Theorem leexp2r 14357

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 7459	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (𝐴↑𝑗) = (𝐴↑𝑀))
2	1	breq1d 5118	. . . . . . 7 ⊢ (𝑗 = 𝑀 → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑𝑀) ≤ (𝐴↑𝑀)))
3	2	imbi2d 343	. . . . . 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 7459	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐴↑𝑗) = (𝐴↑𝑘))
5	4	breq1d 5118	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑𝑘) ≤ (𝐴↑𝑀)))
6	5	imbi2d 343	. . . . . 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 7459	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (𝐴↑𝑗) = (𝐴↑(𝑘 + 1)))
8	7	breq1d 5118	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀)))
9	8	imbi2d 343	. . . . . 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 7459	. . . . . . . 8 ⊢ (𝑗 = 𝑁 → (𝐴↑𝑗) = (𝐴↑𝑁))
11	10	breq1d 5118	. . . . . . 7 ⊢ (𝑗 = 𝑁 → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑𝑁) ≤ (𝐴↑𝑀)))
12	11	imbi2d 343	. . . . . 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 14261	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝐴↑𝑀) ∈ ℝ)
14	13	adantr 486	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑀) ∈ ℝ)
15	14	leidd 11922	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑀) ≤ (𝐴↑𝑀))
16		simprll 791	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ∈ ℝ)
17		1red 11348	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 1 ∈ ℝ)
18		simprlr 792	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑀 ∈ ℕ0)
19		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑘 ∈ (ℤ≥‘𝑀))
20		eluznn0 13084	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0)
21	18, 19, 20	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑘 ∈ ℕ0)
22		reexpcl 14261	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ)
23	16, 21, 22	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) ∈ ℝ)
24		simprrl 793	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 0 ≤ 𝐴)
25		expge0 14281	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴↑𝑘))
26	16, 21, 24, 25	syl3anc 1399	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 0 ≤ (𝐴↑𝑘))
27		simprrr 794	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ≤ 1)
28	16, 17, 23, 26, 27	lemul2ad 12297	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) · 𝐴) ≤ ((𝐴↑𝑘) · 1))
29	16	recnd 11376	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ∈ ℂ)
30		expp1 14251	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
31	29, 21, 30	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
32	23	recnd 11376	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) ∈ ℂ)
33	32	mulridd 11365	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) · 1) = (𝐴↑𝑘))
34	33	eqcomd 2769	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) = ((𝐴↑𝑘) · 1))
35	28, 31, 34	3brtr4d 5143	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑘))
36		peano2nn0 12686	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
37	21, 36	syl 18	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝑘 + 1) ∈ ℕ0)
38		reexpcl 14261	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (𝑘 + 1) ∈ ℕ0) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
39	16, 37, 38	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
40	13	ad2antrl 741	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑀) ∈ ℝ)
41		letr 11443	. . . . . . . . . 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 1399	. . . . . . . . 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 708	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) ≤ (𝐴↑𝑀) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀)))
44	43	ex 418	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → ((𝐴↑𝑘) ≤ (𝐴↑𝑀) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀))))
45	44	a2d 30	. . . . . 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 13077	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀)))
47	46	expd 421	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → ((0 ≤ 𝐴 ∧ 𝐴 ≤ 1) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))))
48	47	com12 33	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑁 ∈ (ℤ≥‘𝑀) → ((0 ≤ 𝐴 ∧ 𝐴 ≤ 1) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))))
49	48	3impia 1135	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((0 ≤ 𝐴 ∧ 𝐴 ≤ 1) → (𝐴↑𝑁) ≤ (𝐴↑𝑀)))
50	49	imp 412	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451  ℂcc 11237  ℝcr 11238  0cc0 11239  1c1 11240   + caddc 11242   · cmul 11244   ≤ cle 11383  ℕ₀cn0 12646  ℤ_≥cuz 13005  ↑cexp 14244

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-n0 12647  df-z 12734  df-uz 13006  df-seq 14185  df-exp 14245

This theorem is used by:  exple1  14360  leexp2rd  14440