Theorem expsgt0 28588

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 7408	. . . . . 6 ⊢ (𝑚 = 0s → (𝐴↑s𝑚) = (𝐴↑s 0s ))
2	1	breq2d 5117	. . . . 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 7408	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝐴↑s𝑚) = (𝐴↑s𝑛))
5	4	breq2d 5117	. . . . 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 7408	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴↑s𝑚) = (𝐴↑s(𝑛 +s 1s )))
8	7	breq2d 5117	. . . . 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 7408	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝐴↑s𝑚) = (𝐴↑s𝑁))
11	10	breq2d 5117	. . . . 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 27963	. . . . . 6 ⊢ 0s <s 1s
14		exps0 28578	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
15	13, 14	breqtrrid 5143	. . . . 5 ⊢ (𝐴 ∈ No → 0s <s (𝐴↑s 0s ))
16	15	adantr 485	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s 0s ))
17		simp2l 1216	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 𝐴 ∈ No )
18		simp1 1152	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 𝑛 ∈ ℕ0s)
19		expscl 28582	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s𝑛) ∈ No )
20	17, 18, 19	syl2anc 595	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → (𝐴↑s𝑛) ∈ No )
21		simp3 1154	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s (𝐴↑s𝑛))
22		simp2r 1217	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s 𝐴)
23	20, 17, 21, 22	mulsgt0d 28296	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s ((𝐴↑s𝑛) ·s 𝐴))
24		expsp1 28580	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴))
25	17, 18, 24	syl2anc 595	. . . . . . 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 5133	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s (𝐴↑s(𝑛 +s 1s )))
27	26	3exp 1135	. . . . 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 28484	. . 3 ⊢ (𝑁 ∈ ℕ0s → ((𝐴 ∈ No ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑁)))
30	29	expd 420	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → ( 0s <s 𝐴 → 0s <s (𝐴↑s𝑁))))
31	30	3imp21 1129	1 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑁))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   class class class wbr 5105  (class class class)co 7400   No csur 27762   <s clts 27763   0_s c0s 27956   1_s c1s 27957   +_s cadds 28110   ·_s cmuls 28257  ℕ_0scn0s 28463  ↑_scexps 28563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-oadd 8445  df-nadd 8640  df-no 27765  df-lts 27766  df-bday 27767  df-les 27867  df-slts 27909  df-cuts 27911  df-0s 27958  df-1s 27959  df-made 27978  df-old 27979  df-left 27981  df-right 27982  df-norec 28089  df-norec2 28100  df-adds 28111  df-negs 28172  df-subs 28173  df-muls 28258  df-seqs 28435  df-n0s 28465  df-nns 28466  df-zs 28530  df-exps 28564

This theorem is referenced by:  pw2gt0divsd  28596  pw2ge0divsd  28597  pw2ltdivmulsd  28601  pw2ltmuldivs2d  28602  pw2ltdivmuls2d  28608  pw2cut  28611  bdayfinbndlem1  28618