ (((2s ·s 𝐴) ∈ ℕ0s ∧ 1s ∈ ℕ0s) → ((2s ·s 𝐴) +s 1s ) ∈ ℕ0s)
21	17, 19, 20	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) ∈ ℕ0s)
22		n0scut 28293	. . . . . 6 ⊢ (((2s ·s 𝐴) +s 1s ) ∈ ℕ0s → ((2s ·s 𝐴) +s 1s ) = ({(((2s ·s 𝐴) +s 1s ) -s 1s )} |s ∅))
23	21, 22	syl 17	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ({(((2s ·s 𝐴) +s 1s ) -s 1s )} |s ∅))
24		2sno 28358	. . . . . . . . . 10 ⊢ 2s ∈ No
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2s ∈ No )
26	25, 2	mulscld 28116	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐴) ∈ No )
27		pncans 28057	. . . . . . . 8 ⊢ (((2s ·s 𝐴) ∈ No ∧ 1s ∈ No ) → (((2s ·s 𝐴) +s 1s ) -s 1s ) = (2s ·s 𝐴))
28	26, 4, 27	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) -s 1s ) = (2s ·s 𝐴))
29	28	sneqd 4620	. . . . . 6 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) -s 1s )} = {(2s ·s 𝐴)})
30	29	oveq1d 7429	. . . . 5 ⊢ (𝜑 → ({(((2s ·s 𝐴) +s 1s ) -s 1s )} |s ∅) = ({(2s ·s 𝐴)} |s ∅))
31	23, 30	eqtrd 2769	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ({(2s ·s 𝐴)} |s ∅))
32		snelpwi 5430	. . . . . 6 ⊢ ((2s ·s 𝐴) ∈ No → {(2s ·s 𝐴)} ∈ 𝒫 No )
33		nulssgt 27798	. . . . . 6 ⊢ ({(2s ·s 𝐴)} ∈ 𝒫 No → {(2s ·s 𝐴)} <<s ∅)
34	26, 32, 33	3syl 18	. . . . 5 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s ∅)
35		slerflex 27763	. . . . . . 7 ⊢ ((2s ·s 𝐴) ∈ No → (2s ·s 𝐴) ≤s (2s ·s 𝐴))
36	26, 35	syl 17	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝐴) ≤s (2s ·s 𝐴))
37		ovex 7447	. . . . . . . . 9 ⊢ (2s ·s 𝐴) ∈ V
38		breq2 5129	. . . . . . . . 9 ⊢ (𝑦 = (2s ·s 𝐴) → (𝑥 ≤s 𝑦 ↔ 𝑥 ≤s (2s ·s 𝐴)))
39	37, 38	rexsn 4664	. . . . . . . 8 ⊢ (∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦 ↔ 𝑥 ≤s (2s ·s 𝐴))
40	39	ralbii 3081	. . . . . . 7 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦 ↔ ∀𝑥 ∈ {(2s ·s 𝐴)}𝑥 ≤s (2s ·s 𝐴))
41		breq1 5128	. . . . . . . 8 ⊢ (𝑥 = (2s ·s 𝐴) → (𝑥 ≤s (2s ·s 𝐴) ↔ (2s ·s 𝐴) ≤s (2s ·s 𝐴)))
42	37, 41	ralsn 4663	. . . . . . 7 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}𝑥 ≤s (2s ·s 𝐴) ↔ (2s ·s 𝐴) ≤s (2s ·s 𝐴))
43	40, 42	bitri 275	. . . . . 6 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦 ↔ (2s ·s 𝐴) ≤s (2s ·s 𝐴))
44	36, 43	sylibr 234	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦)
45		ral0 4495	. . . . . 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 27968	. . . . . . 7 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) ∈ No )
48	26	sltp1d 28003	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) <s ((2s ·s 𝐴) +s 1s ))
49	26, 47, 48	ssltsn 27792	. . . . . 6 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s {((2s ·s 𝐴) +s 1s )})
50	31	sneqd 4620	. . . . . 6 ⊢ (𝜑 → {((2s ·s 𝐴) +s 1s )} = {({(2s ·s 𝐴)} |s ∅)})
51	49, 50	breqtrd 5151	. . . . 5 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s {({(2s ·s 𝐴)} |s ∅)})
52	25, 5	mulscld 28116	. . . . . . 7 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) ∈ No )
53	4	sltp1d 28003	. . . . . . . . . 10 ⊢ (𝜑 → 1s <s ( 1s +s 1s ))
54		1p1e2s 28355	. . . . . . . . . 10 ⊢ ( 1s +s 1s ) = 2s
55	53, 54	breqtrdi 5166	. . . . . . . . 9 ⊢ (𝜑 → 1s <s 2s)
56	4, 25, 26	sltadd2d 27985	. . . . . . . . 9 ⊢ (𝜑 → ( 1s <s 2s ↔ ((2s ·s 𝐴) +s 1s ) <s ((2s ·s 𝐴) +s 2s)))
57	55, 56	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) <s ((2s ·s 𝐴) +s 2s))
58	25, 2, 4	addsdid 28137	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s )))
59		mulsrid 28094	. . . . . . . . . . 11 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
60	24, 59	ax-mp 5	. . . . . . . . . 10 ⊢ (2s ·s 1s ) = 2s
61	60	oveq2i 7425	. . . . . . . . 9 ⊢ ((2s ·s 𝐴) +s (2s ·s 1s )) = ((2s ·s 𝐴) +s 2s)
62	58, 61	eqtrdi 2785	. . . . . . . 8 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s 2s))
63	57, 62	breqtrrd 5153	. . . . . . 7 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) <s (2s ·s (𝐴 +s 1s )))
64	47, 52, 63	ssltsn 27792	. . . . . 6 ⊢ (𝜑 → {((2s ·s 𝐴) +s 1s )} <<s {(2s ·s (𝐴 +s 1s ))})
65	50, 64	eqbrtrrd 5149	. . . . 5 ⊢ (𝜑 → {({(2s ·s 𝐴)} |s ∅)} <<s {(2s ·s (𝐴 +s 1s ))})
66	34, 44, 46, 51, 65	cofcut1d 27910	. . . 4 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s ∅) = ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}))
67	11, 31, 66	3eqtrrd 2774	. . 3 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}) = (𝐴 +s (𝐴 +s 1s )))
68		eqid 2734	. . 3 ⊢ ({𝐴} |s {(𝐴 +s 1s )}) = ({𝐴} |s {(𝐴 +s 1s )})
69	2, 5, 6, 67, 68	halfcut 28371	. 2 ⊢ (𝜑 → ({𝐴} |s {(𝐴 +s 1s )}) = ((𝐴 +s (𝐴 +s 1s )) /su 2s))
70	11	oveq1d 7429	. 2 ⊢ (𝜑 → ((𝐴 +s (𝐴 +s 1s )) /su 2s) = (((2s ·s 𝐴) +s 1s ) /su 2s))
71		2ne0s 28359	. . . . 5 ⊢ 2s ≠ 0s
72	71	a1i 11	. . . 4 ⊢ (𝜑 → 2s ≠ 0s )
73	26, 4, 25, 72	divsdird 28214	. . 3 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) /su 2s) = (((2s ·s 𝐴) /su 2s) +s ( 1s /su 2s)))
74	2, 25, 72	divscan3d 28215	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) /su 2s) = 𝐴)
75	74	oveq1d 7429	. . 3 ⊢ (𝜑 → (((2s ·s 𝐴) /su 2s) +s ( 1s /su 2s)) = (𝐴 +s ( 1s /su 2s)))
76	73, 75	eqtrd 2769	. 2 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) /su 2s) = (𝐴 +s ( 1s /su 2s)))
77	69, 70, 76	3eqtrd 2773	1 ⊢ (𝜑 → ({𝐴} |s {(𝐴 +s 1s )}) = (𝐴 +s ( 1s /su 2s)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050  ∃wrex 3059  ∅c0 4315  𝒫 cpw 4582  {csn 4608   class class class wbr 5125  (class class class)co 7414   No csur 27639   <s cslt 27640   ≤s csle 27744   <<s csslt 27780   |s cscut 27782   0_s c0s 27822   1_s c1s 27823   +_s cadds 27947   -_s csubs 28007   ·_s cmuls 28087   /_su cdivs 28168  ℕ_0scnn0s 28273  ℕ_scnns 28274  2_sc2s 28349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738  ax-dc 10469

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-tp 4613  df-op 4615  df-ot 4617  df-uni 4890  df-int 4929  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7871  df-1st 7997  df-2nd 7998  df-frecs 8289  df-wrecs 8320  df-recs 8394  df-rdg 8433  df-1o 8489  df-2o 8490  df-oadd 8493  df-nadd 8687  df-no 27642  df-slt 27643  df-bday 27644  df-sle 27745  df-sslt 27781  df-scut 27783  df-0s 27824  df-1s 27825  df-made 27841  df-old 27842  df-left 27844  df-right 27845  df-norec 27926  df-norec2 27937  df-adds 27948  df-negs 28008  df-subs 28009  df-muls 28088  df-divs 28169  df-n0s 28275  df-nns 28276  df-2s 28350

This theorem is referenced by: (None)