Theorem pw2cutp1 28343

Description: Simplify pw2cut 28342 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 28278	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
3		1zs 28286	. . . . 5 ⊢ 1s ∈ ℤs
4		zaddscl 28289	. . . . 5 ⊢ ((𝐴 ∈ ℤs ∧ 1s ∈ ℤs) → (𝐴 +s 1s ) ∈ ℤs)
5	1, 3, 4	sylancl 586	. . . 4 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ ℤs)
6	5	znod 28278	. . 3 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ No )
7		pw2cutp1.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
8	2	sltp1d 27929	. . 3 ⊢ (𝜑 → 𝐴 <s (𝐴 +s 1s ))
9		2nns 28311	. . . . . . . . 9 ⊢ 2s ∈ ℕs
10		nnzs 28281	. . . . . . . . 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 28292	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝐴) ∈ ℤs)
14		zaddscl 28289	. . . . . 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 28302	. . . . 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 28310	. . . . . . 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 7405	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ((𝐴 +s 𝐴) +s 1s ))
21		1sno 27746	. . . . . . 7 ⊢ 1s ∈ No
22	21	a1i 11	. . . . . 6 ⊢ (𝜑 → 1s ∈ No )
23	2, 2, 22	addsassd 27920	. . . . 5 ⊢ (𝜑 → ((𝐴 +s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
24	20, 23	eqtrd 2765	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
25	13	znod 28278	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) ∈ No )
26		pncans 27983	. . . . . . 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 4604	. . . . 5 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) -s 1s )} = {(2s ·s 𝐴)})
29		1p1e2s 28309	. . . . . . . . 9 ⊢ ( 1s +s 1s ) = 2s
30		2sno 28312	. . . . . . . . . 10 ⊢ 2s ∈ No
31		mulsrid 28023	. . . . . . . . . 10 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
32	30, 31	ax-mp 5	. . . . . . . . 9 ⊢ (2s ·s 1s ) = 2s
33	29, 32	eqtr4i 2756	. . . . . . . 8 ⊢ ( 1s +s 1s ) = (2s ·s 1s )
34	33	oveq2i 7401	. . . . . . 7 ⊢ ((2s ·s 𝐴) +s ( 1s +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s ))
35	25, 22, 22	addsassd 27920	. . . . . . 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 28066	. . . . . . 7 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s )))
38	34, 35, 37	3eqtr4a 2791	. . . . . 6 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) +s 1s ) = (2s ·s (𝐴 +s 1s )))
39	38	sneqd 4604	. . . . 5 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) +s 1s )} = {(2s ·s (𝐴 +s 1s ))})
40	28, 39	oveq12d 7408	. . . 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 2774	. . 3 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}) = (𝐴 +s (𝐴 +s 1s )))
42	2, 6, 7, 8, 41	pw2cut 28342	. 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 7405	. 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 2768	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 2109  {csn 4592  (class class class)co 7390   No csur 27558   |s cscut 27701   1_s c1s 27742   +_s cadds 27873   -_s csubs 27933   ·_s cmuls 28016   /_su cdivs 28097  ℕ_0scnn0s 28213  ℕ_scnns 28214  ℤ_sczs 28273  2_sc2s 28303  ↑_scexps 28305

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-dc 10406

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-nadd 8633  df-no 27561  df-slt 27562  df-bday 27563  df-sle 27664  df-sslt 27700  df-scut 27702  df-0s 27743  df-1s 27744  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec 27852  df-norec2 27863  df-adds 27874  df-negs 27934  df-subs 27935  df-muls 28017  df-divs 28098  df-seqs 28185  df-n0s 28215  df-nns 28216  df-zs 28274  df-2s 28304  df-exps 28306

This theorem is referenced by: (None)