Theorem expsval 28688

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 4597	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	xpeq2d 5689	. . . . . 6 ⊢ (𝑥 = 𝐴 → (ℕs × {𝑥}) = (ℕs × {𝐴}))
5	1, 2, 4	seqseq123d 28549	. . . . 5 ⊢ (𝑥 = 𝐴 → seqs 1s ( ·s , (ℕs × {𝑥})) = seqs 1s ( ·s , (ℕs × {𝐴})))
6	5	fveq1d 6884	. . . 4 ⊢ (𝑥 = 𝐴 → (seqs 1s ( ·s , (ℕs × {𝑥}))‘𝑦) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦))
7	5	fveq1d 6884	. . . . 5 ⊢ (𝑥 = 𝐴 → (seqs 1s ( ·s , (ℕs × {𝑥}))‘( -us ‘𝑦)) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦)))
8	7	oveq2d 7432	. . . 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 4507	. . 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 4506	. 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 5111	. . . 4 ⊢ (𝑦 = 𝐵 → ( 0s <s 𝑦 ↔ 0s <s 𝐵))
13		fveq2 6882	. . . 4 ⊢ (𝑦 = 𝐵 → (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵))
14		2fveq3 6887	. . . . 5 ⊢ (𝑦 = 𝐵 → (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦)) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵)))
15	14	oveq2d 7432	. . . 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 4514	. . 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 4512	. 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 28676	. 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 28073	. . . 4 ⊢ 1s ∈ No
20	19	elexi 3475	. . 3 ⊢ 1s ∈ V
21		fvex 6895	. . . 4 ⊢ (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵) ∈ V
22		ovex 7449	. . . 4 ⊢ ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵))) ∈ V
23	21, 22	ifex 4536	. . 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 4536	. 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 7576	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 401   = wceq 1570   ∈ wcel 2145  ifcif 4485  {csn 4587   class class class wbr 5107   × cxp 5657  ‘cfv 6537  (class class class)co 7416   No csur 27874   <s clts 27875   0_s c0s 28068   1_s c1s 28069   -u_s cnegs 28282   ·_s cmuls 28369   /_su cdivs 28450  seq_scseqs 28546  ℕ_scnns 28576  ℤ_sczs 28641  ↑_scexps 28675

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2100  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 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-no 27877  df-lts 27878  df-bday 27879  df-slts 28021  df-cuts 28023  df-0s 28070  df-1s 28071  df-seqs 28547  df-exps 28676

This theorem is used by:  expnnsval  28689  exps0  28690