Theorem expsne0 28699

Description: A non-negative surreal integer power is nonzero if its base is nonzero. (Contributed by Scott Fenton, 7-Aug-2025.)

Assertion

Ref	Expression
expsne0	⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ≠ 0s )

Proof of Theorem expsne0

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7424	. . . . . . . . 9 ⊢ (𝑚 = 0s → (𝐴↑s𝑚) = (𝐴↑s 0s ))
2	1	eqeq1d 2764	. . . . . . . 8 ⊢ (𝑚 = 0s → ((𝐴↑s𝑚) = 0s ↔ (𝐴↑s 0s ) = 0s ))
3	2	imbi1d 344	. . . . . . 7 ⊢ (𝑚 = 0s → (((𝐴↑s𝑚) = 0s → 𝐴 = 0s ) ↔ ((𝐴↑s 0s ) = 0s → 𝐴 = 0s )))
4	3	imbi2d 343	. . . . . 6 ⊢ (𝑚 = 0s → ((𝐴 ∈ No → ((𝐴↑s𝑚) = 0s → 𝐴 = 0s )) ↔ (𝐴 ∈ No → ((𝐴↑s 0s ) = 0s → 𝐴 = 0s ))))
5		oveq2 7424	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝐴↑s𝑚) = (𝐴↑s𝑛))
6	5	eqeq1d 2764	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝐴↑s𝑚) = 0s ↔ (𝐴↑s𝑛) = 0s ))
7	6	imbi1d 344	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (((𝐴↑s𝑚) = 0s → 𝐴 = 0s ) ↔ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )))
8	7	imbi2d 343	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐴 ∈ No → ((𝐴↑s𝑚) = 0s → 𝐴 = 0s )) ↔ (𝐴 ∈ No → ((𝐴↑s𝑛) = 0s → 𝐴 = 0s ))))
9		oveq2 7424	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴↑s𝑚) = (𝐴↑s(𝑛 +s 1s )))
10	9	eqeq1d 2764	. . . . . . . 8 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴↑s𝑚) = 0s ↔ (𝐴↑s(𝑛 +s 1s )) = 0s ))
11	10	imbi1d 344	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → (((𝐴↑s𝑚) = 0s → 𝐴 = 0s ) ↔ ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s )))
12	11	imbi2d 343	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴 ∈ No → ((𝐴↑s𝑚) = 0s → 𝐴 = 0s )) ↔ (𝐴 ∈ No → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s ))))
13		oveq2 7424	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝐴↑s𝑚) = (𝐴↑s𝑁))
14	13	eqeq1d 2764	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝐴↑s𝑚) = 0s ↔ (𝐴↑s𝑁) = 0s ))
15	14	imbi1d 344	. . . . . . 7 ⊢ (𝑚 = 𝑁 → (((𝐴↑s𝑚) = 0s → 𝐴 = 0s ) ↔ ((𝐴↑s𝑁) = 0s → 𝐴 = 0s )))
16	15	imbi2d 343	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((𝐴 ∈ No → ((𝐴↑s𝑚) = 0s → 𝐴 = 0s )) ↔ (𝐴 ∈ No → ((𝐴↑s𝑁) = 0s → 𝐴 = 0s ))))
17		1ne0s 28083	. . . . . . . . 9 ⊢ 1s ≠ 0s
18		exps0 28690	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
19	18	neeq1d 3016	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ((𝐴↑s 0s ) ≠ 0s ↔ 1s ≠ 0s ))
20	17, 19	mpbiri 261	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) ≠ 0s )
21	20	neneqd 2962	. . . . . . 7 ⊢ (𝐴 ∈ No → ¬ (𝐴↑s 0s ) = 0s )
22	21	pm2.21d 122	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴↑s 0s ) = 0s → 𝐴 = 0s ))
23		expsp1 28692	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴))
24	23	eqeq1d 2764	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → ((𝐴↑s(𝑛 +s 1s )) = 0s ↔ ((𝐴↑s𝑛) ·s 𝐴) = 0s ))
25		expscl 28694	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s𝑛) ∈ No )
26		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → 𝐴 ∈ No )
27	25, 26	muls0ord 28448	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (((𝐴↑s𝑛) ·s 𝐴) = 0s ↔ ((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s )))
28	24, 27	bitrd 282	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → ((𝐴↑s(𝑛 +s 1s )) = 0s ↔ ((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s )))
29	28	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → ((𝐴↑s(𝑛 +s 1s )) = 0s ↔ ((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s )))
30		simpr 490	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → ((𝐴↑s𝑛) = 0s → 𝐴 = 0s ))
31		idd 25	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → (𝐴 = 0s → 𝐴 = 0s ))
32	30, 31	jaod 873	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → (((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s ) → 𝐴 = 0s ))
33	29, 32	sylbid 243	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s ))
34	33	ex 418	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (((𝐴↑s𝑛) = 0s → 𝐴 = 0s ) → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s )))
35	34	expcom 419	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (𝐴 ∈ No → (((𝐴↑s𝑛) = 0s → 𝐴 = 0s ) → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s ))))
36	35	a2d 30	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ((𝐴 ∈ No → ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → (𝐴 ∈ No → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s ))))
37	4, 8, 12, 16, 22, 36	n0sind 28596	. . . . 5 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → ((𝐴↑s𝑁) = 0s → 𝐴 = 0s )))
38	37	imp 412	. . . 4 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝐴 ∈ No ) → ((𝐴↑s𝑁) = 0s → 𝐴 = 0s ))
39	38	necon3d 2978	. . 3 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝐴 ∈ No ) → (𝐴 ≠ 0s → (𝐴↑s𝑁) ≠ 0s ))
40	39	ex 418	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → (𝐴 ≠ 0s → (𝐴↑s𝑁) ≠ 0s )))
41	40	3imp231 1130	1 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ≠ 0s )

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  (class class class)co 7416   No csur 27874   0_s c0s 28068   1_s c1s 28069   +_s cadds 28222   ·_s cmuls 28369  ℕ_0scn0s 28575  ↑_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-ot 4596  df-uni 4871  df-int 4911  df-iun 4956  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-se 5613  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-lim 6366  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-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-oadd 8462  df-nadd 8657  df-no 27877  df-lts 27878  df-bday 27879  df-les 27979  df-slts 28021  df-cuts 28023  df-0s 28070  df-1s 28071  df-made 28090  df-old 28091  df-left 28093  df-right 28094  df-norec 28201  df-norec2 28212  df-adds 28223  df-negs 28284  df-subs 28285  df-muls 28370  df-seqs 28547  df-n0s 28577  df-nns 28578  df-zs 28642  df-exps 28676

This theorem is used by:  pw2divscld  28702  pw2divmulsd  28703  pw2divscan2d  28705  pw2divsassd  28706  pw2divsrecd  28710  pw2cut  28723  z12zsodd  28745