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Theorem expord2t 6604
Description: The power of a positive number smaller than 1 decreases as its exponent increases.
Assertion
Ref Expression
expord2t |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A < 1)) -> (M < N <-> (A^N) < (A^M)))
Proof of Theorem expord2t
StepHypRef Expression
1 expordt 6602 . . 3 |- ((((1 / A) e. RR /\ M e. NN0 /\ N e. NN0) /\ 1 < (1 / A)) -> (M < N <-> ((1 / A)^M) < ((1 / A)^N)))
2 gt0ne0t 5618 . . . . . . . 8 |- ((A e. RR /\ 0 < A) -> A =/= 0)
3 rerecclt 5803 . . . . . . . 8 |- ((A e. RR /\ A =/= 0) -> (1 / A) e. RR)
42, 3syldan 467 . . . . . . 7 |- ((A e. RR /\ 0 < A) -> (1 / A) e. RR)
54expcom 374 . . . . . 6 |- (0 < A -> (A e. RR -> (1 / A) e. RR))
6 idd 61 . . . . . 6 |- (0 < A -> (M e. NN0 -> M e. NN0))
7 idd 61 . . . . . 6 |- (0 < A -> (N e. NN0 -> N e. NN0))
85, 6, 73anim123d 900 . . . . 5 |- (0 < A -> ((A e. RR /\ M e. NN0 /\ N e. NN0) -> ((1 / A) e. RR /\ M e. NN0 /\ N e. NN0)))
98impcom 351 . . . 4 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 0 < A) -> ((1 / A) e. RR /\ M e. NN0 /\ N e. NN0))
109adantrr 395 . . 3 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A < 1)) -> ((1 / A) e. RR /\ M e. NN0 /\ N e. NN0))
11 reclt1t 5898 . . . . . 6 |- ((A e. RR /\ 0 < A) -> (A < 1 <-> 1 < (1 / A)))
1211biimpa 416 . . . . 5 |- (((A e. RR /\ 0 < A) /\ A < 1) -> 1 < (1 / A))
1312anasss 440 . . . 4 |- ((A e. RR /\ (0 < A /\ A < 1)) -> 1 < (1 / A))
14133ad2antl1 809 . . 3 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A < 1)) -> 1 < (1 / A))
151, 10, 14sylanc 471 . 2 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A < 1)) -> (M < N <-> ((1 / A)^M) < ((1 / A)^N)))
16 ltrect 5884 . . . . 5 |- ((((A^N) e. RR /\ 0 < (A^N)) /\ ((A^M) e. RR /\ 0 < (A^M))) -> ((A^N) < (A^M) <-> (1 / (A^M)) < (1 / (A^N))))
17 reexpclt 6580 . . . . . . 7 |- ((A e. RR /\ N e. NN0) -> (A^N) e. RR)
1817adantr 389 . . . . . 6 |- (((A e. RR /\ N e. NN0) /\ 0 < A) -> (A^N) e. RR)
19183adantl2 804 . . . . 5 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 0 < A) -> (A^N) e. RR)
20 expgt0t 6589 . . . . . . 7 |- ((A e. RR /\ N e. NN0 /\ 0 < A) -> 0 < (A^N))
21203expa 833 . . . . . 6 |- (((A e. RR /\ N e. NN0) /\ 0 < A) -> 0 < (A^N))
22213adantl2 804 . . . . 5 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 0 < A) -> 0 < (A^N))
23 reexpclt 6580 . . . . . . 7 |- ((A e. RR /\ M e. NN0) -> (A^M) e. RR)
2423adantr 389 . . . . . 6 |- (((A e. RR /\ M e. NN0) /\ 0 < A) -> (A^M) e. RR)
25243adantl3 805 . . . . 5 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 0 < A) -> (A^M) e. RR)
26 expgt0t 6589 . . . . . . 7 |- ((A e. RR /\ M e. NN0 /\ 0 < A) -> 0 < (A^M))
27263expa 833 . . . . . 6 |- (((A e. RR /\ M e. NN0) /\ 0 < A) -> 0 < (A^M))
28273adantl3 805 . . . . 5 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 0 < A) -> 0 < (A^M))
2916, 19, 22, 25, 28syl2anc 472 . . . 4 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 0 < A) -> ((A^N) < (A^M) <-> (1 / (A^M)) < (1 / (A^N))))
30 recexpt 6595 . . . . . . . 8 |- ((A e. CC /\ M e. NN0 /\ A =/= 0) -> ((1 / A)^M) = (1 / (A^M)))
31303expa 833 . . . . . . 7 |- (((A e. CC /\ M e. NN0) /\ A =/= 0) -> ((1 / A)^M) = (1 / (A^M)))
32313adantl3 805 . . . . . 6 |- (((A e. CC /\ M e. NN0 /\ N e. NN0) /\ A =/= 0) -> ((1 / A)^M) = (1 / (A^M)))
33 recexpt 6595 . . . . . . . 8 |- ((A e. CC /\ N e. NN0 /\ A =/= 0) -> ((1 / A)^N) = (1 / (A^N)))
34333expa 833 . . . . . . 7 |- (((A e. CC /\ N e. NN0) /\ A =/= 0) -> ((1 / A)^N) = (1 / (A^N)))
35343adantl2 804 . . . . . 6 |- (((A e. CC /\ M e. NN0 /\ N e. NN0) /\ A =/= 0) -> ((1 / A)^N) = (1 / (A^N)))
3632, 35breq12d 2631 . . . . 5 |- (((A e. CC /\ M e. NN0 /\ N e. NN0) /\ A =/= 0) -> (((1 / A)^M) < ((1 / A)^N) <-> (1 / (A^M)) < (1 / (A^N))))
37 recnt 5313 . . . . . . 7 |- (A e. RR -> A e. CC)
38 id 59 . . . . . . 7 |- (M e. NN0 -> M e. NN0)
39 id 59 . . . . . . 7 |- (N e. NN0 -> N e. NN0)
4037, 38, 393anim123i 821 . . . . . 6 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (A e. CC /\ M e. NN0 /\ N e. NN0))
4140adantr 389 . . . . 5 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 0 < A) -> (A e. CC /\ M e. NN0 /\ N e. NN0))
4223ad2antl1 809 . . . . 5 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 0 < A) -> A =/= 0)
4336, 41, 42sylanc 471 . . . 4 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 0 < A) -> (((1 / A)^M) < ((1 / A)^N) <-> (1 / (A^M)) < (1 / (A^N))))
4429, 43bitr4d 531 . . 3 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 0 < A) -> ((A^N) < (A^M) <-> ((1 / A)^M) < ((1 / A)^N)))
4544adantrr 395 . 2 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A < 1)) -> ((A^N) < (A^M) <-> ((1 / A)^M) < ((1 / A)^N)))
4615, 45bitr4d 531 1 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A < 1)) -> (M < N <-> (A^N) < (A^M)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   =/= wne 1585   class class class wbr 2619  (class class class)co 3963  CCcc 5232  RRcr 5233  0cc0 5234  1c1 5235   / cdiv 5294  NN0cn0 5297   < clt 5486  ^cexp 6568
This theorem is referenced by:  expword2it 6605
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-n 5925  df-n0 6100  df-z 6136  df-seq1 6308  df-exp 6569
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