Theorem leexp2r 10347

Description: Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.)

Assertion

Ref	Expression
leexp2r	⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))

Proof of Theorem leexp2r

Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5782	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (𝐴↑𝑗) = (𝐴↑𝑀))
2	1	breq1d 3939	. . . . . . 7 ⊢ (𝑗 = 𝑀 → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑𝑀) ≤ (𝐴↑𝑀)))
3	2	imbi2d 229	. . . . . 6 ⊢ (𝑗 = 𝑀 → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑗) ≤ (𝐴↑𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑀) ≤ (𝐴↑𝑀))))
4		oveq2 5782	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐴↑𝑗) = (𝐴↑𝑘))
5	4	breq1d 3939	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑𝑘) ≤ (𝐴↑𝑀)))
6	5	imbi2d 229	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑗) ≤ (𝐴↑𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑘) ≤ (𝐴↑𝑀))))
7		oveq2 5782	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (𝐴↑𝑗) = (𝐴↑(𝑘 + 1)))
8	7	breq1d 3939	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀)))
9	8	imbi2d 229	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑗) ≤ (𝐴↑𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀))))
10		oveq2 5782	. . . . . . . 8 ⊢ (𝑗 = 𝑁 → (𝐴↑𝑗) = (𝐴↑𝑁))
11	10	breq1d 3939	. . . . . . 7 ⊢ (𝑗 = 𝑁 → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑𝑁) ≤ (𝐴↑𝑀)))
12	11	imbi2d 229	. . . . . 6 ⊢ (𝑗 = 𝑁 → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑗) ≤ (𝐴↑𝑀)) ↔ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))))
13		reexpcl 10310	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝐴↑𝑀) ∈ ℝ)
14	13	adantr 274	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑀) ∈ ℝ)
15	14	leidd 8276	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑀) ≤ (𝐴↑𝑀))
16	15	a1i 9	. . . . . 6 ⊢ (𝑀 ∈ ℤ → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑀) ≤ (𝐴↑𝑀)))
17		simprll 526	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ∈ ℝ)
18		1red 7781	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 1 ∈ ℝ)
19		simprlr 527	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑀 ∈ ℕ0)
20		simpl 108	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑘 ∈ (ℤ≥‘𝑀))
21		eluznn0 9393	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0)
22	19, 20, 21	syl2anc 408	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑘 ∈ ℕ0)
23		reexpcl 10310	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ)
24	17, 22, 23	syl2anc 408	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) ∈ ℝ)
25		simprrl 528	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 0 ≤ 𝐴)
26		expge0 10329	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴↑𝑘))
27	17, 22, 25, 26	syl3anc 1216	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 0 ≤ (𝐴↑𝑘))
28		simprrr 529	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ≤ 1)
29	17, 18, 24, 27, 28	lemul2ad 8698	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) · 𝐴) ≤ ((𝐴↑𝑘) · 1))
30	17	recnd 7794	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ∈ ℂ)
31		expp1 10300	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
32	30, 22, 31	syl2anc 408	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
33	24	recnd 7794	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) ∈ ℂ)
34	33	mulid1d 7783	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) · 1) = (𝐴↑𝑘))
35	34	eqcomd 2145	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) = ((𝐴↑𝑘) · 1))
36	29, 32, 35	3brtr4d 3960	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑘))
37		peano2nn0 9017	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
38	22, 37	syl 14	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝑘 + 1) ∈ ℕ0)
39		reexpcl 10310	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (𝑘 + 1) ∈ ℕ0) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
40	17, 38, 39	syl2anc 408	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
41	13	ad2antrl 481	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑀) ∈ ℝ)
42		letr 7847	. . . . . . . . . 10 ⊢ (((𝐴↑(𝑘 + 1)) ∈ ℝ ∧ (𝐴↑𝑘) ∈ ℝ ∧ (𝐴↑𝑀) ∈ ℝ) → (((𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑘) ∧ (𝐴↑𝑘) ≤ (𝐴↑𝑀)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀)))
43	40, 24, 41, 42	syl3anc 1216	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (((𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑘) ∧ (𝐴↑𝑘) ≤ (𝐴↑𝑀)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀)))
44	36, 43	mpand 425	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) ≤ (𝐴↑𝑀) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀)))
45	44	ex 114	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → ((𝐴↑𝑘) ≤ (𝐴↑𝑀) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀))))
46	45	a2d 26	. . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ((((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑘) ≤ (𝐴↑𝑀)) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀))))
47	3, 6, 9, 12, 16, 46	uzind4 9383	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀)))
48	47	expd 256	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → ((0 ≤ 𝐴 ∧ 𝐴 ≤ 1) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))))
49	48	com12 30	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑁 ∈ (ℤ≥‘𝑀) → ((0 ≤ 𝐴 ∧ 𝐴 ≤ 1) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))))
50	49	3impia 1178	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((0 ≤ 𝐴 ∧ 𝐴 ≤ 1) → (𝐴↑𝑁) ≤ (𝐴↑𝑀)))
51	50	imp 123	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962   = wceq 1331   ∈ wcel 1480   class class class wbr 3929  ‘cfv 5123  (class class class)co 5774  ℂcc 7618  ℝcr 7619  0cc0 7620  1c1 7621   + caddc 7623   · cmul 7625   ≤ cle 7801  ℕ₀cn0 8977  ℤcz 9054  ℤ_≥cuz 9326  ↑cexp 10292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-seqfrec 10219  df-exp 10293

This theorem is referenced by:  exple1  10349  leexp2rd  10454