Theorem leexp2r 8962

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 5463	. . . . . . . 8 ⊢ (𝑗 = 𝑀 → (A↑𝑗) = (A↑𝑀))
2	1	breq1d 3765	. . . . . . 7 ⊢ (𝑗 = 𝑀 → ((A↑𝑗) ≤ (A↑𝑀) ↔ (A↑𝑀) ≤ (A↑𝑀)))
3	2	imbi2d 219	. . . . . 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 5463	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (A↑𝑗) = (A↑𝑘))
5	4	breq1d 3765	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ((A↑𝑗) ≤ (A↑𝑀) ↔ (A↑𝑘) ≤ (A↑𝑀)))
6	5	imbi2d 219	. . . . . 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 5463	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (A↑𝑗) = (A↑(𝑘 + 1)))
8	7	breq1d 3765	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → ((A↑𝑗) ≤ (A↑𝑀) ↔ (A↑(𝑘 + 1)) ≤ (A↑𝑀)))
9	8	imbi2d 219	. . . . . 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 5463	. . . . . . . 8 ⊢ (𝑗 = 𝑁 → (A↑𝑗) = (A↑𝑁))
11	10	breq1d 3765	. . . . . . 7 ⊢ (𝑗 = 𝑁 → ((A↑𝑗) ≤ (A↑𝑀) ↔ (A↑𝑁) ≤ (A↑𝑀)))
12	11	imbi2d 219	. . . . . 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 8926	. . . . . . . . 9 ⊢ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (A↑𝑀) ∈ ℝ)
14	13	adantr 261	. . . . . . . 8 ⊢ (((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1)) → (A↑𝑀) ∈ ℝ)
15	14	leidd 7301	. . . . . . 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 489	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → A ∈ ℝ)
18		1red 6840	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → 1 ∈ ℝ)
19		simprlr 490	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → 𝑀 ∈ ℕ0)
20		simpl 102	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → 𝑘 ∈ (ℤ≥‘𝑀))
21		eluznn0 8313	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0)
22	19, 20, 21	syl2anc 391	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → 𝑘 ∈ ℕ0)
23		reexpcl 8926	. . . . . . . . . . . 12 ⊢ ((A ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (A↑𝑘) ∈ ℝ)
24	17, 22, 23	syl2anc 391	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → (A↑𝑘) ∈ ℝ)
25		simprrl 491	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → 0 ≤ A)
26		expge0 8945	. . . . . . . . . . . 12 ⊢ ((A ∈ ℝ ∧ 𝑘 ∈ ℕ0 ∧ 0 ≤ A) → 0 ≤ (A↑𝑘))
27	17, 22, 25, 26	syl3anc 1134	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → 0 ≤ (A↑𝑘))
28		simprrr 492	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → A ≤ 1)
29	17, 18, 24, 27, 28	lemul2ad 7687	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → ((A↑𝑘) · A) ≤ ((A↑𝑘) · 1))
30	17	recnd 6851	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → A ∈ ℂ)
31		expp1 8916	. . . . . . . . . . 11 ⊢ ((A ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (A↑(𝑘 + 1)) = ((A↑𝑘) · A))
32	30, 22, 31	syl2anc 391	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → (A↑(𝑘 + 1)) = ((A↑𝑘) · A))
33	24	recnd 6851	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → (A↑𝑘) ∈ ℂ)
34	33	mulid1d 6842	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → ((A↑𝑘) · 1) = (A↑𝑘))
35	34	eqcomd 2042	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → (A↑𝑘) = ((A↑𝑘) · 1))
36	29, 32, 35	3brtr4d 3785	. . . . . . . . 9 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → (A↑(𝑘 + 1)) ≤ (A↑𝑘))
37		peano2nn0 7998	. . . . . . . . . . . 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 8926	. . . . . . . . . . 11 ⊢ ((A ∈ ℝ ∧ (𝑘 + 1) ∈ ℕ0) → (A↑(𝑘 + 1)) ∈ ℝ)
40	17, 38, 39	syl2anc 391	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → (A↑(𝑘 + 1)) ∈ ℝ)
41	13	ad2antrl 459	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → (A↑𝑀) ∈ ℝ)
42		letr 6898	. . . . . . . . . 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 1134	. . . . . . . . 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 405	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1))) → ((A↑𝑘) ≤ (A↑𝑀) → (A↑(𝑘 + 1)) ≤ (A↑𝑀)))
45	44	ex 108	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1)) → ((A↑𝑘) ≤ (A↑𝑀) → (A↑(𝑘 + 1)) ≤ (A↑𝑀))))
46	45	a2d 23	. . . . . 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 8307	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) ∧ (0 ≤ A ∧ A ≤ 1)) → (A↑𝑁) ≤ (A↑𝑀)))
48	47	expd 245	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) → ((0 ≤ A ∧ A ≤ 1) → (A↑𝑁) ≤ (A↑𝑀))))
49	48	com12 27	. . 3 ⊢ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0) → (𝑁 ∈ (ℤ≥‘𝑀) → ((0 ≤ A ∧ A ≤ 1) → (A↑𝑁) ≤ (A↑𝑀))))
50	49	3impia 1100	. 2 ⊢ ((A ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ((0 ≤ A ∧ A ≤ 1) → (A↑𝑁) ≤ (A↑𝑀)))
51	50	imp 115	1 ⊢ (((A ∈ ℝ ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) ∧ (0 ≤ A ∧ A ≤ 1)) → (A↑𝑁) ≤ (A↑𝑀))

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242   ∈ wcel 1390   class class class wbr 3755  ‘cfv 4845  (class class class)co 5455  ℂcc 6709  ℝcr 6710  0cc0 6711  1c1 6712   + caddc 6714   · cmul 6716   ≤ cle 6858  ℕ₀cn0 7957  ℤcz 8021  ℤ_≥cuz 8249  ↑cexp 8908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-mulrcl 6782  ax-addcom 6783  ax-mulcom 6784  ax-addass 6785  ax-mulass 6786  ax-distr 6787  ax-i2m1 6788  ax-1rid 6790  ax-0id 6791  ax-rnegex 6792  ax-precex 6793  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-ltwlin 6796  ax-pre-lttrn 6797  ax-pre-apti 6798  ax-pre-ltadd 6799  ax-pre-mulgt0 6800  ax-pre-mulext 6801

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rmo 2308  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-if 3326  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-frec 5918  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863  df-sub 6981  df-neg 6982  df-reap 7359  df-ap 7366  df-div 7434  df-inn 7696  df-n0 7958  df-z 8022  df-uz 8250  df-iseq 8893  df-iexp 8909

This theorem is referenced by:  exple1  8964  leexp2rd  9063