Theorem expsp1 28339

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) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))

Proof of Theorem expsp1

Step	Hyp	Ref	Expression
1		eln0s 28274	. 2 ⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ))
2		1sno 27759	. . . . . . 7 ⊢ 1s ∈ No
3	2	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → 1s ∈ No )
4		dfnns2 28284	. . . . . . 7 ⊢ ℕs = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
5	4	a1i 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)
7	3, 5, 6	seqsp1 28228	. . . . 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 28265	. . . . . . 7 ⊢ (𝑁 ∈ ℕs → (𝑁 +s 1s ) ∈ ℕs)
9		fvconst2g 7142	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ (𝑁 +s 1s ) ∈ ℕs) → ((ℕs × {𝐴})‘(𝑁 +s 1s )) = 𝐴)
10	8, 9	sylan2 593	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → ((ℕs × {𝐴})‘(𝑁 +s 1s )) = 𝐴)
11	10	oveq2d 7369	. . . . 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))
12	7, 11	eqtrd 2764	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
13		expsnnval 28336	. . . . 5 ⊢ ((𝐴 ∈ No ∧ (𝑁 +s 1s ) ∈ ℕs) → (𝐴↑s(𝑁 +s 1s )) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )))
14	8, 13	sylan2 593	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s(𝑁 +s 1s )) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )))
15		expsnnval 28336	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s𝑁) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁))
16	15	oveq1d 7368	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → ((𝐴↑s𝑁) ·s 𝐴) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
17	12, 14, 16	3eqtr4d 2774	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
18		mulslid 28068	. . . . 5 ⊢ (𝐴 ∈ No → ( 1s ·s 𝐴) = 𝐴)
19	18	adantr 480	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → ( 1s ·s 𝐴) = 𝐴)
20		oveq2 7361	. . . . . 6 ⊢ (𝑁 = 0s → (𝐴↑s𝑁) = (𝐴↑s 0s ))
21		exps0 28337	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
22	20, 21	sylan9eqr 2786	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s𝑁) = 1s )
23	22	oveq1d 7368	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → ((𝐴↑s𝑁) ·s 𝐴) = ( 1s ·s 𝐴))
24		oveq1 7360	. . . . . . 7 ⊢ (𝑁 = 0s → (𝑁 +s 1s ) = ( 0s +s 1s ))
25		addslid 27898	. . . . . . . 8 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
26	2, 25	ax-mp 5	. . . . . . 7 ⊢ ( 0s +s 1s ) = 1s
27	24, 26	eqtrdi 2780	. . . . . 6 ⊢ (𝑁 = 0s → (𝑁 +s 1s ) = 1s )
28	27	oveq2d 7369	. . . . 5 ⊢ (𝑁 = 0s → (𝐴↑s(𝑁 +s 1s )) = (𝐴↑s 1s ))
29		exps1 28338	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴↑s 1s ) = 𝐴)
30	28, 29	sylan9eqr 2786	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s(𝑁 +s 1s )) = 𝐴)
31	19, 23, 30	3eqtr4rd 2775	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
32	17, 31	jaodan 959	. 2 ⊢ ((𝐴 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s )) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
33	1, 32	sylan2b 594	1 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  Vcvv 3438  {csn 4579   ↦ cmpt 5176   × cxp 5621   “ cima 5626  ‘cfv 6486  (class class class)co 7353  ωcom 7806  reccrdg 8338   No csur 27567   0_s c0s 27754   1_s c1s 27755   +_s cadds 27889   ·_s cmuls 28032  seq_scseqs 28200  ℕ_0scnn0s 28229  ℕ_scnns 28230  ↑_scexps 28322

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 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-nadd 8591  df-no 27570  df-slt 27571  df-bday 27572  df-sle 27673  df-sslt 27710  df-scut 27712  df-0s 27756  df-1s 27757  df-made 27775  df-old 27776  df-left 27778  df-right 27779  df-norec 27868  df-norec2 27879  df-adds 27890  df-negs 27950  df-subs 27951  df-muls 28033  df-seqs 28201  df-n0s 28231  df-nns 28232  df-zs 28290  df-exps 28323

This theorem is referenced by:  expscllem  28340  expadds  28345  expsne0  28346  expsgt0  28347  pw2recs  28348  pw2cut  28366  zs12zodd  28377  zs12bday  28379