Theorem expsval 28629

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 2763	. . . . . 6 ⊢ (𝑥 = 𝐴 → 1s = 1s )
2		eqidd 2763	. . . . . 6 ⊢ (𝑥 = 𝐴 → ·s = ·s )
3		sneq 4598	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	xpeq2d 5690	. . . . . 6 ⊢ (𝑥 = 𝐴 → (ℕs × {𝑥}) = (ℕs × {𝐴}))
5	1, 2, 4	seqseq123d 28490	. . . . 5 ⊢ (𝑥 = 𝐴 → seqs 1s ( ·s , (ℕs × {𝑥})) = seqs 1s ( ·s , (ℕs × {𝐴})))
6	5	fveq1d 6883	. . . 4 ⊢ (𝑥 = 𝐴 → (seqs 1s ( ·s , (ℕs × {𝑥}))‘𝑦) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦))
7	5	fveq1d 6883	. . . . 5 ⊢ (𝑥 = 𝐴 → (seqs 1s ( ·s , (ℕs × {𝑥}))‘( -us ‘𝑦)) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦)))
8	7	oveq2d 7428	. . . 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 4508	. . 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 4507	. 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 2766	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = 0s ↔ 𝐵 = 0s ))
12		breq2 5112	. . . 4 ⊢ (𝑦 = 𝐵 → ( 0s <s 𝑦 ↔ 0s <s 𝐵))
13		fveq2 6881	. . . 4 ⊢ (𝑦 = 𝐵 → (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵))
14		2fveq3 6886	. . . . 5 ⊢ (𝑦 = 𝐵 → (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦)) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵)))
15	14	oveq2d 7428	. . . 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 4515	. . 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 4513	. 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 28617	. 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 28014	. . . 4 ⊢ 1s ∈ No
20	19	elexi 3476	. . 3 ⊢ 1s ∈ V
21		fvex 6894	. . . 4 ⊢ (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵) ∈ V
22		ovex 7445	. . . 4 ⊢ ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵))) ∈ V
23	21, 22	ifex 4537	. . 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 4537	. 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 7572	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))))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ifcif 4486  {csn 4588   class class class wbr 5108   × cxp 5658  ‘cfv 6536  (class class class)co 7412   No csur 27815   <s clts 27816   0_s c0s 28009   1_s c1s 28010   -u_s cnegs 28223   ·_s cmuls 28310   /_su cdivs 28391  seq_scseqs 28487  ℕ_scnns 28517  ℤ_sczs 28582  ↑_scexps 28616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-no 27818  df-lts 27819  df-bday 27820  df-slts 27962  df-cuts 27964  df-0s 28011  df-1s 28012  df-seqs 28488  df-exps 28617

This theorem is used by:  expnnsval  28630  exps0  28631