Theorem leexp2r 10979

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 6066	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (𝐴↑𝑗) = (𝐴↑𝑀))
2	1	breq1d 4124	. . . . . . 7 ⊢ (𝑗 = 𝑀 → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑𝑀) ≤ (𝐴↑𝑀)))
3	2	imbi2d 230	. . . . . 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 6066	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝐴↑𝑗) = (𝐴↑𝑘))
5	4	breq1d 4124	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑𝑘) ≤ (𝐴↑𝑀)))
6	5	imbi2d 230	. . . . . 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 6066	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (𝐴↑𝑗) = (𝐴↑(𝑘 + 1)))
8	7	breq1d 4124	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀)))
9	8	imbi2d 230	. . . . . 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 6066	. . . . . . . 8 ⊢ (𝑗 = 𝑁 → (𝐴↑𝑗) = (𝐴↑𝑁))
11	10	breq1d 4124	. . . . . . 7 ⊢ (𝑗 = 𝑁 → ((𝐴↑𝑗) ≤ (𝐴↑𝑀) ↔ (𝐴↑𝑁) ≤ (𝐴↑𝑀)))
12	11	imbi2d 230	. . . . . 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 10942	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝐴↑𝑀) ∈ ℝ)
14	13	adantr 276	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑀) ∈ ℝ)
15	14	leidd 8805	. . . . . . 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 539	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ∈ ℝ)
18		1red 8305	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 1 ∈ ℝ)
19		simprlr 540	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑀 ∈ ℕ0)
20		simpl 109	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑘 ∈ (ℤ≥‘𝑀))
21		eluznn0 9949	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0)
22	19, 20, 21	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝑘 ∈ ℕ0)
23		reexpcl 10942	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ)
24	17, 22, 23	syl2anc 411	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) ∈ ℝ)
25		simprrl 541	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 0 ≤ 𝐴)
26		expge0 10961	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 0 ≤ (𝐴↑𝑘))
27	17, 22, 25, 26	syl3anc 1274	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 0 ≤ (𝐴↑𝑘))
28		simprrr 542	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ≤ 1)
29	17, 18, 24, 27, 28	lemul2ad 9231	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) · 𝐴) ≤ ((𝐴↑𝑘) · 1))
30	17	recnd 8318	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → 𝐴 ∈ ℂ)
31		expp1 10932	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
32	30, 22, 31	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) = ((𝐴↑𝑘) · 𝐴))
33	24	recnd 8318	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) ∈ ℂ)
34	33	mulridd 8307	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) · 1) = (𝐴↑𝑘))
35	34	eqcomd 2240	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑘) = ((𝐴↑𝑘) · 1))
36	29, 32, 35	3brtr4d 4146	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑘))
37		peano2nn0 9553	. . . . . . . . . . . 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 10942	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ (𝑘 + 1) ∈ ℕ0) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
40	17, 38, 39	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑(𝑘 + 1)) ∈ ℝ)
41	13	ad2antrl 490	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → (𝐴↑𝑀) ∈ ℝ)
42		letr 8372	. . . . . . . . . 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 1274	. . . . . . . . 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 429	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1))) → ((𝐴↑𝑘) ≤ (𝐴↑𝑀) → (𝐴↑(𝑘 + 1)) ≤ (𝐴↑𝑀)))
45	44	ex 115	. . . . . . 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 9938	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀)))
48	47	expd 258	. . . 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 1227	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((0 ≤ 𝐴 ∧ 𝐴 ≤ 1) → (𝐴↑𝑁) ≤ (𝐴↑𝑀)))
51	50	imp 124	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ (0 ≤ 𝐴 ∧ 𝐴 ≤ 1)) → (𝐴↑𝑁) ≤ (𝐴↑𝑀))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205   class class class wbr 4114  ‘cfv 5357  (class class class)co 6058  ℂcc 8141  ℝcr 8142  0cc0 8143  1c1 8144   + caddc 8146   · cmul 8148   ≤ cle 8325  ℕ₀cn0 9513  ℤcz 9594  ℤ_≥cuz 9871  ↑cexp 10924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-seqfrec 10834  df-exp 10925

This theorem is referenced by:  exple1  10981  leexp2rd  11090