Theorem expsgt0 28757

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 7416	. . . . . 6 ⊢ (𝑚 = 0s → (𝐴↑s𝑚) = (𝐴↑s 0s ))
2	1	breq2d 5114	. . . . 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 7416	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝐴↑s𝑚) = (𝐴↑s𝑛))
5	4	breq2d 5114	. . . . 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 7416	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴↑s𝑚) = (𝐴↑s(𝑛 +s 1s )))
8	7	breq2d 5114	. . . . 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 7416	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝐴↑s𝑚) = (𝐴↑s𝑁))
11	10	breq2d 5114	. . . . 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 28132	. . . . . 6 ⊢ 0s <s 1s
14		exps0 28747	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
15	13, 14	breqtrrid 5142	. . . . 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 28751	. . . . . . . . 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 28465	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s ((𝐴↑s𝑛) ·s 𝐴))
24		expsp1 28749	. . . . . . . 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 5132	. . . . . 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 28653	. . 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 2145   class class class wbr 5102  (class class class)co 7408   No csur 27931   <s clts 27932   0_s c0s 28125   1_s c1s 28126   +_s cadds 28279   ·_s cmuls 28426  ℕ_0scn0s 28632  ↑_scexps 28732

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  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 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-oadd 8458  df-nadd 8653  df-no 27934  df-lts 27935  df-bday 27936  df-les 28036  df-slts 28078  df-cuts 28080  df-0s 28127  df-1s 28128  df-made 28147  df-old 28148  df-left 28150  df-right 28151  df-norec 28258  df-norec2 28269  df-adds 28280  df-negs 28341  df-subs 28342  df-muls 28427  df-seqs 28604  df-n0s 28634  df-nns 28635  df-zs 28699  df-exps 28733

This theorem is used by:  pw2gt0divsd  28765  pw2ge0divsd  28766  pw2ltdivmulsd  28770  pw2ltmuldivs2d  28771  pw2ltdivmuls2d  28777  pw2cut  28780  bdayfinbndlem1  28787