Theorem expsgt0 28416

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

Assertion

Ref	Expression
expsgt0	⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑁))

Proof of Theorem expsgt0

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

Step	Hyp	Ref	Expression
1		oveq2 7440	. . . . . 6 ⊢ (𝑚 = 0s → (𝐴↑s𝑚) = (𝐴↑s 0s ))
2	1	breq2d 5154	. . . . 5 ⊢ (𝑚 = 0s → ( 0s <s (𝐴↑s𝑚) ↔ 0s <s (𝐴↑s 0s )))
3	2	imbi2d 340	. . . 4 ⊢ (𝑚 = 0s → (((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑚)) ↔ ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s 0s ))))
4		oveq2 7440	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝐴↑s𝑚) = (𝐴↑s𝑛))
5	4	breq2d 5154	. . . . 5 ⊢ (𝑚 = 𝑛 → ( 0s <s (𝐴↑s𝑚) ↔ 0s <s (𝐴↑s𝑛)))
6	5	imbi2d 340	. . . 4 ⊢ (𝑚 = 𝑛 → (((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑚)) ↔ ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑛))))
7		oveq2 7440	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴↑s𝑚) = (𝐴↑s(𝑛 +s 1s )))
8	7	breq2d 5154	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → ( 0s <s (𝐴↑s𝑚) ↔ 0s <s (𝐴↑s(𝑛 +s 1s ))))
9	8	imbi2d 340	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → (((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑚)) ↔ ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s(𝑛 +s 1s )))))
10		oveq2 7440	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝐴↑s𝑚) = (𝐴↑s𝑁))
11	10	breq2d 5154	. . . . 5 ⊢ (𝑚 = 𝑁 → ( 0s <s (𝐴↑s𝑚) ↔ 0s <s (𝐴↑s𝑁)))
12	11	imbi2d 340	. . . 4 ⊢ (𝑚 = 𝑁 → (((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑚)) ↔ ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑁))))
13		0slt1s 27875	. . . . . 6 ⊢ 0s <s 1s
14		exps0 28411	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
15	13, 14	breqtrrid 5180	. . . . 5 ⊢ (𝐴 ∈ No → 0s <s (𝐴↑s 0s ))
16	15	adantr 480	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s 0s ))
17		simp2l 1199	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 𝐴 ∈ No )
18		simp1 1136	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 𝑛 ∈ ℕ0s)
19		expscl 28414	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s𝑛) ∈ No )
20	17, 18, 19	syl2anc 584	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → (𝐴↑s𝑛) ∈ No )
21		simp3 1138	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s (𝐴↑s𝑛))
22		simp2r 1200	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s 𝐴)
23	20, 17, 21, 22	mulsgt0d 28172	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s ((𝐴↑s𝑛) ·s 𝐴))
24		expsp1 28413	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴))
25	17, 18, 24	syl2anc 584	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴))
26	23, 25	breqtrrd 5170	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s (𝐴↑s(𝑛 +s 1s )))
27	26	3exp 1119	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → ((𝐴 ∈ No ∧ 0s <s 𝐴) → ( 0s <s (𝐴↑s𝑛) → 0s <s (𝐴↑s(𝑛 +s 1s )))))
28	27	a2d 29	. . . 4 ⊢ (𝑛 ∈ ℕ0s → (((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑛)) → ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s(𝑛 +s 1s )))))
29	3, 6, 9, 12, 16, 28	n0sind 28338	. . 3 ⊢ (𝑁 ∈ ℕ0s → ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑁)))
30	29	expd 415	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → ( 0s <s 𝐴 → 0s <s (𝐴↑s𝑁))))
31	30	3imp21 1113	1 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   class class class wbr 5142  (class class class)co 7432   No csur 27685   <s cslt 27686   0_s c0s 27868   1_s c1s 27869   +_s cadds 27993   ·_s cmuls 28133  ℕ_0scnn0s 28319  ↑_scexps 28397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-1s 27871  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec 27972  df-norec2 27983  df-adds 27994  df-negs 28054  df-subs 28055  df-muls 28134  df-seqs 28291  df-n0s 28321  df-nns 28322  df-zs 28366  df-exps 28398

This theorem is referenced by:  cutpw2  28418  pw2bday  28419  pw2cut  28421  zs12bday  28425