Theorem pw2cutp1 28391

Description: Simplify pw2cut 28390 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 28317	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
3		1zs 28325	. . . . 5 ⊢ 1s ∈ ℤs
4		zaddscl 28328	. . . . 5 ⊢ ((𝐴 ∈ ℤs ∧ 1s ∈ ℤs) → (𝐴 +s 1s ) ∈ ℤs)
5	1, 3, 4	sylancl 586	. . . 4 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ ℤs)
6	5	znod 28317	. . 3 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ No )
7		pw2cutp1.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
8	2	sltp1d 27968	. . 3 ⊢ (𝜑 → 𝐴 <s (𝐴 +s 1s ))
9		2nns 28351	. . . . . . . . 9 ⊢ 2s ∈ ℕs
10		nnzs 28320	. . . . . . . . 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 28331	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝐴) ∈ ℤs)
14		zaddscl 28328	. . . . . 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 28341	. . . . 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 28350	. . . . . . 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 7370	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ((𝐴 +s 𝐴) +s 1s ))
21		1sno 27781	. . . . . . 7 ⊢ 1s ∈ No
22	21	a1i 11	. . . . . 6 ⊢ (𝜑 → 1s ∈ No )
23	2, 2, 22	addsassd 27959	. . . . 5 ⊢ (𝜑 → ((𝐴 +s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
24	20, 23	eqtrd 2768	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
25	13	znod 28317	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) ∈ No )
26		pncans 28022	. . . . . . 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 4589	. . . . 5 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) -s 1s )} = {(2s ·s 𝐴)})
29		1p1e2s 28349	. . . . . . . . 9 ⊢ ( 1s +s 1s ) = 2s
30		2sno 28352	. . . . . . . . . 10 ⊢ 2s ∈ No
31		mulsrid 28062	. . . . . . . . . 10 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
32	30, 31	ax-mp 5	. . . . . . . . 9 ⊢ (2s ·s 1s ) = 2s
33	29, 32	eqtr4i 2759	. . . . . . . 8 ⊢ ( 1s +s 1s ) = (2s ·s 1s )
34	33	oveq2i 7366	. . . . . . 7 ⊢ ((2s ·s 𝐴) +s ( 1s +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s ))
35	25, 22, 22	addsassd 27959	. . . . . . 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 28105	. . . . . . 7 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s )))
38	34, 35, 37	3eqtr4a 2794	. . . . . 6 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) +s 1s ) = (2s ·s (𝐴 +s 1s )))
39	38	sneqd 4589	. . . . 5 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) +s 1s )} = {(2s ·s (𝐴 +s 1s ))})
40	28, 39	oveq12d 7373	. . . 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 2777	. . 3 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}) = (𝐴 +s (𝐴 +s 1s )))
42	2, 6, 7, 8, 41	pw2cut 28390	. 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 7370	. 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 2771	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 1541   ∈ wcel 2113  {csn 4577  (class class class)co 7355   No csur 27588   |s cscut 27732   1_s c1s 27777   +_s cadds 27912   -_s csubs 27972   ·_s cmuls 28055   /_su cdivs 28136  ℕ_0scnn0s 28252  ℕ_scnns 28253  ℤ_sczs 28312  2_sc2s 28343  ↑_scexps 28345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-dc 10347

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-oadd 8398  df-nadd 8590  df-no 27591  df-slt 27592  df-bday 27593  df-sle 27694  df-sslt 27731  df-scut 27733  df-0s 27778  df-1s 27779  df-made 27798  df-old 27799  df-left 27801  df-right 27802  df-norec 27891  df-norec2 27902  df-adds 27913  df-negs 27973  df-subs 27974  df-muls 28056  df-divs 28137  df-seqs 28224  df-n0s 28254  df-nns 28255  df-zs 28313  df-2s 28344  df-exps 28346

This theorem is referenced by: (None)