Theorem expsval 28425

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 4591	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	xpeq2d 5655	. . . . . 6 ⊢ (𝑥 = 𝐴 → (ℕs × {𝑥}) = (ℕs × {𝐴}))
5	1, 2, 4	seqseq123d 28286	. . . . 5 ⊢ (𝑥 = 𝐴 → seqs 1s ( ·s , (ℕs × {𝑥})) = seqs 1s ( ·s , (ℕs × {𝐴})))
6	5	fveq1d 6837	. . . 4 ⊢ (𝑥 = 𝐴 → (seqs 1s ( ·s , (ℕs × {𝑥}))‘𝑦) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦))
7	5	fveq1d 6837	. . . . 5 ⊢ (𝑥 = 𝐴 → (seqs 1s ( ·s , (ℕs × {𝑥}))‘( -us ‘𝑦)) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦)))
8	7	oveq2d 7376	. . . 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 4502	. . 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 4501	. 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 5103	. . . 4 ⊢ (𝑦 = 𝐵 → ( 0s <s 𝑦 ↔ 0s <s 𝐵))
13		fveq2 6835	. . . 4 ⊢ (𝑦 = 𝐵 → (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵))
14		2fveq3 6840	. . . . 5 ⊢ (𝑦 = 𝐵 → (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦)) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵)))
15	14	oveq2d 7376	. . . 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 4509	. . 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 4507	. 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 28413	. 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 27810	. . . 4 ⊢ 1s ∈ No
20	19	elexi 3464	. . 3 ⊢ 1s ∈ V
21		fvex 6848	. . . 4 ⊢ (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵) ∈ V
22		ovex 7393	. . . 4 ⊢ ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵))) ∈ V
23	21, 22	ifex 4531	. . 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 4531	. 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 7520	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 4480  {csn 4581   class class class wbr 5099   × cxp 5623  ‘cfv 6493  (class class class)co 7360   No csur 27611   <s clts 27612   0_s c0s 27805   1_s c1s 27806   -u_s cnegs 28019   ·_s cmuls 28106   /_su cdivs 28187  seq_scseqs 28283  ℕ_scnns 28313  ℤ_sczs 28378  ↑_scexps 28412

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

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 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-no 27614  df-lts 27615  df-bday 27616  df-slts 27758  df-cuts 27760  df-0s 27807  df-1s 27808  df-seqs 28284  df-exps 28413

This theorem is referenced by:  expnnsval  28426  exps0  28427