Theorem pw2cutp1 28877

Description: Simplify pw2cut 28876 in the case of successors of surreal integers. (Contributed by Scott Fenton, 11-Nov-2025.)

Hypotheses

Ref	Expression
pw2cutp1.1	⊢ (𝜑 → 𝐴 ∈ ℤs)
pw2cutp1.3	⊢ (𝜑 → 𝑁 ∈ ℕ0s)

Assertion

Ref	Expression
pw2cutp1	⊢ (𝜑 → ({(𝐴 /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))}) = (((2s ·s 𝐴) +s 1s ) /su (2s↑s(𝑁 +s 1s ))))

Proof of Theorem pw2cutp1

Step	Hyp	Ref	Expression
1		pw2cutp1.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℤs)
2	1	znod 28799	. . 3 ⊢ (𝜑 → 𝐴 ∈ No)
3		1zs 28807	. . . . 5 ⊢ 1s ∈ ℤs
4		zaddscl 28810	. . . . 5 ⊢ ((𝐴 ∈ ℤs ∧ 1s ∈ ℤs) → (𝐴 +s 1s ) ∈ ℤs)
5	1, 3, 4	sylancl 598	. . . 4 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ ℤs)
6	5	znod 28799	. . 3 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ No)
7		pw2cutp1.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
8	2	ltsp1d 28431	. . 3 ⊢ (𝜑 → 𝐴 <s (𝐴 +s 1s ))
9		2nns 28834	. . . . . . . . 9 ⊢ 2s ∈ ℕs
10		nnzs 28802	. . . . . . . . 9 ⊢ (2s ∈ ℕs → 2s ∈ ℤs)
11	9, 10	ax-mp 5	. . . . . . . 8 ⊢ 2s ∈ ℤs
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2s ∈ ℤs)
13	12, 1	zmulscld 28813	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝐴) ∈ ℤs)
14		zaddscl 28810	. . . . . 6 ⊢ (((2s ·s 𝐴) ∈ ℤs ∧ 1s ∈ ℤs) → ((2s ·s 𝐴) +s 1s ) ∈ ℤs)
15	13, 3, 14	sylancl 598	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) ∈ ℤs)
16		zcuts 28823	. . . . 5 ⊢ (((2s ·s 𝐴) +s 1s ) ∈ ℤs → ((2s ·s 𝐴) +s 1s ) = ({(((2s ·s 𝐴) +s 1s ) −s 1s )} |s {(((2s ·s 𝐴) +s 1s ) +s 1s )}))
17	15, 16	syl 18	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ({(((2s ·s 𝐴) +s 1s ) −s 1s )} |s {(((2s ·s 𝐴) +s 1s ) +s 1s )}))
18		no2times 28833	. . . . . . 7 ⊢ (𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴))
19	2, 18	syl 18	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝐴) = (𝐴 +s 𝐴))
20	19	oveq1d 7458	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ((𝐴 +s 𝐴) +s 1s ))
21		1no 28226	. . . . . . 7 ⊢ 1s ∈ No
22	21	a1i 11	. . . . . 6 ⊢ (𝜑 → 1s ∈ No)
23	2, 2, 22	addsassd 28422	. . . . 5 ⊢ (𝜑 → ((𝐴 +s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
24	20, 23	eqtrd 2798	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
25	13	znod 28799	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) ∈ No)
26		pncans 28488	. . . . . . 7 ⊢ (((2s ·s 𝐴) ∈ No ∧ 1s ∈ No) → (((2s ·s 𝐴) +s 1s ) −s 1s ) = (2s ·s 𝐴))
27	25, 21, 26	sylancl 598	. . . . . 6 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) −s 1s ) = (2s ·s 𝐴))
28	27	sneqd 4596	. . . . 5 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) −s 1s )} = {(2s ·s 𝐴)})
29		1p1e2s 28832	. . . . . . . . 9 ⊢ ( 1s +s 1s ) = 2s
30		2no 28835	. . . . . . . . . 10 ⊢ 2s ∈ No
31		mulsrid 28529	. . . . . . . . . 10 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
32	30, 31	ax-mp 5	. . . . . . . . 9 ⊢ (2s ·s 1s ) = 2s
33	29, 32	eqtr4i 2789	. . . . . . . 8 ⊢ ( 1s +s 1s ) = (2s ·s 1s )
34	33	oveq2i 7454	. . . . . . 7 ⊢ ((2s ·s 𝐴) +s ( 1s +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s ))
35	25, 22, 22	addsassd 28422	. . . . . . 7 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) +s 1s ) = ((2s ·s 𝐴) +s ( 1s +s 1s )))
36	30	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2s ∈ No)
37	36, 2, 22	addsdid 28572	. . . . . . 7 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s )))
38	34, 35, 37	3eqtr4a 2824	. . . . . 6 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) +s 1s ) = (2s ·s (𝐴 +s 1s )))
39	38	sneqd 4596	. . . . 5 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) +s 1s )} = {(2s ·s (𝐴 +s 1s ))})
40	28, 39	oveq12d 7461	. . . 4 ⊢ (𝜑 → ({(((2s ·s 𝐴) +s 1s ) −s 1s )} |s {(((2s ·s 𝐴) +s 1s ) +s 1s )}) = ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}))
41	17, 24, 40	3eqtr3rd 2807	. . 3 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}) = (𝐴 +s (𝐴 +s 1s )))
42	2, 6, 7, 8, 41	pw2cut 28876	. 2 ⊢ (𝜑 → ({(𝐴 /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))}) = ((𝐴 +s (𝐴 +s 1s )) /su (2s↑s(𝑁 +s 1s ))))
43	24	oveq1d 7458	. 2 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) /su (2s↑s(𝑁 +s 1s ))) = ((𝐴 +s (𝐴 +s 1s )) /su (2s↑s(𝑁 +s 1s ))))
44	42, 43	eqtr4d 2801	1 ⊢ (𝜑 → ({(𝐴 /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))}) = (((2s ·s 𝐴) +s 1s ) /su (2s↑s(𝑁 +s 1s ))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {csn 4588  (class class class)co 7443  Nocsur 28027   |s ccuts 28175   1_s c1s 28222   +_s cadds 28375   −_s csubs 28436   ·_s cmuls 28522   /_su cdivs 28603  ℕ_0scn0s 28728  ℕ_scnns 28729  ℤ_sczs 28794  2_sc2s 28826  ↑_scexps 28828

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-seqs 28700  df-n0s 28730  df-nns 28731  df-zs 28795  df-2s 28827  df-exps 28829

This theorem is used by:  bdaypw2n0bndlem  28879  bdayfinbndlem1  28883