Theorem expsgt0 28853

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 7451	. . . . . 6 ⊢ (𝑚 = 0s → (𝐴↑s𝑚) = (𝐴↑s 0s ))
2	1	breq2d 5118	. . . . 5 ⊢ (𝑚 = 0s → ( 0s <s (𝐴↑s𝑚) ↔ 0s <s (𝐴↑s 0s )))
3	2	imbi2d 343	. . . 4 ⊢ (𝑚 = 0s → (((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑚)) ↔ ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s 0s ))))
4		oveq2 7451	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝐴↑s𝑚) = (𝐴↑s𝑛))
5	4	breq2d 5118	. . . . 5 ⊢ (𝑚 = 𝑛 → ( 0s <s (𝐴↑s𝑚) ↔ 0s <s (𝐴↑s𝑛)))
6	5	imbi2d 343	. . . 4 ⊢ (𝑚 = 𝑛 → (((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑚)) ↔ ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑛))))
7		oveq2 7451	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴↑s𝑚) = (𝐴↑s(𝑛 +s 1s )))
8	7	breq2d 5118	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → ( 0s <s (𝐴↑s𝑚) ↔ 0s <s (𝐴↑s(𝑛 +s 1s ))))
9	8	imbi2d 343	. . . 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 7451	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝐴↑s𝑚) = (𝐴↑s𝑁))
11	10	breq2d 5118	. . . . 5 ⊢ (𝑚 = 𝑁 → ( 0s <s (𝐴↑s𝑚) ↔ 0s <s (𝐴↑s𝑁)))
12	11	imbi2d 343	. . . 4 ⊢ (𝑚 = 𝑁 → (((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑚)) ↔ ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑁))))
13		0lt1s 28228	. . . . . 6 ⊢ 0s <s 1s
14		exps0 28843	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
15	13, 14	breqtrrid 5146	. . . . 5 ⊢ (𝐴 ∈ No → 0s <s (𝐴↑s 0s ))
16	15	adantr 486	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s 0s ))
17		simp2l 1218	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 𝐴 ∈ No)
18		simp1 1154	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 𝑛 ∈ ℕ0s)
19		expscl 28847	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s𝑛) ∈ No)
20	17, 18, 19	syl2anc 596	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → (𝐴↑s𝑛) ∈ No)
21		simp3 1156	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s (𝐴↑s𝑛))
22		simp2r 1219	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s 𝐴)
23	20, 17, 21, 22	mulsgt0d 28561	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s ((𝐴↑s𝑛) ·s 𝐴))
24		expsp1 28845	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴))
25	17, 18, 24	syl2anc 596	. . . . . . 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 5136	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s (𝐴↑s(𝑛 +s 1s )))
27	26	3exp 1137	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → ((𝐴 ∈ No ∧ 0s <s 𝐴) → ( 0s <s (𝐴↑s𝑛) → 0s <s (𝐴↑s(𝑛 +s 1s )))))
28	27	a2d 30	. . . 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 28749	. . 3 ⊢ (𝑁 ∈ ℕ0s → ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑁)))
30	29	expd 421	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → ( 0s <s 𝐴 → 0s <s (𝐴↑s𝑁))))
31	30	3imp21 1131	1 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   class class class wbr 5106  (class class class)co 7443  Nocsur 28027   <s clts 28028   0_s c0s 28221   1_s c1s 28222   +_s cadds 28375   ·_s cmuls 28522  ℕ_0scn0s 28728  ↑_scexps 28828

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-1s 28224  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-norec2 28365  df-adds 28376  df-negs 28437  df-subs 28438  df-muls 28523  df-seqs 28700  df-n0s 28730  df-nns 28731  df-zs 28795  df-exps 28829

This theorem is used by:  pw2gt0divsd  28861  pw2ge0divsd  28862  pw2ltdivmulsd  28866  pw2ltmuldivs2d  28867  pw2ltdivmuls2d  28873  pw2cut  28876  bdayfinbndlem1  28883