Theorem expsp1 28437

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 28369	. 2 ⊢ (𝑁 ∈ ℕ0s ↔ (𝑁 ∈ ℕs ∨ 𝑁 = 0s ))
2		1no 27818	. . . . . . 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 28319	. . . . 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 28358	. . . . . . 7 ⊢ (𝑁 ∈ ℕs → (𝑁 +s 1s ) ∈ ℕs)
9		fvconst2g 7158	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ (𝑁 +s 1s ) ∈ ℕs) → ((ℕs × {𝐴})‘(𝑁 +s 1s )) = 𝐴)
10	8, 9	sylan2 594	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → ((ℕs × {𝐴})‘(𝑁 +s 1s )) = 𝐴)
11	10	oveq2d 7384	. . . . 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 2772	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
13		expnnsval 28434	. . . . 5 ⊢ ((𝐴 ∈ No ∧ (𝑁 +s 1s ) ∈ ℕs) → (𝐴↑s(𝑁 +s 1s )) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )))
14	8, 13	sylan2 594	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s(𝑁 +s 1s )) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘(𝑁 +s 1s )))
15		expnnsval 28434	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s𝑁) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁))
16	15	oveq1d 7383	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → ((𝐴↑s𝑁) ·s 𝐴) = ((seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑁) ·s 𝐴))
17	12, 14, 16	3eqtr4d 2782	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕs) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
18		mulslid 28150	. . . . 5 ⊢ (𝐴 ∈ No → ( 1s ·s 𝐴) = 𝐴)
19	18	adantr 480	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → ( 1s ·s 𝐴) = 𝐴)
20		oveq2 7376	. . . . . 6 ⊢ (𝑁 = 0s → (𝐴↑s𝑁) = (𝐴↑s 0s ))
21		exps0 28435	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
22	20, 21	sylan9eqr 2794	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s𝑁) = 1s )
23	22	oveq1d 7383	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → ((𝐴↑s𝑁) ·s 𝐴) = ( 1s ·s 𝐴))
24		oveq1 7375	. . . . . . 7 ⊢ (𝑁 = 0s → (𝑁 +s 1s ) = ( 0s +s 1s ))
25		addslid 27976	. . . . . . . 8 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
26	2, 25	ax-mp 5	. . . . . . 7 ⊢ ( 0s +s 1s ) = 1s
27	24, 26	eqtrdi 2788	. . . . . 6 ⊢ (𝑁 = 0s → (𝑁 +s 1s ) = 1s )
28	27	oveq2d 7384	. . . . 5 ⊢ (𝑁 = 0s → (𝐴↑s(𝑁 +s 1s )) = (𝐴↑s 1s ))
29		exps1 28436	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴↑s 1s ) = 𝐴)
30	28, 29	sylan9eqr 2794	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s(𝑁 +s 1s )) = 𝐴)
31	19, 23, 30	3eqtr4rd 2783	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝑁 = 0s ) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
32	17, 31	jaodan 960	. 2 ⊢ ((𝐴 ∈ No ∧ (𝑁 ∈ ℕs ∨ 𝑁 = 0s )) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))
33	1, 32	sylan2b 595	1 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s(𝑁 +s 1s )) = ((𝐴↑s𝑁) ·s 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  Vcvv 3442  {csn 4582   ↦ cmpt 5181   × cxp 5630   “ cima 5635  ‘cfv 6500  (class class class)co 7368  ωcom 7818  reccrdg 8350   No csur 27619   0_s c0s 27813   1_s c1s 27814   +_s cadds 27967   ·_s cmuls 28114  seq_scseqs 28291  ℕ_0scn0s 28320  ℕ_scnns 28321  ↑_scexps 28420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-nadd 8604  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-1s 27816  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029  df-subs 28030  df-muls 28115  df-seqs 28292  df-n0s 28322  df-nns 28323  df-zs 28387  df-exps 28421

This theorem is referenced by:  expscllem  28438  expadds  28443  expsne0  28444  expsgt0  28445  pw2recs  28446  pw2cut  28468  bdaypw2n0bndlem  28471  bdayfinbndlem1  28475  z12zsodd  28490