Theorem expsp1 28345

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 28280	. 2 ⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ))
2		1sno 27764	. . . . . . 7 ⊢ 1s ∈ No
3	2	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → 1s ∈ No )
4		dfnns2 28290	. . . . . . 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 28234	. . . . 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 28271	. . . . . . 7 ⊢ (𝑁 ∈ ℕs → (𝑁 +s 1s ) ∈ ℕs)
9		fvconst2g 7131	. . . . . . 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 7357	. . . . 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 2765	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
13		expsnnval 28342	. . . . 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 28342	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s𝑁) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁))
16	15	oveq1d 7356	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → ((𝐴↑s𝑁) ·s 𝐴) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
17	12, 14, 16	3eqtr4d 2775	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
18		mulslid 28074	. . . . 5 ⊢ (𝐴 ∈ No → ( 1s ·s 𝐴) = 𝐴)
19	18	adantr 480	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → ( 1s ·s 𝐴) = 𝐴)
20		oveq2 7349	. . . . . 6 ⊢ (𝑁 = 0s → (𝐴↑s𝑁) = (𝐴↑s 0s ))
21		exps0 28343	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
22	20, 21	sylan9eqr 2787	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s𝑁) = 1s )
23	22	oveq1d 7356	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → ((𝐴↑s𝑁) ·s 𝐴) = ( 1s ·s 𝐴))
24		oveq1 7348	. . . . . . 7 ⊢ (𝑁 = 0s → (𝑁 +s 1s ) = ( 0s +s 1s ))
25		addslid 27904	. . . . . . . 8 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
26	2, 25	ax-mp 5	. . . . . . 7 ⊢ ( 0s +s 1s ) = 1s
27	24, 26	eqtrdi 2781	. . . . . 6 ⊢ (𝑁 = 0s → (𝑁 +s 1s ) = 1s )
28	27	oveq2d 7357	. . . . 5 ⊢ (𝑁 = 0s → (𝐴↑s(𝑁 +s 1s )) = (𝐴↑s 1s ))
29		exps1 28344	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴↑s 1s ) = 𝐴)
30	28, 29	sylan9eqr 2787	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s(𝑁 +s 1s )) = 𝐴)
31	19, 23, 30	3eqtr4rd 2776	. . 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 1541   ∈ wcel 2110  Vcvv 3434  {csn 4574   ↦ cmpt 5170   × cxp 5612   “ cima 5617  ‘cfv 6477  (class class class)co 7341  ωcom 7791  reccrdg 8323   No csur 27571   0_s c0s 27759   1_s c1s 27760   +_s cadds 27895   ·_s cmuls 28038  seq_scseqs 28206  ℕ_0scnn0s 28235  ℕ_scnns 28236  ↑_scexps 28328

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-ot 4583  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-oadd 8384  df-nadd 8576  df-no 27574  df-slt 27575  df-bday 27576  df-sle 27677  df-sslt 27714  df-scut 27716  df-0s 27761  df-1s 27762  df-made 27781  df-old 27782  df-left 27784  df-right 27785  df-norec 27874  df-norec2 27885  df-adds 27896  df-negs 27956  df-subs 27957  df-muls 28039  df-seqs 28207  df-n0s 28237  df-nns 28238  df-zs 28296  df-exps 28329

This theorem is referenced by:  expscllem  28346  expadds  28351  expsne0  28352  expsgt0  28353  pw2recs  28354  pw2cut  28373  zs12zodd  28385  zs12bday  28387