Theorem expsp1 28430

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 28376	. 2 ⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ))
2		1sno 27890	. . . . . . 7 ⊢ 1s ∈ No
3	2	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → 1s ∈ No )
4		dfnns2 28380	. . . . . . 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 28335	. . . . 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 28371	. . . . . . 7 ⊢ (𝑁 ∈ ℕs → (𝑁 +s 1s ) ∈ ℕs)
9		fvconst2g 7239	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ (𝑁 +s 1s ) ∈ ℕs) → ((ℕs × {𝐴})‘(𝑁 +s 1s )) = 𝐴)
10	8, 9	sylan2 592	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → ((ℕs × {𝐴})‘(𝑁 +s 1s )) = 𝐴)
11	10	oveq2d 7464	. . . . 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 2780	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
13		expsnnval 28427	. . . . 5 ⊢ ((𝐴 ∈ No ∧ (𝑁 +s 1s ) ∈ ℕs) → (𝐴↑s(𝑁 +s 1s )) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )))
14	8, 13	sylan2 592	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s(𝑁 +s 1s )) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )))
15		expsnnval 28427	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s𝑁) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁))
16	15	oveq1d 7463	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → ((𝐴↑s𝑁) ·s 𝐴) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
17	12, 14, 16	3eqtr4d 2790	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
18		mulslid 28186	. . . . 5 ⊢ (𝐴 ∈ No → ( 1s ·s 𝐴) = 𝐴)
19	18	adantr 480	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → ( 1s ·s 𝐴) = 𝐴)
20		oveq2 7456	. . . . . 6 ⊢ (𝑁 = 0s → (𝐴↑s𝑁) = (𝐴↑s 0s ))
21		exps0 28428	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
22	20, 21	sylan9eqr 2802	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s𝑁) = 1s )
23	22	oveq1d 7463	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → ((𝐴↑s𝑁) ·s 𝐴) = ( 1s ·s 𝐴))
24		oveq1 7455	. . . . . . 7 ⊢ (𝑁 = 0s → (𝑁 +s 1s ) = ( 0s +s 1s ))
25		addslid 28019	. . . . . . . 8 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
26	2, 25	ax-mp 5	. . . . . . 7 ⊢ ( 0s +s 1s ) = 1s
27	24, 26	eqtrdi 2796	. . . . . 6 ⊢ (𝑁 = 0s → (𝑁 +s 1s ) = 1s )
28	27	oveq2d 7464	. . . . 5 ⊢ (𝑁 = 0s → (𝐴↑s(𝑁 +s 1s )) = (𝐴↑s 1s ))
29		exps1 28429	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴↑s 1s ) = 𝐴)
30	28, 29	sylan9eqr 2802	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s(𝑁 +s 1s )) = 𝐴)
31	19, 23, 30	3eqtr4rd 2791	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
32	17, 31	jaodan 958	. 2 ⊢ ((𝐴 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s )) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
33	1, 32	sylan2b 593	1 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  Vcvv 3488  {csn 4648   ↦ cmpt 5249   × cxp 5698   “ cima 5703  ‘cfv 6573  (class class class)co 7448  ωcom 7903  reccrdg 8465   No csur 27702   0_s c0s 27885   1_s c1s 27886   +_s cadds 28010   ·_s cmuls 28150  seq_scseqs 28307  ℕ_0scnn0s 28336  ℕ_scnns 28337  ↑_scexps 28414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-muls 28151  df-seqs 28308  df-n0s 28338  df-nns 28339  df-zs 28383  df-exps 28415

This theorem is referenced by:  expscl  28431  expsne0  28432  expsgt0  28433  cutpw2  28435  pw2cut  28438  zs12bday  28442