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Theorem expsp1 28413
Description: Value of a surreal number raised to a non-negative integer power plus one. (Contributed by Scott Fenton, 6-Aug-2025.)
Assertion
Ref Expression
expsp1 ((𝐴 No 𝑁 ∈ ℕ0s) → (𝐴s(𝑁 +s 1s )) = ((𝐴s𝑁) ·s 𝐴))

Proof of Theorem expsp1
StepHypRef Expression
1 eln0s 28359 . 2 (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs𝑁 = 0s ))
2 1sno 27873 . . . . . . 7 1s No
32a1i 11 . . . . . 6 ((𝐴 No 𝑁 ∈ ℕs) → 1s No )
4 dfnns2 28363 . . . . . . 7 s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
54a1i 11 . . . . . 6 ((𝐴 No 𝑁 ∈ ℕs) → ℕs = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω))
6 simpr 484 . . . . . 6 ((𝐴 No 𝑁 ∈ ℕs) → 𝑁 ∈ ℕs)
73, 5, 6seqsp1 28318 . . . . 5 ((𝐴 No 𝑁 ∈ ℕs) → (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s ((ℕs × {𝐴})‘(𝑁 +s 1s ))))
8 peano2nns 28354 . . . . . . 7 (𝑁 ∈ ℕs → (𝑁 +s 1s ) ∈ ℕs)
9 fvconst2g 7223 . . . . . . 7 ((𝐴 No ∧ (𝑁 +s 1s ) ∈ ℕs) → ((ℕs × {𝐴})‘(𝑁 +s 1s )) = 𝐴)
108, 9sylan2 593 . . . . . 6 ((𝐴 No 𝑁 ∈ ℕs) → ((ℕs × {𝐴})‘(𝑁 +s 1s )) = 𝐴)
1110oveq2d 7448 . . . . 5 ((𝐴 No 𝑁 ∈ ℕs) → ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s ((ℕs × {𝐴})‘(𝑁 +s 1s ))) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
127, 11eqtrd 2776 . . . 4 ((𝐴 No 𝑁 ∈ ℕs) → (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
13 expsnnval 28410 . . . . 5 ((𝐴 No ∧ (𝑁 +s 1s ) ∈ ℕs) → (𝐴s(𝑁 +s 1s )) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )))
148, 13sylan2 593 . . . 4 ((𝐴 No 𝑁 ∈ ℕs) → (𝐴s(𝑁 +s 1s )) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )))
15 expsnnval 28410 . . . . 5 ((𝐴 No 𝑁 ∈ ℕs) → (𝐴s𝑁) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁))
1615oveq1d 7447 . . . 4 ((𝐴 No 𝑁 ∈ ℕs) → ((𝐴s𝑁) ·s 𝐴) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
1712, 14, 163eqtr4d 2786 . . 3 ((𝐴 No 𝑁 ∈ ℕs) → (𝐴s(𝑁 +s 1s )) = ((𝐴s𝑁) ·s 𝐴))
18 mulslid 28169 . . . . 5 (𝐴 No → ( 1s ·s 𝐴) = 𝐴)
1918adantr 480 . . . 4 ((𝐴 No 𝑁 = 0s ) → ( 1s ·s 𝐴) = 𝐴)
20 oveq2 7440 . . . . . 6 (𝑁 = 0s → (𝐴s𝑁) = (𝐴s 0s ))
21 exps0 28411 . . . . . 6 (𝐴 No → (𝐴s 0s ) = 1s )
2220, 21sylan9eqr 2798 . . . . 5 ((𝐴 No 𝑁 = 0s ) → (𝐴s𝑁) = 1s )
2322oveq1d 7447 . . . 4 ((𝐴 No 𝑁 = 0s ) → ((𝐴s𝑁) ·s 𝐴) = ( 1s ·s 𝐴))
24 oveq1 7439 . . . . . . 7 (𝑁 = 0s → (𝑁 +s 1s ) = ( 0s +s 1s ))
25 addslid 28002 . . . . . . . 8 ( 1s No → ( 0s +s 1s ) = 1s )
262, 25ax-mp 5 . . . . . . 7 ( 0s +s 1s ) = 1s
2724, 26eqtrdi 2792 . . . . . 6 (𝑁 = 0s → (𝑁 +s 1s ) = 1s )
2827oveq2d 7448 . . . . 5 (𝑁 = 0s → (𝐴s(𝑁 +s 1s )) = (𝐴s 1s ))
29 exps1 28412 . . . . 5 (𝐴 No → (𝐴s 1s ) = 𝐴)
3028, 29sylan9eqr 2798 . . . 4 ((𝐴 No 𝑁 = 0s ) → (𝐴s(𝑁 +s 1s )) = 𝐴)
3119, 23, 303eqtr4rd 2787 . . 3 ((𝐴 No 𝑁 = 0s ) → (𝐴s(𝑁 +s 1s )) = ((𝐴s𝑁) ·s 𝐴))
3217, 31jaodan 959 . 2 ((𝐴 No ∧ (𝑁 ∈ ℕs𝑁 = 0s )) → (𝐴s(𝑁 +s 1s )) = ((𝐴s𝑁) ·s 𝐴))
331, 32sylan2b 594 1 ((𝐴 No 𝑁 ∈ ℕ0s) → (𝐴s(𝑁 +s 1s )) = ((𝐴s𝑁) ·s 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1539  wcel 2107  Vcvv 3479  {csn 4625  cmpt 5224   × cxp 5682  cima 5687  cfv 6560  (class class class)co 7432  ωcom 7888  reccrdg 8450   No csur 27685   0s c0s 27868   1s c1s 27869   +s cadds 27993   ·s cmuls 28133  seqscseqs 28290  0scnn0s 28319  scnns 28320  scexps 28397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-1s 27871  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec 27972  df-norec2 27983  df-adds 27994  df-negs 28054  df-subs 28055  df-muls 28134  df-seqs 28291  df-n0s 28321  df-nns 28322  df-zs 28366  df-exps 28398
This theorem is referenced by:  expscl  28414  expsne0  28415  expsgt0  28416  cutpw2  28418  pw2cut  28421  zs12bday  28425
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