Theorem expsp1 28845

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 28777	. 2 ⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ))
2		1no 28226	. . . . . . 7 ⊢ 1s ∈ No
3	2	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → 1s ∈ No)
4		dfnns2 28788	. . . . . . 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 490	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → 𝑁 ∈ ℕs)
7	3, 5, 6	seqsp1 28727	. . . . 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 28766	. . . . . . 7 ⊢ (𝑁 ∈ ℕs → (𝑁 +s 1s ) ∈ ℕs)
9		fvconst2g 7231	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ (𝑁 +s 1s ) ∈ ℕs) → ((ℕs × {𝐴})‘(𝑁 +s 1s )) = 𝐴)
10	8, 9	sylan2 605	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → ((ℕs × {𝐴})‘(𝑁 +s 1s )) = 𝐴)
11	10	oveq2d 7459	. . . . 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 2798	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
13		expnnsval 28842	. . . . 5 ⊢ ((𝐴 ∈ No ∧ (𝑁 +s 1s ) ∈ ℕs) → (𝐴↑s(𝑁 +s 1s )) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )))
14	8, 13	sylan2 605	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s(𝑁 +s 1s )) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )))
15		expnnsval 28842	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s𝑁) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁))
16	15	oveq1d 7458	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → ((𝐴↑s𝑁) ·s 𝐴) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
17	12, 14, 16	3eqtr4d 2808	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
18		mulslid 28558	. . . . 5 ⊢ (𝐴 ∈ No → ( 1s ·s 𝐴) = 𝐴)
19	18	adantr 486	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → ( 1s ·s 𝐴) = 𝐴)
20		oveq2 7451	. . . . . 6 ⊢ (𝑁 = 0s → (𝐴↑s𝑁) = (𝐴↑s 0s ))
21		exps0 28843	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
22	20, 21	sylan9eqr 2820	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s𝑁) = 1s )
23	22	oveq1d 7458	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → ((𝐴↑s𝑁) ·s 𝐴) = ( 1s ·s 𝐴))
24		oveq1 7450	. . . . . . 7 ⊢ (𝑁 = 0s → (𝑁 +s 1s ) = ( 0s +s 1s ))
25		addslid 28384	. . . . . . . 8 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
26	2, 25	ax-mp 5	. . . . . . 7 ⊢ ( 0s +s 1s ) = 1s
27	24, 26	eqtrdi 2814	. . . . . 6 ⊢ (𝑁 = 0s → (𝑁 +s 1s ) = 1s )
28	27	oveq2d 7459	. . . . 5 ⊢ (𝑁 = 0s → (𝐴↑s(𝑁 +s 1s )) = (𝐴↑s 1s ))
29		exps1 28844	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴↑s 1s ) = 𝐴)
30	28, 29	sylan9eqr 2820	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s(𝑁 +s 1s )) = 𝐴)
31	19, 23, 30	3eqtr4rd 2809	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
32	17, 31	jaodan 972	. 2 ⊢ ((𝐴 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s )) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
33	1, 32	sylan2b 606	1 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  Vcvv 3454  {csn 4588   ↦ cmpt 5189   × cxp 5653   “ cima 5658  ‘cfv 6558  (class class class)co 7443  ωcom 7901  reccrdg 8445  Nocsur 28027   0_s c0s 28221   1_s c1s 28222   +_s cadds 28375   ·_s cmuls 28522  seq_scseqs 28699  ℕ_0scn0s 28728  ℕ_scnns 28729  ↑_scexps 28828

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-1s 28224  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-norec2 28365  df-adds 28376  df-negs 28437  df-subs 28438  df-muls 28523  df-seqs 28700  df-n0s 28730  df-nns 28731  df-zs 28795  df-exps 28829

This theorem is used by:  expscllem  28846  expadds  28851  expsne0  28852  expsgt0  28853  pw2recs  28854  pw2cut  28876  bdaypw2n0bndlem  28879  bdayfinbndlem1  28883  z12zsodd  28898