Theorem pw2cutp1 28478

Description: Simplify pw2cut 28477 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 28400	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
3		1zs 28408	. . . . 5 ⊢ 1s ∈ ℤs
4		zaddscl 28411	. . . . 5 ⊢ ((𝐴 ∈ ℤs ∧ 1s ∈ ℤs) → (𝐴 +s 1s ) ∈ ℤs)
5	1, 3, 4	sylancl 592	. . . 4 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ ℤs)
6	5	znod 28400	. . 3 ⊢ (𝜑 → (𝐴 +s 1s ) ∈ No )
7		pw2cutp1.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
8	2	ltsp1d 28032	. . 3 ⊢ (𝜑 → 𝐴 <s (𝐴 +s 1s ))
9		2nns 28435	. . . . . . . . 9 ⊢ 2s ∈ ℕs
10		nnzs 28403	. . . . . . . . 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 28414	. . . . . 6 ⊢ (𝜑 → (2s ·s 𝐴) ∈ ℤs)
14		zaddscl 28411	. . . . . 6 ⊢ (((2s ·s 𝐴) ∈ ℤs ∧ 1s ∈ ℤs) → ((2s ·s 𝐴) +s 1s ) ∈ ℤs)
15	13, 3, 14	sylancl 592	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) ∈ ℤs)
16		zcuts 28424	. . . . 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 28434	. . . . . . 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 7378	. . . . 5 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = ((𝐴 +s 𝐴) +s 1s ))
21		1no 27827	. . . . . . 7 ⊢ 1s ∈ No
22	21	a1i 11	. . . . . 6 ⊢ (𝜑 → 1s ∈ No )
23	2, 2, 22	addsassd 28023	. . . . 5 ⊢ (𝜑 → ((𝐴 +s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
24	20, 23	eqtrd 2775	. . . 4 ⊢ (𝜑 → ((2s ·s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
25	13	znod 28400	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) ∈ No )
26		pncans 28089	. . . . . . 7 ⊢ (((2s ·s 𝐴) ∈ No ∧ 1s ∈ No ) → (((2s ·s 𝐴) +s 1s ) -s 1s ) = (2s ·s 𝐴))
27	25, 21, 26	sylancl 592	. . . . . 6 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) -s 1s ) = (2s ·s 𝐴))
28	27	sneqd 4574	. . . . 5 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) -s 1s )} = {(2s ·s 𝐴)})
29		1p1e2s 28433	. . . . . . . . 9 ⊢ ( 1s +s 1s ) = 2s
30		2no 28436	. . . . . . . . . 10 ⊢ 2s ∈ No
31		mulsrid 28130	. . . . . . . . . 10 ⊢ (2s ∈ No → (2s ·s 1s ) = 2s)
32	30, 31	ax-mp 5	. . . . . . . . 9 ⊢ (2s ·s 1s ) = 2s
33	29, 32	eqtr4i 2766	. . . . . . . 8 ⊢ ( 1s +s 1s ) = (2s ·s 1s )
34	33	oveq2i 7374	. . . . . . 7 ⊢ ((2s ·s 𝐴) +s ( 1s +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s ))
35	25, 22, 22	addsassd 28023	. . . . . . 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 28173	. . . . . . 7 ⊢ (𝜑 → (2s ·s (𝐴 +s 1s )) = ((2s ·s 𝐴) +s (2s ·s 1s )))
38	34, 35, 37	3eqtr4a 2801	. . . . . 6 ⊢ (𝜑 → (((2s ·s 𝐴) +s 1s ) +s 1s ) = (2s ·s (𝐴 +s 1s )))
39	38	sneqd 4574	. . . . 5 ⊢ (𝜑 → {(((2s ·s 𝐴) +s 1s ) +s 1s )} = {(2s ·s (𝐴 +s 1s ))})
40	28, 39	oveq12d 7381	. . . 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 2784	. . 3 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s (𝐴 +s 1s ))}) = (𝐴 +s (𝐴 +s 1s )))
42	2, 6, 7, 8, 41	pw2cut 28477	. 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 7378	. 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 2778	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 1547   ∈ wcel 2119  {csn 4562  (class class class)co 7363   No csur 27628   |s ccuts 27776   1_s c1s 27823   +_s cadds 27976   -_s csubs 28037   ·_s cmuls 28123   /_su cdivs 28204  ℕ_0scn0s 28329  ℕ_scnns 28330  ℤ_sczs 28395  2_sc2s 28427  ↑_scexps 28429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-dc 10366

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-oadd 8406  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-1s 27825  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec 27955  df-norec2 27966  df-adds 27977  df-negs 28038  df-subs 28039  df-muls 28124  df-divs 28205  df-seqs 28301  df-n0s 28331  df-nns 28332  df-zs 28396  df-2s 28428  df-exps 28430

This theorem is referenced by:  bdaypw2n0bndlem  28480  bdayfinbndlem1  28484