Theorem expword2it 6536

Description: Weak ordering relationship for exponentiation. (Contributed by Paul Chapman, 14-Jan-2008.)

Assertion

Ref	Expression
expword2it	|- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A <_ 1 /\ M < N)) -> (A^N) <_ (A^M))

Proof of Theorem expword2it

Step	Hyp	Ref	Expression
1		expord2t 6535	. . . . . . . . . . . . . . . 16 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A < 1)) -> (M < N <-> (A^N) < (A^M)))
2	1	biimpd 153	. . . . . . . . . . . . . . 15 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A < 1)) -> (M < N -> (A^N) < (A^M)))
3	2	ex 373	. . . . . . . . . . . . . 14 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> ((0 < A /\ A < 1) -> (M < N -> (A^N) < (A^M))))
4	3	exp3a 375	. . . . . . . . . . . . 13 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (0 < A -> (A < 1 -> (M < N -> (A^N) < (A^M)))))
5	4	com34 36	. . . . . . . . . . . 12 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (0 < A -> (M < N -> (A < 1 -> (A^N) < (A^M)))))
6	5	imp3a 361	. . . . . . . . . . 11 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> ((0 < A /\ M < N) -> (A < 1 -> (A^N) < (A^M))))
7	6	imp 350	. . . . . . . . . 10 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ M < N)) -> (A < 1 -> (A^N) < (A^M)))
8		1expt 6516	. . . . . . . . . . . . . . . . 17 |- (M e. NN0 -> (1^M) = 1)
9	8	adantr 389	. . . . . . . . . . . . . . . 16 |- ((M e. NN0 /\ N e. NN0) -> (1^M) = 1)
10		1expt 6516	. . . . . . . . . . . . . . . . 17 |- (N e. NN0 -> (1^N) = 1)
11	10	adantl 388	. . . . . . . . . . . . . . . 16 |- ((M e. NN0 /\ N e. NN0) -> (1^N) = 1)
12	9, 11	eqtr4d 1502	. . . . . . . . . . . . . . 15 |- ((M e. NN0 /\ N e. NN0) -> (1^M) = (1^N))
13	12	3adant1 795	. . . . . . . . . . . . . 14 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (1^M) = (1^N))
14	13	adantr 389	. . . . . . . . . . . . 13 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ A = 1) -> (1^M) = (1^N))
15		opreq1 3953	. . . . . . . . . . . . . 14 |- (A = 1 -> (A^M) = (1^M))
16	15	adantl 388	. . . . . . . . . . . . 13 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ A = 1) -> (A^M) = (1^M))
17		opreq1 3953	. . . . . . . . . . . . . 14 |- (A = 1 -> (A^N) = (1^N))
18	17	adantl 388	. . . . . . . . . . . . 13 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ A = 1) -> (A^N) = (1^N))
19	14, 16, 18	3eqtr4rd 1510	. . . . . . . . . . . 12 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ A = 1) -> (A^N) = (A^M))
20	19	ex 373	. . . . . . . . . . 11 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (A = 1 -> (A^N) = (A^M)))
21	20	adantr 389	. . . . . . . . . 10 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ M < N)) -> (A = 1 -> (A^N) = (A^M)))
22	7, 21	orim12d 563	. . . . . . . . 9 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ M < N)) -> ((A < 1 \/ A = 1) -> ((A^N) < (A^M) \/ (A^N) = (A^M))))
23		1re 5407	. . . . . . . . . . . 12 |- 1 e. RR
24		leloet 5491	. . . . . . . . . . . 12 |- ((A e. RR /\ 1 e. RR) -> (A <_ 1 <-> (A < 1 \/ A = 1)))
25	23, 24	mpan2 694	. . . . . . . . . . 11 |- (A e. RR -> (A <_ 1 <-> (A < 1 \/ A = 1)))
26	25	3ad2ant1 798	. . . . . . . . . 10 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (A <_ 1 <-> (A < 1 \/ A = 1)))
27	26	adantr 389	. . . . . . . . 9 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ M < N)) -> (A <_ 1 <-> (A < 1 \/ A = 1)))
28		reexpclt 6512	. . . . . . . . . . . . 13 |- ((A e. RR /\ N e. NN0) -> (A^N) e. RR)
29	28	3adant2 796	. . . . . . . . . . . 12 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (A^N) e. RR)
30		reexpclt 6512	. . . . . . . . . . . . 13 |- ((A e. RR /\ M e. NN0) -> (A^M) e. RR)
31	30	3adant3 797	. . . . . . . . . . . 12 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (A^M) e. RR)
32	29, 31	jca 288	. . . . . . . . . . 11 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> ((A^N) e. RR /\ (A^M) e. RR))
33		leloet 5491	. . . . . . . . . . 11 |- (((A^N) e. RR /\ (A^M) e. RR) -> ((A^N) <_ (A^M) <-> ((A^N) < (A^M) \/ (A^N) = (A^M))))
34	32, 33	syl 10	. . . . . . . . . 10 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> ((A^N) <_ (A^M) <-> ((A^N) < (A^M) \/ (A^N) = (A^M))))
35	34	adantr 389	. . . . . . . . 9 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ M < N)) -> ((A^N) <_ (A^M) <-> ((A^N) < (A^M) \/ (A^N) = (A^M))))
36	22, 27, 35	3imtr4d 541	. . . . . . . 8 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ M < N)) -> (A <_ 1 -> (A^N) <_ (A^M)))
37	36	ex 373	. . . . . . 7 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> ((0 < A /\ M < N) -> (A <_ 1 -> (A^N) <_ (A^M))))
38	37	exp3a 375	. . . . . 6 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (0 < A -> (M < N -> (A <_ 1 -> (A^N) <_ (A^M)))))
39	38	com34 36	. . . . 5 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (0 < A -> (A <_ 1 -> (M < N -> (A^N) <_ (A^M)))))
40	39	imp3a 361	. . . 4 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> ((0 < A /\ A <_ 1) -> (M < N -> (A^N) <_ (A^M))))
41	40	imp3a 361	. . 3 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (((0 < A /\ A <_ 1) /\ M < N) -> (A^N) <_ (A^M)))
42	41	imp 350	. 2 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ ((0 < A /\ A <_ 1) /\ M < N)) -> (A^N) <_ (A^M))
43		df-3an 775	. 2 |- ((0 < A /\ A <_ 1 /\ M < N) <-> ((0 < A /\ A <_ 1) /\ M < N))
44	42, 43	sylan2b 452	1 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A <_ 1 /\ M < N)) -> (A^N) <_ (A^M))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   class class class wbr 2609  (class class class)co 3948  RRcr 5205  0cc0 5206  1c1 5207   <_ cle 5267  NN0cn0 5269   < clt 5458  ^cexp 6500

This theorem is referenced by:  sin01bndlem2 7410  cos01bndlem2 7412  sin01gt0 7418

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-n0 6047  df-z 6083  df-seq1 6245  df-exp 6501

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