Theorem addhalfcut 28428

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 28339	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
3		1sno 27881	. . . . 5 ⊢ 1s ∈ No
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 1s ∈ No )
5	2, 4	addscld 28022	. . 3 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ No )
6	2	sltp1d 28057	. . 3 ⊢ (𝜑 → 𝐴 <s (𝐴 +s 1s ))
7		no2times 28410	. . . . . . 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 7460	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ((𝐴 +s 𝐴) +s 1s ))
10	2, 2, 4	addsassd 28048	. . . . 5 ⊢ (𝜑 → ((𝐴 +s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
11	9, 10	eqtr2d 2775	. . . 4 ⊢ (𝜑 → (𝐴 +s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s 1s ))
12		2nns 28411	. . . . . . . . . 10 ⊢ 2s ∈ ℕs
13		nnn0s 28341	. . . . . . . . . 10 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
14	12, 13	ax-mp 5	. . . . . . . . 9 ⊢ 2s ∈ ℕ0s
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2s ∈ ℕ0s)
16		n0mulscl 28357	. . . . . . . 8 ⊢ ((2s ∈ ℕ0s ∧ 𝐴 ∈ ℕ0s) → (2s ·s 𝐴) ∈ ℕ0s)
17	15, 1, 16	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) ∈ ℕ0s)
18		1n0s 28360	. . . . . . . 8 ⊢ 1s ∈ ℕ0s
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1s ∈ ℕ0s)
20		n0addscl 28356	. . . . . . 7 ⊢ (((2s ·s 𝐴) ∈ ℕ0s ∧ 1s ∈ ℕ0s) → ((2s ·s 𝐴) +s 1s ) ∈ ℕ0s)
21	17, 19, 20	syl2anc 583	. . . . . 6 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) ∈ ℕ0s)
22		n0scut 28347	. . . . . 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 28412	. . . . . . . . . 10 ⊢ 2s ∈ No
25	24	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2s ∈ No )
26	25, 2	mulscld 28170	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐴) ∈ No )
27		pncans 28111	. . . . . . . 8 ⊢ (((2s ·s 𝐴) ∈ No ∧ 1s ∈ No ) → (((2s ·s 𝐴) +s 1s ) -s 1s ) = (2s ·s 𝐴))
28	26, 4, 27	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) -s 1s ) = (2s ·s 𝐴))
29	28	sneqd 4660	. . . . . 6 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) -s 1s )} = {(2s ·s 𝐴)})
30	29	oveq1d 7460	. . . . 5 ⊢ (𝜑 → ({(((2s ·s 𝐴) +s 1s ) -s 1s )} |s ∅) = ({(2s ·s 𝐴)} |s ∅))
31	23, 30	eqtrd 2774	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ({(2s ·s 𝐴)} |s ∅))
32		snelpwi 5466	. . . . . 6 ⊢ ((2s ·s 𝐴) ∈ No → {(2s ·s 𝐴)} ∈ 𝒫 No )
33		nulssgt 27852	. . . . . 6 ⊢ ({(2s ·s 𝐴)} ∈ 𝒫 No → {(2s ·s 𝐴)} <<s ∅)
34	26, 32, 33	3syl 18	. . . . 5 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s ∅)
35		slerflex 27817	. . . . . . 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 7478	. . . . . . . . 9 ⊢ (2s ·s 𝐴) ∈ V
38		breq2 5173	. . . . . . . . 9 ⊢ (𝑦 = (2s ·s 𝐴) → (𝑥 ≤s 𝑦 ↔ 𝑥 ≤s (2s ·s 𝐴)))
39	37, 38	rexsn 4706	. . . . . . . 8 ⊢ (∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦 ↔ 𝑥 ≤s (2s ·s 𝐴))
40	39	ralbii 3095	. . . . . . 7 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(2s ·s 𝐴)}𝑥 ≤s 𝑦 ↔ ∀𝑥 ∈ {(2s ·s 𝐴)}𝑥 ≤s (2s ·s 𝐴))
41		breq1 5172	. . . . . . . 8 ⊢ (𝑥 = (2s ·s 𝐴) → (𝑥 ≤s (2s ·s 𝐴) ↔ (2s ·s 𝐴) ≤s (2s ·s 𝐴)))
42	37, 41	ralsn 4705	. . . . . . 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 4532	. . . . . 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 28022	. . . . . . 7 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) ∈ No )
48	26	sltp1d 28057	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) <s ((2s ·s 𝐴) +s 1s ))
49	26, 47, 48	ssltsn 27846	. . . . . 6 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s {((2s ·s 𝐴) +s 1s )})
50	31	sneqd 4660	. . . . . 6 ⊢ (𝜑 → {((2s ·s 𝐴) +s 1s )} = {({(2s ·s 𝐴)} |s ∅)})
51	49, 50	breqtrd 5195	. . . . 5 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s {({(2s ·s 𝐴)} |s ∅)})
52	25, 5	mulscld 28170	. . . . . . 7 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) ∈ No )
53	4	sltp1d 28057	. . . . . . . . . 10 ⊢ (𝜑 → 1s <s ( 1s +s 1s ))
54		1p1e2s 28409	. . . . . . . . . 10 ⊢ ( 1s +s 1s ) = 2s
55	53, 54	breqtrdi 5210	. . . . . . . . 9 ⊢ (𝜑 → 1s <s 2s)
56	4, 25, 26	sltadd2d 28039	. . . . . . . . 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 28191	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s )))
59		mulsrid 28148	. . . . . . . . . . 11 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
60	24, 59	ax-mp 5	. . . . . . . . . 10 ⊢ (2s ·s 1s ) = 2s
61	60	oveq2i 7456	. . . . . . . . 9 ⊢ ((2s ·s 𝐴) +s (2s ·s 1s )) = ((2s ·s 𝐴) +s 2s)
62	58, 61	eqtrdi 2790	. . . . . . . 8 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s 2s))
63	57, 62	breqtrrd 5197	. . . . . . 7 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) <s (2s ·s (𝐴 +s 1s )))
64	47, 52, 63	ssltsn 27846	. . . . . 6 ⊢ (𝜑 → {((2s ·s 𝐴) +s 1s )} <<s {(2s ·s (𝐴 +s 1s ))})
65	50, 64	eqbrtrrd 5193	. . . . 5 ⊢ (𝜑 → {({(2s ·s 𝐴)} |s ∅)} <<s {(2s ·s (𝐴 +s 1s ))})
66	34, 44, 46, 51, 65	cofcut1d 27964	. . . 4 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s ∅) = ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}))
67	11, 31, 66	3eqtrrd 2779	. . 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 28425	. 2 ⊢ (𝜑 → ({𝐴} |s {(𝐴 +s 1s )}) = ((𝐴 +s (𝐴 +s 1s )) /su 2s))
70	11	oveq1d 7460	. 2 ⊢ (𝜑 → ((𝐴 +s (𝐴 +s 1s )) /su 2s) = (((2s ·s 𝐴) +s 1s ) /su 2s))
71		2ne0s 28413	. . . . 5 ⊢ 2s ≠ 0s
72	71	a1i 11	. . . 4 ⊢ (𝜑 → 2s ≠ 0s )
73	26, 4, 25, 72	divsdird 28268	. . 3 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) /su 2s) = (((2s ·s 𝐴) /su 2s) +s ( 1s /su 2s)))
74	2, 25, 72	divscan3d 28269	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) /su 2s) = 𝐴)
75	74	oveq1d 7460	. . 3 ⊢ (𝜑 → (((2s ·s 𝐴) /su 2s) +s ( 1s /su 2s)) = (𝐴 +s ( 1s /su 2s)))
76	73, 75	eqtrd 2774	. 2 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) /su 2s) = (𝐴 +s ( 1s /su 2s)))
77	69, 70, 76	3eqtrd 2778	1 ⊢ (𝜑 → ({𝐴} |s {(𝐴 +s 1s )}) = (𝐴 +s ( 1s /su 2s)))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2103   ≠ wne 2942  ∀wral 3063  ∃wrex 3072  ∅c0 4347  𝒫 cpw 4622  {csn 4648   class class class wbr 5169  (class class class)co 7445   No csur 27693   <s cslt 27694   ≤s csle 27798   <<s csslt 27834   |s cscut 27836   0_s c0s 27876   1_s c1s 27877   +_s cadds 28001   -_s csubs 28061   ·_s cmuls 28141   /_su cdivs 28222  ℕ_0scnn0s 28327  ℕ_scnns 28328  2_sc2s 28403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-dc 10511

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-se 5655  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-2o 8519  df-oadd 8522  df-nadd 8718  df-no 27696  df-slt 27697  df-bday 27698  df-sle 27799  df-sslt 27835  df-scut 27837  df-0s 27878  df-1s 27879  df-made 27895  df-old 27896  df-left 27898  df-right 27899  df-norec 27980  df-norec2 27991  df-adds 28002  df-negs 28062  df-subs 28063  df-muls 28142  df-divs 28223  df-n0s 28329  df-nns 28330  df-2s 28404

This theorem is referenced by: (None)