Theorem expsval 28438

Description: The value of surreal exponentiation. (Contributed by Scott Fenton, 24-Jul-2025.)

Assertion

Ref	Expression
expsval	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ ℤs) → (𝐴↑s𝐵) = if(𝐵 = 0s , 1s , if( 0s <s 𝐵, (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵))))))

Proof of Theorem expsval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2738	. . . . . 6 ⊢ (𝑥 = 𝐴 → 1s = 1s )
2		eqidd 2738	. . . . . 6 ⊢ (𝑥 = 𝐴 → ·s = ·s )
3		sneq 4592	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	xpeq2d 5664	. . . . . 6 ⊢ (𝑥 = 𝐴 → (ℕs × {𝑥}) = (ℕs × {𝐴}))
5	1, 2, 4	seqseq123d 28299	. . . . 5 ⊢ (𝑥 = 𝐴 → seqs 1s ( ·s , (ℕs × {𝑥})) = seqs 1s ( ·s , (ℕs × {𝐴})))
6	5	fveq1d 6846	. . . 4 ⊢ (𝑥 = 𝐴 → (seqs 1s ( ·s , (ℕs × {𝑥}))‘𝑦) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦))
7	5	fveq1d 6846	. . . . 5 ⊢ (𝑥 = 𝐴 → (seqs 1s ( ·s , (ℕs × {𝑥}))‘( -us ‘𝑦)) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦)))
8	7	oveq2d 7386	. . . 4 ⊢ (𝑥 = 𝐴 → ( 1s /su (seqs 1s ( ·s , (ℕs × {𝑥}))‘( -us ‘𝑦))) = ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦))))
9	6, 8	ifeq12d 4503	. . 3 ⊢ (𝑥 = 𝐴 → if( 0s <s 𝑦, (seqs 1s ( ·s , (ℕs × {𝑥}))‘𝑦), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝑥}))‘( -us ‘𝑦)))) = if( 0s <s 𝑦, (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦)))))
10	9	ifeq2d 4502	. 2 ⊢ (𝑥 = 𝐴 → if(𝑦 = 0s , 1s , if( 0s <s 𝑦, (seqs 1s ( ·s , (ℕs × {𝑥}))‘𝑦), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝑥}))‘( -us ‘𝑦))))) = if(𝑦 = 0s , 1s , if( 0s <s 𝑦, (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦))))))
11		eqeq1 2741	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = 0s ↔ 𝐵 = 0s ))
12		breq2 5104	. . . 4 ⊢ (𝑦 = 𝐵 → ( 0s <s 𝑦 ↔ 0s <s 𝐵))
13		fveq2 6844	. . . 4 ⊢ (𝑦 = 𝐵 → (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵))
14		2fveq3 6849	. . . . 5 ⊢ (𝑦 = 𝐵 → (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦)) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵)))
15	14	oveq2d 7386	. . . 4 ⊢ (𝑦 = 𝐵 → ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦))) = ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵))))
16	12, 13, 15	ifbieq12d 4510	. . 3 ⊢ (𝑦 = 𝐵 → if( 0s <s 𝑦, (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦)))) = if( 0s <s 𝐵, (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵)))))
17	11, 16	ifbieq2d 4508	. 2 ⊢ (𝑦 = 𝐵 → if(𝑦 = 0s , 1s , if( 0s <s 𝑦, (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦))))) = if(𝐵 = 0s , 1s , if( 0s <s 𝐵, (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵))))))
18		df-exps 28426	. 2 ⊢ ↑s = (𝑥 ∈ No , 𝑦 ∈ ℤs ↦ if(𝑦 = 0s , 1s , if( 0s <s 𝑦, (seqs 1s ( ·s , (ℕs × {𝑥}))‘𝑦), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝑥}))‘( -us ‘𝑦))))))
19		1no 27823	. . . 4 ⊢ 1s ∈ No
20	19	elexi 3465	. . 3 ⊢ 1s ∈ V
21		fvex 6857	. . . 4 ⊢ (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵) ∈ V
22		ovex 7403	. . . 4 ⊢ ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵))) ∈ V
23	21, 22	ifex 4532	. . 3 ⊢ if( 0s <s 𝐵, (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵)))) ∈ V
24	20, 23	ifex 4532	. 2 ⊢ if(𝐵 = 0s , 1s , if( 0s <s 𝐵, (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵))))) ∈ V
25	10, 17, 18, 24	ovmpo 7530	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ ℤs) → (𝐴↑s𝐵) = if(𝐵 = 0s , 1s , if( 0s <s 𝐵, (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵), ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵))))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ifcif 4481  {csn 4582   class class class wbr 5100   × cxp 5632  ‘cfv 6502  (class class class)co 7370   No csur 27624   <s clts 27625   0_s c0s 27818   1_s c1s 27819   -u_s cnegs 28032   ·_s cmuls 28119   /_su cdivs 28200  seq_scseqs 28296  ℕ_scnns 28326  ℤ_sczs 28391  ↑_scexps 28425

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 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

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-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-no 27627  df-lts 27628  df-bday 27629  df-slts 27771  df-cuts 27773  df-0s 27820  df-1s 27821  df-seqs 28297  df-exps 28426

This theorem is referenced by:  expnnsval  28439  exps0  28440