Theorem expsgt0 28445

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 7376	. . . . . 6 ⊢ (𝑚 = 0s → (𝐴↑s𝑚) = (𝐴↑s 0s ))
2	1	breq2d 5112	. . . . 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 7376	. . . . . 6 ⊢ (𝑚 = 𝑛 → (𝐴↑s𝑚) = (𝐴↑s𝑛))
5	4	breq2d 5112	. . . . 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 7376	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴↑s𝑚) = (𝐴↑s(𝑛 +s 1s )))
8	7	breq2d 5112	. . . . 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 7376	. . . . . 6 ⊢ (𝑚 = 𝑁 → (𝐴↑s𝑚) = (𝐴↑s𝑁))
11	10	breq2d 5112	. . . . 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		0lt1s 27820	. . . . . 6 ⊢ 0s <s 1s
14		exps0 28435	. . . . . 6 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
15	13, 14	breqtrrid 5138	. . . . 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 1201	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 𝐴 ∈ No )
18		simp1 1137	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 𝑛 ∈ ℕ0s)
19		expscl 28439	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s𝑛) ∈ No )
20	17, 18, 19	syl2anc 585	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → (𝐴↑s𝑛) ∈ No )
21		simp3 1139	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s (𝐴↑s𝑛))
22		simp2r 1202	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s 𝐴)
23	20, 17, 21, 22	mulsgt0d 28153	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s ((𝐴↑s𝑛) ·s 𝐴))
24		expsp1 28437	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴))
25	17, 18, 24	syl2anc 585	. . . . . . 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 5128	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝐴 ∈ No ∧ 0s <s 𝐴) ∧ 0s <s (𝐴↑s𝑛)) → 0s <s (𝐴↑s(𝑛 +s 1s )))
27	26	3exp 1120	. . . . 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 28341	. . 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 1114	1 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s ∧ 0s <s 𝐴) → 0s <s (𝐴↑s𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5100  (class class class)co 7368   No csur 27619   <s clts 27620   0_s c0s 27813   1_s c1s 27814   +_s cadds 27967   ·_s cmuls 28114  ℕ_0scn0s 28320  ↑_scexps 28420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-nadd 8604  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-1s 27816  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029  df-subs 28030  df-muls 28115  df-seqs 28292  df-n0s 28322  df-nns 28323  df-zs 28387  df-exps 28421

This theorem is referenced by:  pw2gt0divsd  28453  pw2ge0divsd  28454  pw2ltdivmulsd  28458  pw2ltmuldivs2d  28459  pw2ltdivmuls2d  28465  pw2cut  28468  bdayfinbndlem1  28475