Theorem expsne0 28605

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 7418	. . . . . . . . 9 ⊢ (𝑚 = 0s → (𝐴↑s𝑚) = (𝐴↑s 0s ))
2	1	eqeq1d 2763	. . . . . . . 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 7418	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝐴↑s𝑚) = (𝐴↑s𝑛))
6	5	eqeq1d 2763	. . . . . . . 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 7418	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴↑s𝑚) = (𝐴↑s(𝑛 +s 1s )))
10	9	eqeq1d 2763	. . . . . . . 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 7418	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝐴↑s𝑚) = (𝐴↑s𝑁))
14	13	eqeq1d 2763	. . . . . . . 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 27989	. . . . . . . . 9 ⊢ 1s ≠ 0s
18		exps0 28596	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
19	18	neeq1d 3015	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ((𝐴↑s 0s ) ≠ 0s ↔ 1s ≠ 0s ))
20	17, 19	mpbiri 261	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) ≠ 0s )
21	20	neneqd 2961	. . . . . . 7 ⊢ (𝐴 ∈ No → ¬ (𝐴↑s 0s ) = 0s )
22	21	pm2.21d 122	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴↑s 0s ) = 0s → 𝐴 = 0s ))
23		expsp1 28598	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴))
24	23	eqeq1d 2763	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → ((𝐴↑s(𝑛 +s 1s )) = 0s ↔ ((𝐴↑s𝑛) ·s 𝐴) = 0s ))
25		expscl 28600	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s𝑛) ∈ No )
26		simpl 487	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → 𝐴 ∈ No )
27	25, 26	muls0ord 28354	. . . . . . . . . . . 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 485	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → ((𝐴↑s(𝑛 +s 1s )) = 0s ↔ ((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s )))
30		simpr 489	. . . . . . . . . . 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 872	. . . . . . . . . 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 417	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (((𝐴↑s𝑛) = 0s → 𝐴 = 0s ) → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s )))
35	34	expcom 418	. . . . . . 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 28502	. . . . 5 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → ((𝐴↑s𝑁) = 0s → 𝐴 = 0s )))
38	37	imp 411	. . . 4 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝐴 ∈ No ) → ((𝐴↑s𝑁) = 0s → 𝐴 = 0s ))
39	38	necon3d 2977	. . 3 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝐴 ∈ No ) → (𝐴 ≠ 0s → (𝐴↑s𝑁) ≠ 0s ))
40	39	ex 417	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → (𝐴 ≠ 0s → (𝐴↑s𝑁) ≠ 0s )))
41	40	3imp231 1128	1 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ≠ 0s )

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  (class class class)co 7410   No csur 27780   0_s c0s 27974   1_s c1s 27975   +_s cadds 28128   ·_s cmuls 28275  ℕ_0scn0s 28481  ↑_scexps 28581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  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-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  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 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-nadd 8651  df-no 27783  df-lts 27784  df-bday 27785  df-les 27885  df-slts 27927  df-cuts 27929  df-0s 27976  df-1s 27977  df-made 27996  df-old 27997  df-left 27999  df-right 28000  df-norec 28107  df-norec2 28118  df-adds 28129  df-negs 28190  df-subs 28191  df-muls 28276  df-seqs 28453  df-n0s 28483  df-nns 28484  df-zs 28548  df-exps 28582

This theorem is referenced by:  pw2divscld  28608  pw2divmulsd  28609  pw2divscan2d  28611  pw2divsassd  28612  pw2divsrecd  28616  pw2cut  28629  z12zsodd  28651