Theorem expsval 28426

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 2741	. . . . . 6 ⊢ (𝑥 = 𝐴 → 1s = 1s )
2		eqidd 2741	. . . . . 6 ⊢ (𝑥 = 𝐴 → ·s = ·s )
3		sneq 4658	. . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
4	3	xpeq2d 5730	. . . . . 6 ⊢ (𝑥 = 𝐴 → (ℕs × {𝑥}) = (ℕs × {𝐴}))
5	1, 2, 4	seqseq123d 28310	. . . . 5 ⊢ (𝑥 = 𝐴 → seqs 1s ( ·s , (ℕs × {𝑥})) = seqs 1s ( ·s , (ℕs × {𝐴})))
6	5	fveq1d 6922	. . . 4 ⊢ (𝑥 = 𝐴 → (seqs 1s ( ·s , (ℕs × {𝑥}))‘𝑦) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦))
7	5	fveq1d 6922	. . . . 5 ⊢ (𝑥 = 𝐴 → (seqs 1s ( ·s , (ℕs × {𝑥}))‘( -us ‘𝑦)) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦)))
8	7	oveq2d 7464	. . . 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 4569	. . 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 4568	. 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 2744	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = 0s ↔ 𝐵 = 0s ))
12		breq2 5170	. . . 4 ⊢ (𝑦 = 𝐵 → ( 0s <s 𝑦 ↔ 0s <s 𝐵))
13		fveq2 6920	. . . 4 ⊢ (𝑦 = 𝐵 → (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝑦) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵))
14		2fveq3 6925	. . . . 5 ⊢ (𝑦 = 𝐵 → (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝑦)) = (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵)))
15	14	oveq2d 7464	. . . 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 4576	. . 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 4574	. 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 28415	. 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		1sno 27890	. . . 4 ⊢ 1s ∈ No
20	19	elexi 3511	. . 3 ⊢ 1s ∈ V
21		fvex 6933	. . . 4 ⊢ (seqs 1s ( ·s , (ℕs × {𝐴}))‘𝐵) ∈ V
22		ovex 7481	. . . 4 ⊢ ( 1s /su (seqs 1s ( ·s , (ℕs × {𝐴}))‘( -us ‘𝐵))) ∈ V
23	21, 22	ifex 4598	. . 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 4598	. 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 7610	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 1537   ∈ wcel 2108  ifcif 4548  {csn 4648   class class class wbr 5166   × cxp 5698  ‘cfv 6573  (class class class)co 7448   No csur 27702   <s cslt 27703   0_s c0s 27885   1_s c1s 27886   -u_s cnegs 28069   ·_s cmuls 28150   /_su cdivs 28231  seq_scseqs 28307  ℕ_scnns 28337  ℤ_sczs 28382  ↑_scexps 28414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-seqs 28308  df-exps 28415

This theorem is referenced by:  expsnnval  28427  exps0  28428