Theorem leexp2r 14115

Description: Weak ordering relationship for exponentiation of a fixed real base in [0, 1] to integer exponents. (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 7377	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (𝐴↑𝑗) = (𝐴↑𝑀))
2	1	breq1d 5112	. . . . . . 7 ⊢ (𝑗 = 𝑀 → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑𝑀) ≤ (𝐴↑𝑀)))
3	2	imbi2d 340	. . . . . 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 7377	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐴↑𝑗) = (𝐴↑𝑘))
5	4	breq1d 5112	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑𝑘) ≤ (𝐴↑𝑀)))
6	5	imbi2d 340	. . . . . 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 7377	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (𝐴↑𝑗) = (𝐴↑(𝑘 + 1)))
8	7	breq1d 5112	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀)))
9	8	imbi2d 340	. . . . . 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 7377	. . . . . . . 8 ⊢ (𝑗 = 𝑁 → (𝐴↑𝑗) = (𝐴↑𝑁))
11	10	breq1d 5112	. . . . . . 7 ⊢ (𝑗 = 𝑁 → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑𝑁) ≤ (𝐴↑𝑀)))
12	11	imbi2d 340	. . . . . 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 14019	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝐴↑𝑀) ∈ ℝ)
14	13	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑀) ∈ ℝ)
15	14	leidd 11720	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑀) ≤ (𝐴↑𝑀))
16		simprll 778	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ∈ ℝ)
17		1red 11151	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 1 ∈ ℝ)
18		simprlr 779	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑀 ∈ ℕ0)
19		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑘 ∈ (ℤ≥‘𝑀))
20		eluznn0 12852	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0)
21	18, 19, 20	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑘 ∈ ℕ0)
22		reexpcl 14019	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ)
23	16, 21, 22	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) ∈ ℝ)
24		simprrl 780	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 0 ≤ 𝐴)
25		expge0 14039	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴↑𝑘))
26	16, 21, 24, 25	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 0 ≤ (𝐴↑𝑘))
27		simprrr 781	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ≤ 1)
28	16, 17, 23, 26, 27	lemul2ad 12099	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) · 𝐴) ≤ ((𝐴↑𝑘) · 1))
29	16	recnd 11178	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ∈ ℂ)
30		expp1 14009	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
31	29, 21, 30	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
32	23	recnd 11178	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) ∈ ℂ)
33	32	mulridd 11167	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) · 1) = (𝐴↑𝑘))
34	33	eqcomd 2735	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) = ((𝐴↑𝑘) · 1))
35	28, 31, 34	3brtr4d 5134	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑘))
36		peano2nn0 12458	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
37	21, 36	syl 17	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝑘 + 1) ∈ ℕ0)
38		reexpcl 14019	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (𝑘 + 1) ∈ ℕ0) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
39	16, 37, 38	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
40	13	ad2antrl 728	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑀) ∈ ℝ)
41		letr 11244	. . . . . . . . . 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)))
42	39, 23, 40, 41	syl3anc 1373	. . . . . . . . 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)))
43	35, 42	mpand 695	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) ≤ (𝐴↑𝑀) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀)))
44	43	ex 412	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → ((𝐴↑𝑘) ≤ (𝐴↑𝑀) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀))))
45	44	a2d 29	. . . . . 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))))
46	3, 6, 9, 12, 15, 45	uzind4i 12845	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀)))
47	46	expd 415	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → ((0 ≤ 𝐴 ∧ 𝐴 ≤ 1) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))))
48	47	com12 32	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑁 ∈ (ℤ≥‘𝑀) → ((0 ≤ 𝐴 ∧ 𝐴 ≤ 1) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))))
49	48	3impia 1117	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((0 ≤ 𝐴 ∧ 𝐴 ≤ 1) → (𝐴↑𝑁) ≤ (𝐴↑𝑀)))
50	49	imp 406	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  ℂcc 11042  ℝcr 11043  0cc0 11044  1c1 11045   + caddc 11047   · cmul 11049   ≤ cle 11185  ℕ₀cn0 12418  ℤ_≥cuz 12769  ↑cexp 14002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-n0 12419  df-z 12506  df-uz 12770  df-seq 13943  df-exp 14003

This theorem is referenced by:  exple1  14118  leexp2rd  14196