Theorem pw2cutp1 28388

Description: Simplify pw2cut 28387 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 28323	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
3		1zs 28331	. . . . 5 ⊢ 1s ∈ ℤs
4		zaddscl 28334	. . . . 5 ⊢ ((𝐴 ∈ ℤs ∧ 1s ∈ ℤs) → (𝐴 +s 1s ) ∈ ℤs)
5	1, 3, 4	sylancl 586	. . . 4 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ ℤs)
6	5	znod 28323	. . 3 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ No )
7		pw2cutp1.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
8	2	sltp1d 27974	. . 3 ⊢ (𝜑 → 𝐴 <s (𝐴 +s 1s ))
9		2nns 28356	. . . . . . . . 9 ⊢ 2s ∈ ℕs
10		nnzs 28326	. . . . . . . . 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 28337	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝐴) ∈ ℤs)
14		zaddscl 28334	. . . . . 6 ⊢ (((2s ·s 𝐴) ∈ ℤs ∧ 1s ∈ ℤs) → ((2s ·s 𝐴) +s 1s ) ∈ ℤs)
15	13, 3, 14	sylancl 586	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) ∈ ℤs)
16		zscut 28347	. . . . 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 17	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ({(((2s ·s 𝐴) +s 1s ) -s 1s )} |s {(((2s ·s 𝐴) +s 1s ) +s 1s )}))
18		no2times 28355	. . . . . . 7 ⊢ (𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴))
19	2, 18	syl 17	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝐴) = (𝐴 +s 𝐴))
20	19	oveq1d 7420	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ((𝐴 +s 𝐴) +s 1s ))
21		1sno 27791	. . . . . . 7 ⊢ 1s ∈ No
22	21	a1i 11	. . . . . 6 ⊢ (𝜑 → 1s ∈ No )
23	2, 2, 22	addsassd 27965	. . . . 5 ⊢ (𝜑 → ((𝐴 +s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
24	20, 23	eqtrd 2770	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
25	13	znod 28323	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) ∈ No )
26		pncans 28028	. . . . . . 7 ⊢ (((2s ·s 𝐴) ∈ No ∧ 1s ∈ No ) → (((2s ·s 𝐴) +s 1s ) -s 1s ) = (2s ·s 𝐴))
27	25, 21, 26	sylancl 586	. . . . . 6 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) -s 1s ) = (2s ·s 𝐴))
28	27	sneqd 4613	. . . . 5 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) -s 1s )} = {(2s ·s 𝐴)})
29		1p1e2s 28354	. . . . . . . . 9 ⊢ ( 1s +s 1s ) = 2s
30		2sno 28357	. . . . . . . . . 10 ⊢ 2s ∈ No
31		mulsrid 28068	. . . . . . . . . 10 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
32	30, 31	ax-mp 5	. . . . . . . . 9 ⊢ (2s ·s 1s ) = 2s
33	29, 32	eqtr4i 2761	. . . . . . . 8 ⊢ ( 1s +s 1s ) = (2s ·s 1s )
34	33	oveq2i 7416	. . . . . . 7 ⊢ ((2s ·s 𝐴) +s ( 1s +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s ))
35	25, 22, 22	addsassd 27965	. . . . . . 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 28111	. . . . . . 7 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s )))
38	34, 35, 37	3eqtr4a 2796	. . . . . 6 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) +s 1s ) = (2s ·s (𝐴 +s 1s )))
39	38	sneqd 4613	. . . . 5 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) +s 1s )} = {(2s ·s (𝐴 +s 1s ))})
40	28, 39	oveq12d 7423	. . . 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 2779	. . 3 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}) = (𝐴 +s (𝐴 +s 1s )))
42	2, 6, 7, 8, 41	pw2cut 28387	. 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 7420	. 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 2773	1 ⊢ (𝜑 → ({(𝐴 /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))}) = (((2s ·s 𝐴) +s 1s ) /su (2s↑s(𝑁 +s 1s ))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  {csn 4601  (class class class)co 7405   No csur 27603   |s cscut 27746   1_s c1s 27787   +_s cadds 27918   -_s csubs 27978   ·_s cmuls 28061   /_su cdivs 28142  ℕ_0scnn0s 28258  ℕ_scnns 28259  ℤ_sczs 28318  2_sc2s 28348  ↑_scexps 28350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-dc 10460

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-nadd 8678  df-no 27606  df-slt 27607  df-bday 27608  df-sle 27709  df-sslt 27745  df-scut 27747  df-0s 27788  df-1s 27789  df-made 27807  df-old 27808  df-left 27810  df-right 27811  df-norec 27897  df-norec2 27908  df-adds 27919  df-negs 27979  df-subs 27980  df-muls 28062  df-divs 28143  df-seqs 28230  df-n0s 28260  df-nns 28261  df-zs 28319  df-2s 28349  df-exps 28351

This theorem is referenced by: (None)