Theorem expsne0 28536

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 7404	. . . . . . . . 9 ⊢ (𝑚 = 0s → (𝐴↑s𝑚) = (𝐴↑s 0s ))
2	1	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑚 = 0s → ((𝐴↑s𝑚) = 0s ↔ (𝐴↑s 0s ) = 0s ))
3	2	imbi1d 343	. . . . . . 7 ⊢ (𝑚 = 0s → (((𝐴↑s𝑚) = 0s → 𝐴 = 0s ) ↔ ((𝐴↑s 0s ) = 0s → 𝐴 = 0s )))
4	3	imbi2d 342	. . . . . 6 ⊢ (𝑚 = 0s → ((𝐴 ∈ No → ((𝐴↑s𝑚) = 0s → 𝐴 = 0s )) ↔ (𝐴 ∈ No → ((𝐴↑s 0s ) = 0s → 𝐴 = 0s ))))
5		oveq2 7404	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝐴↑s𝑚) = (𝐴↑s𝑛))
6	5	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝐴↑s𝑚) = 0s ↔ (𝐴↑s𝑛) = 0s ))
7	6	imbi1d 343	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (((𝐴↑s𝑚) = 0s → 𝐴 = 0s ) ↔ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )))
8	7	imbi2d 342	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐴 ∈ No → ((𝐴↑s𝑚) = 0s → 𝐴 = 0s )) ↔ (𝐴 ∈ No → ((𝐴↑s𝑛) = 0s → 𝐴 = 0s ))))
9		oveq2 7404	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴↑s𝑚) = (𝐴↑s(𝑛 +s 1s )))
10	9	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴↑s𝑚) = 0s ↔ (𝐴↑s(𝑛 +s 1s )) = 0s ))
11	10	imbi1d 343	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → (((𝐴↑s𝑚) = 0s → 𝐴 = 0s ) ↔ ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s )))
12	11	imbi2d 342	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴 ∈ No → ((𝐴↑s𝑚) = 0s → 𝐴 = 0s )) ↔ (𝐴 ∈ No → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s ))))
13		oveq2 7404	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝐴↑s𝑚) = (𝐴↑s𝑁))
14	13	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝐴↑s𝑚) = 0s ↔ (𝐴↑s𝑁) = 0s ))
15	14	imbi1d 343	. . . . . . 7 ⊢ (𝑚 = 𝑁 → (((𝐴↑s𝑚) = 0s → 𝐴 = 0s ) ↔ ((𝐴↑s𝑁) = 0s → 𝐴 = 0s )))
16	15	imbi2d 342	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((𝐴 ∈ No → ((𝐴↑s𝑚) = 0s → 𝐴 = 0s )) ↔ (𝐴 ∈ No → ((𝐴↑s𝑁) = 0s → 𝐴 = 0s ))))
17		1ne0s 27920	. . . . . . . . 9 ⊢ 1s ≠ 0s
18		exps0 28527	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
19	18	neeq1d 3017	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ((𝐴↑s 0s ) ≠ 0s ↔ 1s ≠ 0s ))
20	17, 19	mpbiri 260	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) ≠ 0s )
21	20	neneqd 2963	. . . . . . 7 ⊢ (𝐴 ∈ No → ¬ (𝐴↑s 0s ) = 0s )
22	21	pm2.21d 121	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴↑s 0s ) = 0s → 𝐴 = 0s ))
23		expsp1 28529	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴))
24	23	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → ((𝐴↑s(𝑛 +s 1s )) = 0s ↔ ((𝐴↑s𝑛) ·s 𝐴) = 0s ))
25		expscl 28531	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s𝑛) ∈ No )
26		simpl 486	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → 𝐴 ∈ No )
27	25, 26	muls0ord 28285	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (((𝐴↑s𝑛) ·s 𝐴) = 0s ↔ ((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s )))
28	24, 27	bitrd 281	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → ((𝐴↑s(𝑛 +s 1s )) = 0s ↔ ((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s )))
29	28	adantr 484	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → ((𝐴↑s(𝑛 +s 1s )) = 0s ↔ ((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s )))
30		simpr 488	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → ((𝐴↑s𝑛) = 0s → 𝐴 = 0s ))
31		idd 24	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → (𝐴 = 0s → 𝐴 = 0s ))
32	30, 31	jaod 870	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → (((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s ) → 𝐴 = 0s ))
33	29, 32	sylbid 242	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s ))
34	33	ex 416	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (((𝐴↑s𝑛) = 0s → 𝐴 = 0s ) → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s )))
35	34	expcom 417	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (𝐴 ∈ No → (((𝐴↑s𝑛) = 0s → 𝐴 = 0s ) → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s ))))
36	35	a2d 29	. . . . . 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 28433	. . . . 5 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → ((𝐴↑s𝑁) = 0s → 𝐴 = 0s )))
38	37	imp 410	. . . 4 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝐴 ∈ No ) → ((𝐴↑s𝑁) = 0s → 𝐴 = 0s ))
39	38	necon3d 2979	. . 3 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝐴 ∈ No ) → (𝐴 ≠ 0s → (𝐴↑s𝑁) ≠ 0s ))
40	39	ex 416	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → (𝐴 ≠ 0s → (𝐴↑s𝑁) ≠ 0s )))
41	40	3imp231 1126	1 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ≠ 0s )

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  (class class class)co 7396   No csur 27711   0_s c0s 27905   1_s c1s 27906   +_s cadds 28059   ·_s cmuls 28206  ℕ_0scn0s 28412  ↑_scexps 28512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-nadd 8636  df-no 27714  df-lts 27715  df-bday 27716  df-les 27816  df-slts 27858  df-cuts 27860  df-0s 27907  df-1s 27908  df-made 27927  df-old 27928  df-left 27930  df-right 27931  df-norec 28038  df-norec2 28049  df-adds 28060  df-negs 28121  df-subs 28122  df-muls 28207  df-seqs 28384  df-n0s 28414  df-nns 28415  df-zs 28479  df-exps 28513

This theorem is referenced by:  pw2divscld  28539  pw2divmulsd  28540  pw2divscan2d  28542  pw2divsassd  28543  pw2divsrecd  28547  pw2cut  28560  z12zsodd  28582