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Theorem expsp1 28427
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 28373 . 2 (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs𝑁 = 0s ))
2 1sno 27887 . . . . . . 7 1s No
32a1i 11 . . . . . 6 ((𝐴 No 𝑁 ∈ ℕs) → 1s No )
4 dfnns2 28377 . . . . . . 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 28332 . . . . 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 28368 . . . . . . 7 (𝑁 ∈ ℕs → (𝑁 +s 1s ) ∈ ℕs)
9 fvconst2g 7222 . . . . . . 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 7447 . . . . 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 2775 . . . 4 ((𝐴 No 𝑁 ∈ ℕs) → (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
13 expsnnval 28424 . . . . 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 28424 . . . . 5 ((𝐴 No 𝑁 ∈ ℕs) → (𝐴s𝑁) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁))
1615oveq1d 7446 . . . 4 ((𝐴 No 𝑁 ∈ ℕs) → ((𝐴s𝑁) ·s 𝐴) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
1712, 14, 163eqtr4d 2785 . . 3 ((𝐴 No 𝑁 ∈ ℕs) → (𝐴s(𝑁 +s 1s )) = ((𝐴s𝑁) ·s 𝐴))
18 mulslid 28183 . . . . 5 (𝐴 No → ( 1s ·s 𝐴) = 𝐴)
1918adantr 480 . . . 4 ((𝐴 No 𝑁 = 0s ) → ( 1s ·s 𝐴) = 𝐴)
20 oveq2 7439 . . . . . 6 (𝑁 = 0s → (𝐴s𝑁) = (𝐴s 0s ))
21 exps0 28425 . . . . . 6 (𝐴 No → (𝐴s 0s ) = 1s )
2220, 21sylan9eqr 2797 . . . . 5 ((𝐴 No 𝑁 = 0s ) → (𝐴s𝑁) = 1s )
2322oveq1d 7446 . . . 4 ((𝐴 No 𝑁 = 0s ) → ((𝐴s𝑁) ·s 𝐴) = ( 1s ·s 𝐴))
24 oveq1 7438 . . . . . . 7 (𝑁 = 0s → (𝑁 +s 1s ) = ( 0s +s 1s ))
25 addslid 28016 . . . . . . . 8 ( 1s No → ( 0s +s 1s ) = 1s )
262, 25ax-mp 5 . . . . . . 7 ( 0s +s 1s ) = 1s
2724, 26eqtrdi 2791 . . . . . 6 (𝑁 = 0s → (𝑁 +s 1s ) = 1s )
2827oveq2d 7447 . . . . 5 (𝑁 = 0s → (𝐴s(𝑁 +s 1s )) = (𝐴s 1s ))
29 exps1 28426 . . . . 5 (𝐴 No → (𝐴s 1s ) = 𝐴)
3028, 29sylan9eqr 2797 . . . 4 ((𝐴 No 𝑁 = 0s ) → (𝐴s(𝑁 +s 1s )) = 𝐴)
3119, 23, 303eqtr4rd 2786 . . 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 1537  wcel 2106  Vcvv 3478  {csn 4631  cmpt 5231   × cxp 5687  cima 5692  cfv 6563  (class class class)co 7431  ωcom 7887  reccrdg 8448   No csur 27699   0s c0s 27882   1s c1s 27883   +s cadds 28007   ·s cmuls 28147  seqscseqs 28304  0scnn0s 28333  scnns 28334  scexps 28411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148  df-seqs 28305  df-n0s 28335  df-nns 28336  df-zs 28380  df-exps 28412
This theorem is referenced by:  expscl  28428  expsne0  28429  expsgt0  28430  cutpw2  28432  pw2cut  28435  zs12bday  28439
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