Theorem expsne0 28495

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 7389	. . . . . . . . 9 ⊢ (𝑚 = 0s → (𝐴↑s𝑚) = (𝐴↑s 0s ))
2	1	eqeq1d 2754	. . . . . . . 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 7389	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝐴↑s𝑚) = (𝐴↑s𝑛))
6	5	eqeq1d 2754	. . . . . . . 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 7389	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴↑s𝑚) = (𝐴↑s(𝑛 +s 1s )))
10	9	eqeq1d 2754	. . . . . . . 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 7389	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝐴↑s𝑚) = (𝐴↑s𝑁))
14	13	eqeq1d 2754	. . . . . . . 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 27879	. . . . . . . . 9 ⊢ 1s ≠ 0s
18		exps0 28486	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
19	18	neeq1d 3006	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ((𝐴↑s 0s ) ≠ 0s ↔ 1s ≠ 0s ))
20	17, 19	mpbiri 260	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) ≠ 0s )
21	20	neneqd 2952	. . . . . . 7 ⊢ (𝐴 ∈ No → ¬ (𝐴↑s 0s ) = 0s )
22	21	pm2.21d 121	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴↑s 0s ) = 0s → 𝐴 = 0s ))
23		expsp1 28488	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴))
24	23	eqeq1d 2754	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → ((𝐴↑s(𝑛 +s 1s )) = 0s ↔ ((𝐴↑s𝑛) ·s 𝐴) = 0s ))
25		expscl 28490	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s𝑛) ∈ No )
26		simpl 485	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → 𝐴 ∈ No )
27	25, 26	muls0ord 28244	. . . . . . . . . . . 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 483	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → ((𝐴↑s(𝑛 +s 1s )) = 0s ↔ ((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s )))
30		simpr 487	. . . . . . . . . . 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 868	. . . . . . . . . 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 415	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (((𝐴↑s𝑛) = 0s → 𝐴 = 0s ) → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s )))
35	34	expcom 416	. . . . . . 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 28392	. . . . 5 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → ((𝐴↑s𝑁) = 0s → 𝐴 = 0s )))
38	37	imp 409	. . . 4 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝐴 ∈ No ) → ((𝐴↑s𝑁) = 0s → 𝐴 = 0s ))
39	38	necon3d 2968	. . 3 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝐴 ∈ No ) → (𝐴 ≠ 0s → (𝐴↑s𝑁) ≠ 0s ))
40	39	ex 415	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → (𝐴 ≠ 0s → (𝐴↑s𝑁) ≠ 0s )))
41	40	3imp231 1121	1 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ≠ 0s )

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 856   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  (class class class)co 7381   No csur 27670   0_s c0s 27864   1_s c1s 27865   +_s cadds 28018   ·_s cmuls 28165  ℕ_0scn0s 28371  ↑_scexps 28471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-ot 4581  df-uni 4856  df-int 4896  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-2o 8422  df-oadd 8425  df-nadd 8620  df-no 27673  df-lts 27674  df-bday 27675  df-les 27775  df-slts 27817  df-cuts 27819  df-0s 27866  df-1s 27867  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27997  df-norec2 28008  df-adds 28019  df-negs 28080  df-subs 28081  df-muls 28166  df-seqs 28343  df-n0s 28373  df-nns 28374  df-zs 28438  df-exps 28472

This theorem is referenced by:  pw2divscld  28498  pw2divmulsd  28499  pw2divscan2d  28501  pw2divsassd  28502  pw2divsrecd  28506  pw2cut  28519  z12zsodd  28541