Theorem expsne0 28432

Description: A non-negative surreal integer power is non-zero if its base is non-zero. (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 7456	. . . . . . . . 9 ⊢ (𝑚 = 0s → (𝐴↑s𝑚) = (𝐴↑s 0s ))
2	1	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑚 = 0s → ((𝐴↑s𝑚) = 0s ↔ (𝐴↑s 0s ) = 0s ))
3	2	imbi1d 341	. . . . . . 7 ⊢ (𝑚 = 0s → (((𝐴↑s𝑚) = 0s → 𝐴 = 0s ) ↔ ((𝐴↑s 0s ) = 0s → 𝐴 = 0s )))
4	3	imbi2d 340	. . . . . 6 ⊢ (𝑚 = 0s → ((𝐴 ∈ No → ((𝐴↑s𝑚) = 0s → 𝐴 = 0s )) ↔ (𝐴 ∈ No → ((𝐴↑s 0s ) = 0s → 𝐴 = 0s ))))
5		oveq2 7456	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝐴↑s𝑚) = (𝐴↑s𝑛))
6	5	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝐴↑s𝑚) = 0s ↔ (𝐴↑s𝑛) = 0s ))
7	6	imbi1d 341	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (((𝐴↑s𝑚) = 0s → 𝐴 = 0s ) ↔ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )))
8	7	imbi2d 340	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐴 ∈ No → ((𝐴↑s𝑚) = 0s → 𝐴 = 0s )) ↔ (𝐴 ∈ No → ((𝐴↑s𝑛) = 0s → 𝐴 = 0s ))))
9		oveq2 7456	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴↑s𝑚) = (𝐴↑s(𝑛 +s 1s )))
10	9	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴↑s𝑚) = 0s ↔ (𝐴↑s(𝑛 +s 1s )) = 0s ))
11	10	imbi1d 341	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → (((𝐴↑s𝑚) = 0s → 𝐴 = 0s ) ↔ ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s )))
12	11	imbi2d 340	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴 ∈ No → ((𝐴↑s𝑚) = 0s → 𝐴 = 0s )) ↔ (𝐴 ∈ No → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s ))))
13		oveq2 7456	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → (𝐴↑s𝑚) = (𝐴↑s𝑁))
14	13	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝐴↑s𝑚) = 0s ↔ (𝐴↑s𝑁) = 0s ))
15	14	imbi1d 341	. . . . . . 7 ⊢ (𝑚 = 𝑁 → (((𝐴↑s𝑚) = 0s → 𝐴 = 0s ) ↔ ((𝐴↑s𝑁) = 0s → 𝐴 = 0s )))
16	15	imbi2d 340	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((𝐴 ∈ No → ((𝐴↑s𝑚) = 0s → 𝐴 = 0s )) ↔ (𝐴 ∈ No → ((𝐴↑s𝑁) = 0s → 𝐴 = 0s ))))
17		0slt1s 27892	. . . . . . . . . 10 ⊢ 0s <s 1s
18		sgt0ne0 27897	. . . . . . . . . 10 ⊢ ( 0s <s 1s → 1s ≠ 0s )
19	17, 18	ax-mp 5	. . . . . . . . 9 ⊢ 1s ≠ 0s
20		exps0 28428	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
21	20	neeq1d 3006	. . . . . . . . 9 ⊢ (𝐴 ∈ No → ((𝐴↑s 0s ) ≠ 0s ↔ 1s ≠ 0s ))
22	19, 21	mpbiri 258	. . . . . . . 8 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) ≠ 0s )
23	22	neneqd 2951	. . . . . . 7 ⊢ (𝐴 ∈ No → ¬ (𝐴↑s 0s ) = 0s )
24	23	pm2.21d 121	. . . . . 6 ⊢ (𝐴 ∈ No → ((𝐴↑s 0s ) = 0s → 𝐴 = 0s ))
25		expsp1 28430	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴))
26	25	eqeq1d 2742	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → ((𝐴↑s(𝑛 +s 1s )) = 0s ↔ ((𝐴↑s𝑛) ·s 𝐴) = 0s ))
27		expscl 28431	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s𝑛) ∈ No )
28		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → 𝐴 ∈ No )
29	27, 28	muls0ord 28229	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (((𝐴↑s𝑛) ·s 𝐴) = 0s ↔ ((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s )))
30	26, 29	bitrd 279	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → ((𝐴↑s(𝑛 +s 1s )) = 0s ↔ ((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s )))
31	30	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → ((𝐴↑s(𝑛 +s 1s )) = 0s ↔ ((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s )))
32		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → ((𝐴↑s𝑛) = 0s → 𝐴 = 0s ))
33		idd 24	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → (𝐴 = 0s → 𝐴 = 0s ))
34	32, 33	jaod 858	. . . . . . . . . 10 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → (((𝐴↑s𝑛) = 0s ∨ 𝐴 = 0s ) → 𝐴 = 0s ))
35	31, 34	sylbid 240	. . . . . . . . 9 ⊢ (((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) ∧ ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s ))
36	35	ex 412	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (((𝐴↑s𝑛) = 0s → 𝐴 = 0s ) → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s )))
37	36	expcom 413	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (𝐴 ∈ No → (((𝐴↑s𝑛) = 0s → 𝐴 = 0s ) → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s ))))
38	37	a2d 29	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ((𝐴 ∈ No → ((𝐴↑s𝑛) = 0s → 𝐴 = 0s )) → (𝐴 ∈ No → ((𝐴↑s(𝑛 +s 1s )) = 0s → 𝐴 = 0s ))))
39	4, 8, 12, 16, 24, 38	n0sind 28355	. . . . 5 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → ((𝐴↑s𝑁) = 0s → 𝐴 = 0s )))
40	39	imp 406	. . . 4 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝐴 ∈ No ) → ((𝐴↑s𝑁) = 0s → 𝐴 = 0s ))
41	40	necon3d 2967	. . 3 ⊢ ((𝑁 ∈ ℕ0s ∧ 𝐴 ∈ No ) → (𝐴 ≠ 0s → (𝐴↑s𝑁) ≠ 0s ))
42	41	ex 412	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → (𝐴 ≠ 0s → (𝐴↑s𝑁) ≠ 0s )))
43	42	3imp231 1113	1 ⊢ ((𝐴 ∈ No ∧ 𝐴 ≠ 0s ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ≠ 0s )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   class class class wbr 5166  (class class class)co 7448   No csur 27702   <s cslt 27703   0_s c0s 27885   1_s c1s 27886   +_s cadds 28010   ·_s cmuls 28150  ℕ_0scnn0s 28336  ↑_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-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  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-se 5653  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-lim 6400  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-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-muls 28151  df-seqs 28308  df-n0s 28338  df-nns 28339  df-zs 28383  df-exps 28415

This theorem is referenced by:  cutpw2  28435  pw2bday  28436  pw2cut  28438  zs12bday  28442