Theorem leexp2r 14079

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 7365	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (𝐴↑𝑗) = (𝐴↑𝑀))
2	1	breq1d 5115	. . . . . . 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 7365	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐴↑𝑗) = (𝐴↑𝑘))
5	4	breq1d 5115	. . . . . . 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 7365	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (𝐴↑𝑗) = (𝐴↑(𝑘 + 1)))
8	7	breq1d 5115	. . . . . . 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 7365	. . . . . . . 8 ⊢ (𝑗 = 𝑁 → (𝐴↑𝑗) = (𝐴↑𝑁))
11	10	breq1d 5115	. . . . . . 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 13984	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝐴↑𝑀) ∈ ℝ)
14	13	adantr 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑀) ∈ ℝ)
15	14	leidd 11721	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑀) ≤ (𝐴↑𝑀))
16		simprll 777	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ∈ ℝ)
17		1red 11156	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 1 ∈ ℝ)
18		simprlr 778	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑀 ∈ ℕ0)
19		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑘 ∈ (ℤ≥‘𝑀))
20		eluznn0 12842	. . . . . . . . . . . . 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 13984	. . . . . . . . . . . 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 779	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 0 ≤ 𝐴)
25		expge0 14004	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴↑𝑘))
26	16, 21, 24, 25	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 0 ≤ (𝐴↑𝑘))
27		simprrr 780	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ≤ 1)
28	16, 17, 23, 26, 27	lemul2ad 12095	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) · 𝐴) ≤ ((𝐴↑𝑘) · 1))
29	16	recnd 11183	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ∈ ℂ)
30		expp1 13974	. . . . . . . . . . 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 11183	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) ∈ ℂ)
33	32	mulid1d 11172	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) · 1) = (𝐴↑𝑘))
34	33	eqcomd 2742	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) = ((𝐴↑𝑘) · 1))
35	28, 31, 34	3brtr4d 5137	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑘))
36		peano2nn0 12453	. . . . . . . . . . . 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 13984	. . . . . . . . . . 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 726	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑀) ∈ ℝ)
41		letr 11249	. . . . . . . . . 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 1371	. . . . . . . . 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 693	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) ≤ (𝐴↑𝑀) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀)))
44	43	ex 413	. . . . . . 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 12835	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀)))
47	46	expd 416	. . . 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 407	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  ℂcc 11049  ℝcr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   ≤ cle 11190  ℕ₀cn0 12413  ℤ_≥cuz 12763  ↑cexp 13967

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-seq 13907  df-exp 13968

This theorem is referenced by:  exple1  14081  leexp2rd  14158