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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 𝑀 0 𝑁 (ℤ𝑀)) (0 ≤ A A ≤ 1)) → (A𝑁) ≤ (A𝑀))

Proof of Theorem leexp2r
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5463 . . . . . . . 8 (𝑗 = 𝑀 → (A𝑗) = (A𝑀))
21breq1d 3765 . . . . . . 7 (𝑗 = 𝑀 → ((A𝑗) ≤ (A𝑀) ↔ (A𝑀) ≤ (A𝑀)))
32imbi2d 219 . . . . . 6 (𝑗 = 𝑀 → ((((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A𝑗) ≤ (A𝑀)) ↔ (((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A𝑀) ≤ (A𝑀))))
4 oveq2 5463 . . . . . . . 8 (𝑗 = 𝑘 → (A𝑗) = (A𝑘))
54breq1d 3765 . . . . . . 7 (𝑗 = 𝑘 → ((A𝑗) ≤ (A𝑀) ↔ (A𝑘) ≤ (A𝑀)))
65imbi2d 219 . . . . . 6 (𝑗 = 𝑘 → ((((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A𝑗) ≤ (A𝑀)) ↔ (((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A𝑘) ≤ (A𝑀))))
7 oveq2 5463 . . . . . . . 8 (𝑗 = (𝑘 + 1) → (A𝑗) = (A↑(𝑘 + 1)))
87breq1d 3765 . . . . . . 7 (𝑗 = (𝑘 + 1) → ((A𝑗) ≤ (A𝑀) ↔ (A↑(𝑘 + 1)) ≤ (A𝑀)))
98imbi2d 219 . . . . . 6 (𝑗 = (𝑘 + 1) → ((((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A𝑗) ≤ (A𝑀)) ↔ (((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A↑(𝑘 + 1)) ≤ (A𝑀))))
10 oveq2 5463 . . . . . . . 8 (𝑗 = 𝑁 → (A𝑗) = (A𝑁))
1110breq1d 3765 . . . . . . 7 (𝑗 = 𝑁 → ((A𝑗) ≤ (A𝑀) ↔ (A𝑁) ≤ (A𝑀)))
1211imbi2d 219 . . . . . 6 (𝑗 = 𝑁 → ((((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A𝑗) ≤ (A𝑀)) ↔ (((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A𝑁) ≤ (A𝑀))))
13 reexpcl 8926 . . . . . . . . 9 ((A 𝑀 0) → (A𝑀) ℝ)
1413adantr 261 . . . . . . . 8 (((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A𝑀) ℝ)
1514leidd 7301 . . . . . . 7 (((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A𝑀) ≤ (A𝑀))
1615a1i 9 . . . . . 6 (𝑀 ℤ → (((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A𝑀) ≤ (A𝑀)))
17 simprll 489 . . . . . . . . . . 11 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → A ℝ)
18 1red 6840 . . . . . . . . . . 11 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → 1 ℝ)
19 simprlr 490 . . . . . . . . . . . . 13 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → 𝑀 0)
20 simpl 102 . . . . . . . . . . . . 13 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → 𝑘 (ℤ𝑀))
21 eluznn0 8313 . . . . . . . . . . . . 13 ((𝑀 0 𝑘 (ℤ𝑀)) → 𝑘 0)
2219, 20, 21syl2anc 391 . . . . . . . . . . . 12 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → 𝑘 0)
23 reexpcl 8926 . . . . . . . . . . . 12 ((A 𝑘 0) → (A𝑘) ℝ)
2417, 22, 23syl2anc 391 . . . . . . . . . . 11 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → (A𝑘) ℝ)
25 simprrl 491 . . . . . . . . . . . 12 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → 0 ≤ A)
26 expge0 8945 . . . . . . . . . . . 12 ((A 𝑘 0 0 ≤ A) → 0 ≤ (A𝑘))
2717, 22, 25, 26syl3anc 1134 . . . . . . . . . . 11 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → 0 ≤ (A𝑘))
28 simprrr 492 . . . . . . . . . . 11 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → A ≤ 1)
2917, 18, 24, 27, 28lemul2ad 7687 . . . . . . . . . 10 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → ((A𝑘) · A) ≤ ((A𝑘) · 1))
3017recnd 6851 . . . . . . . . . . 11 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → A ℂ)
31 expp1 8916 . . . . . . . . . . 11 ((A 𝑘 0) → (A↑(𝑘 + 1)) = ((A𝑘) · A))
3230, 22, 31syl2anc 391 . . . . . . . . . 10 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → (A↑(𝑘 + 1)) = ((A𝑘) · A))
3324recnd 6851 . . . . . . . . . . . 12 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → (A𝑘) ℂ)
3433mulid1d 6842 . . . . . . . . . . 11 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → ((A𝑘) · 1) = (A𝑘))
3534eqcomd 2042 . . . . . . . . . 10 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → (A𝑘) = ((A𝑘) · 1))
3629, 32, 353brtr4d 3785 . . . . . . . . 9 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → (A↑(𝑘 + 1)) ≤ (A𝑘))
37 peano2nn0 7998 . . . . . . . . . . . 12 (𝑘 0 → (𝑘 + 1) 0)
3822, 37syl 14 . . . . . . . . . . 11 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → (𝑘 + 1) 0)
39 reexpcl 8926 . . . . . . . . . . 11 ((A (𝑘 + 1) 0) → (A↑(𝑘 + 1)) ℝ)
4017, 38, 39syl2anc 391 . . . . . . . . . 10 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → (A↑(𝑘 + 1)) ℝ)
4113ad2antrl 459 . . . . . . . . . 10 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → (A𝑀) ℝ)
42 letr 6898 . . . . . . . . . 10 (((A↑(𝑘 + 1)) (A𝑘) (A𝑀) ℝ) → (((A↑(𝑘 + 1)) ≤ (A𝑘) (A𝑘) ≤ (A𝑀)) → (A↑(𝑘 + 1)) ≤ (A𝑀)))
4340, 24, 41, 42syl3anc 1134 . . . . . . . . 9 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → (((A↑(𝑘 + 1)) ≤ (A𝑘) (A𝑘) ≤ (A𝑀)) → (A↑(𝑘 + 1)) ≤ (A𝑀)))
4436, 43mpand 405 . . . . . . . 8 ((𝑘 (ℤ𝑀) ((A 𝑀 0) (0 ≤ A A ≤ 1))) → ((A𝑘) ≤ (A𝑀) → (A↑(𝑘 + 1)) ≤ (A𝑀)))
4544ex 108 . . . . . . 7 (𝑘 (ℤ𝑀) → (((A 𝑀 0) (0 ≤ A A ≤ 1)) → ((A𝑘) ≤ (A𝑀) → (A↑(𝑘 + 1)) ≤ (A𝑀))))
4645a2d 23 . . . . . 6 (𝑘 (ℤ𝑀) → ((((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A𝑘) ≤ (A𝑀)) → (((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A↑(𝑘 + 1)) ≤ (A𝑀))))
473, 6, 9, 12, 16, 46uzind4 8307 . . . . 5 (𝑁 (ℤ𝑀) → (((A 𝑀 0) (0 ≤ A A ≤ 1)) → (A𝑁) ≤ (A𝑀)))
4847expd 245 . . . 4 (𝑁 (ℤ𝑀) → ((A 𝑀 0) → ((0 ≤ A A ≤ 1) → (A𝑁) ≤ (A𝑀))))
4948com12 27 . . 3 ((A 𝑀 0) → (𝑁 (ℤ𝑀) → ((0 ≤ A A ≤ 1) → (A𝑁) ≤ (A𝑀))))
50493impia 1100 . 2 ((A 𝑀 0 𝑁 (ℤ𝑀)) → ((0 ≤ A A ≤ 1) → (A𝑁) ≤ (A𝑀)))
5150imp 115 1 (((A 𝑀 0 𝑁 (ℤ𝑀)) (0 ≤ A A ≤ 1)) → (A𝑁) ≤ (A𝑀))
 Colors of variables: wff set class 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
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