Theorem addhalfcut 28451

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 28317	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
3		1no 27802	. . . . 5 ⊢ 1s ∈ No
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 1s ∈ No )
5	2, 4	addscld 27972	. . 3 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ No )
6	2	ltsp1d 28007	. . 3 ⊢ (𝜑 → 𝐴 <s (𝐴 +s 1s ))
7		no2times 28409	. . . . . . 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 7382	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ((𝐴 +s 𝐴) +s 1s ))
10	2, 2, 4	addsassd 27998	. . . . 5 ⊢ (𝜑 → ((𝐴 +s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
11	9, 10	eqtr2d 2773	. . . 4 ⊢ (𝜑 → (𝐴 +s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s 1s ))
12		2nns 28410	. . . . . . . . . 10 ⊢ 2s ∈ ℕs
13		nnn0s 28319	. . . . . . . . . 10 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
14	12, 13	ax-mp 5	. . . . . . . . 9 ⊢ 2s ∈ ℕ0s
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2s ∈ ℕ0s)
16		n0mulscl 28337	. . . . . . . 8 ⊢ ((2s ∈ ℕ0s ∧ 𝐴 ∈ ℕ0s) → (2s ·s 𝐴) ∈ ℕ0s)
17	15, 1, 16	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) ∈ ℕ0s)
18		1n0s 28340	. . . . . . . 8 ⊢ 1s ∈ ℕ0s
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1s ∈ ℕ0s)
20		n0addscl 28336	. . . . . . 7 ⊢ (((2s ·s 𝐴) ∈ ℕ0s ∧ 1s ∈ ℕ0s) → ((2s ·s 𝐴) +s 1s ) ∈ ℕ0s)
21	17, 19, 20	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) ∈ ℕ0s)
22		n0cut 28326	. . . . . 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		2no 28411	. . . . . . . . . 10 ⊢ 2s ∈ No
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2s ∈ No )
26	25, 2	mulscld 28127	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐴) ∈ No )
27		pncans 28064	. . . . . . . 8 ⊢ (((2s ·s 𝐴) ∈ No ∧ 1s ∈ No ) → (((2s ·s 𝐴) +s 1s ) -s 1s ) = (2s ·s 𝐴))
28	26, 4, 27	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) -s 1s ) = (2s ·s 𝐴))
29	28	sneqd 4580	. . . . . 6 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) -s 1s )} = {(2s ·s 𝐴)})
30	29	oveq1d 7382	. . . . 5 ⊢ (𝜑 → ({(((2s ·s 𝐴) +s 1s ) -s 1s )} |s ∅) = ({(2s ·s 𝐴)} |s ∅))
31	23, 30	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ({(2s ·s 𝐴)} |s ∅))
32		snelpwi 5397	. . . . . 6 ⊢ ((2s ·s 𝐴) ∈ No → {(2s ·s 𝐴)} ∈ 𝒫 No )
33		nulsgts 27768	. . . . . 6 ⊢ ({(2s ·s 𝐴)} ∈ 𝒫 No → {(2s ·s 𝐴)} <<s ∅)
34	26, 32, 33	3syl 18	. . . . 5 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s ∅)
35		lesid 27731	. . . . . . 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 7400	. . . . . . . . 9 ⊢ (2s ·s 𝐴) ∈ V
38		breq2 5090	. . . . . . . . 9 ⊢ (𝑦 = (2s ·s 𝐴) → (𝑥 ≤s 𝑦 ↔ 𝑥 ≤s (2s ·s 𝐴)))
39	37, 38	rexsn 4627	. . . . . . . 8 ⊢ (∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦 ↔ 𝑥 ≤s (2s ·s 𝐴))
40	39	ralbii 3084	. . . . . . 7 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦 ↔ ∀𝑥 ∈ {(2s ·s 𝐴)}𝑥 ≤s (2s ·s 𝐴))
41		breq1 5089	. . . . . . . 8 ⊢ (𝑥 = (2s ·s 𝐴) → (𝑥 ≤s (2s ·s 𝐴) ↔ (2s ·s 𝐴) ≤s (2s ·s 𝐴)))
42	37, 41	ralsn 4626	. . . . . . 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 4439	. . . . . 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 27972	. . . . . . 7 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) ∈ No )
48	26	ltsp1d 28007	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) <s ((2s ·s 𝐴) +s 1s ))
49	26, 47, 48	sltssn 27762	. . . . . 6 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s {((2s ·s 𝐴) +s 1s )})
50	31	sneqd 4580	. . . . . 6 ⊢ (𝜑 → {((2s ·s 𝐴) +s 1s )} = {({(2s ·s 𝐴)} |s ∅)})
51	49, 50	breqtrd 5112	. . . . 5 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s {({(2s ·s 𝐴)} |s ∅)})
52	25, 5	mulscld 28127	. . . . . . 7 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) ∈ No )
53	4	ltsp1d 28007	. . . . . . . . . 10 ⊢ (𝜑 → 1s <s ( 1s +s 1s ))
54		1p1e2s 28408	. . . . . . . . . 10 ⊢ ( 1s +s 1s ) = 2s
55	53, 54	breqtrdi 5127	. . . . . . . . 9 ⊢ (𝜑 → 1s <s 2s)
56	4, 25, 26	ltadds2d 27989	. . . . . . . . 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 28148	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s )))
59		mulsrid 28105	. . . . . . . . . . 11 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
60	24, 59	ax-mp 5	. . . . . . . . . 10 ⊢ (2s ·s 1s ) = 2s
61	60	oveq2i 7378	. . . . . . . . 9 ⊢ ((2s ·s 𝐴) +s (2s ·s 1s )) = ((2s ·s 𝐴) +s 2s)
62	58, 61	eqtrdi 2788	. . . . . . . 8 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s 2s))
63	57, 62	breqtrrd 5114	. . . . . . 7 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) <s (2s ·s (𝐴 +s 1s )))
64	47, 52, 63	sltssn 27762	. . . . . 6 ⊢ (𝜑 → {((2s ·s 𝐴) +s 1s )} <<s {(2s ·s (𝐴 +s 1s ))})
65	50, 64	eqbrtrrd 5110	. . . . 5 ⊢ (𝜑 → {({(2s ·s 𝐴)} |s ∅)} <<s {(2s ·s (𝐴 +s 1s ))})
66	34, 44, 46, 51, 65	cofcut1d 27913	. . . 4 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s ∅) = ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}))
67	11, 31, 66	3eqtrrd 2777	. . 3 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}) = (𝐴 +s (𝐴 +s 1s )))
68		eqid 2737	. . 3 ⊢ ({𝐴} |s {(𝐴 +s 1s )}) = ({𝐴} |s {(𝐴 +s 1s )})
69	2, 5, 6, 67, 68	halfcut 28450	. 2 ⊢ (𝜑 → ({𝐴} |s {(𝐴 +s 1s )}) = ((𝐴 +s (𝐴 +s 1s )) /su 2s))
70	11	oveq1d 7382	. 2 ⊢ (𝜑 → ((𝐴 +s (𝐴 +s 1s )) /su 2s) = (((2s ·s 𝐴) +s 1s ) /su 2s))
71		2ne0s 28412	. . . . 5 ⊢ 2s ≠ 0s
72	71	a1i 11	. . . 4 ⊢ (𝜑 → 2s ≠ 0s )
73	26, 4, 25, 72	divsdird 28227	. . 3 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) /su 2s) = (((2s ·s 𝐴) /su 2s) +s ( 1s /su 2s)))
74	2, 25, 72	divscan3d 28228	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) /su 2s) = 𝐴)
75	74	oveq1d 7382	. . 3 ⊢ (𝜑 → (((2s ·s 𝐴) /su 2s) +s ( 1s /su 2s)) = (𝐴 +s ( 1s /su 2s)))
76	73, 75	eqtrd 2772	. 2 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) /su 2s) = (𝐴 +s ( 1s /su 2s)))
77	69, 70, 76	3eqtrd 2776	1 ⊢ (𝜑 → ({𝐴} |s {(𝐴 +s 1s )}) = (𝐴 +s ( 1s /su 2s)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  ∅c0 4274  𝒫 cpw 4542  {csn 4568   class class class wbr 5086  (class class class)co 7367   No csur 27603   <s clts 27604   ≤s cles 27708   <<s cslts 27749   |s ccuts 27751   0_s c0s 27797   1_s c1s 27798   +_s cadds 27951   -_s csubs 28012   ·_s cmuls 28098   /_su cdivs 28179  ℕ_0scn0s 28304  ℕ_scnns 28305  2_sc2s 28402

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376  ax-un 7689  ax-dc 10368

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 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6266  df-ord 6327  df-on 6328  df-lim 6329  df-suc 6330  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-nadd 8602  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-1s 27800  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-norec2 27941  df-adds 27952  df-negs 28013  df-subs 28014  df-muls 28099  df-divs 28180  df-n0s 28306  df-nns 28307  df-2s 28403

This theorem is referenced by: (None)