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Theorem expord2t 6554
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 6552 . . 3 |- ((((1 / A) e. RR /\ M e. NN0 /\ N e. NN0) /\ 1 < (1 / A)) -> (M < N <-> ((1 / A)^M) < ((1 / A)^N)))
2 gt0ne0t 5606 . . . . . . . 8 |- ((A e. RR /\ 0 < A) -> A =/= 0)
3 rerecclt 5773 . . . . . . . 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 899 . . . . 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 5860 . . . . . 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 808 . . 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 5846 . . . . 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 6530 . . . . . . 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 803 . . . . 5 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 0 < A) -> (A^N) e. RR)
20 expgt0t 6539 . . . . . . 7 |- ((A e. RR /\ N e. NN0 /\ 0 < A) -> 0 < (A^N))
21203expa 832 . . . . . 6 |- (((A e. RR /\ N e. NN0) /\ 0 < A) -> 0 < (A^N))
22213adantl2 803 . . . . 5 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 0 < A) -> 0 < (A^N))
23 reexpclt 6530 . . . . . . 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 804 . . . . 5 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ 0 < A) -> (A^M) e. RR)
26 expgt0t 6539 . . . . . . 7 |- ((A e. RR /\ M e. NN0 /\ 0 < A) -> 0 < (A^M))
27263expa 832 . . . . . 6 |- (((A e. RR /\ M e. NN0) /\ 0 < A) -> 0 < (A^M))
28273adantl3 804 . . . . 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 6545 . . . . . . . 8 |- ((A e. CC /\ M e. NN0 /\ A =/= 0) -> ((1 / A)^M) = (1 / (A^M)))
31303expa 832 . . . . . . 7 |- (((A e. CC /\ M e. NN0) /\ A =/= 0) -> ((1 / A)^M) = (1 / (A^M)))
32313adantl3 804 . . . . . 6 |- (((A e. CC /\ M e. NN0 /\ N e. NN0) /\ A =/= 0) -> ((1 / A)^M) = (1 / (A^M)))
33 recexpt 6545 . . . . . . . 8 |- ((A e. CC /\ N e. NN0 /\ A =/= 0) -> ((1 / A)^N) = (1 / (A^N)))
34333expa 832 . . . . . . 7 |- (((A e. CC /\ N e. NN0) /\ A =/= 0) -> ((1 / A)^N) = (1 / (A^N)))
35343adantl2 803 . . . . . 6 |- (((A e. CC /\ M e. NN0 /\ N e. NN0) /\ A =/= 0) -> ((1 / A)^N) = (1 / (A^N)))
3632, 35breq12d 2628 . . . . 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 5300 . . . . . . 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 820 . . . . . 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 808 . . . . 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 530 . . 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 530 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 774   = wceq 955   e. wcel 957   =/= wne 1584   class class class wbr 2616  (class class class)co 3960  CCcc 5219  RRcr 5220  0cc0 5221  1c1 5222   / cdiv 5281  NN0cn0 5284   < clt 5473  ^cexp 6518
This theorem is referenced by:  expword2it 6555
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-inf2 4612
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-nel 1587  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-int 2531  df-iun 2565  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-om 3129  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-f1 3192  df-fo 3193  df-f1o 3194  df-fv 3195  df-rdg 3929  df-opr 3962  df-oprab 3963  df-1st 4076  df-2nd 4077  df-1o 4130  df-oadd 4132  df-omul 4133  df-er 4258  df-ec 4260  df-qs 4263  df-en 4364  df-dom 4365  df-sdom 4366  df-ni 4987  df-pli 4988  df-mi 4989  df-lti 4990  df-plpq 5022  df-mpq 5023  df-enq 5024  df-nq 5025  df-plq 5026  df-mq 5027  df-rq 5028  df-ltq 5029  df-1q 5030  df-np 5073  df-1p 5074  df-plp 5075  df-mp 5076  df-ltp 5077  df-plpr 5151  df-mpr 5152  df-enr 5153  df-nr 5154  df-plr 5155  df-mr 5156  df-ltr 5157  df-0r 5158  df-1r 5159  df-m1r 5160  df-c 5227  df-0 5228  df-1 5229  df-i 5230  df-r 5231  df-plus 5232  df-mul 5233  df-lt 5234  df-sub 5343  df-neg 5345  df-pnf 5474  df-mnf 5475  df-xr 5476  df-ltxr 5477  df-le 5478  df-div 5686  df-n 5887  df-n0 6061  df-z 6097  df-seq1 6263  df-exp 6519
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