Theorem expword2i 7850

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

Assertion

Ref	Expression
expword2i	|- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A <_ 1 /\ M < N)) -> (A^N) <_ (A^M))

Proof of Theorem expword2i

Step	Hyp	Ref	Expression
1		expord2 7849	. . . . . . . . . . . . . . . 16 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A < 1)) -> (M < N <-> (A^N) < (A^M)))
2	1	biimpd 170	. . . . . . . . . . . . . . 15 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ A < 1)) -> (M < N -> (A^N) < (A^M)))
3	2	ex 402	. . . . . . . . . . . . . 14 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> ((0 < A /\ A < 1) -> (M < N -> (A^N) < (A^M))))
4	3	exp3a 405	. . . . . . . . . . . . 13 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (0 < A -> (A < 1 -> (M < N -> (A^N) < (A^M)))))
5	4	com34 40	. . . . . . . . . . . 12 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (0 < A -> (M < N -> (A < 1 -> (A^N) < (A^M)))))
6	5	imp3a 388	. . . . . . . . . . 11 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> ((0 < A /\ M < N) -> (A < 1 -> (A^N) < (A^M))))
7	6	imp 377	. . . . . . . . . 10 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ M < N)) -> (A < 1 -> (A^N) < (A^M)))
8		1exp 7827	. . . . . . . . . . . . . . . . 17 |- (M e. NN0 -> (1^M) = 1)
9	8	adantr 425	. . . . . . . . . . . . . . . 16 |- ((M e. NN0 /\ N e. NN0) -> (1^M) = 1)
10		1exp 7827	. . . . . . . . . . . . . . . . 17 |- (N e. NN0 -> (1^N) = 1)
11	10	adantl 424	. . . . . . . . . . . . . . . 16 |- ((M e. NN0 /\ N e. NN0) -> (1^N) = 1)
12	9, 11	eqtr4d 1928	. . . . . . . . . . . . . . 15 |- ((M e. NN0 /\ N e. NN0) -> (1^M) = (1^N))
13	12	3adant1 894	. . . . . . . . . . . . . 14 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (1^M) = (1^N))
14	13	adantr 425	. . . . . . . . . . . . 13 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ A = 1) -> (1^M) = (1^N))
15		opreq1 4889	. . . . . . . . . . . . . 14 |- (A = 1 -> (A^M) = (1^M))
16	15	adantl 424	. . . . . . . . . . . . 13 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ A = 1) -> (A^M) = (1^M))
17		opreq1 4889	. . . . . . . . . . . . . 14 |- (A = 1 -> (A^N) = (1^N))
18	17	adantl 424	. . . . . . . . . . . . 13 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ A = 1) -> (A^N) = (1^N))
19	14, 16, 18	3eqtr4rd 1939	. . . . . . . . . . . 12 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ A = 1) -> (A^N) = (A^M))
20	19	ex 402	. . . . . . . . . . 11 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (A = 1 -> (A^N) = (A^M)))
21	20	adantr 425	. . . . . . . . . 10 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ M < N)) -> (A = 1 -> (A^N) = (A^M)))
22	7, 21	orim12d 624	. . . . . . . . 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 6598	. . . . . . . . . . . 12 |- 1 e. RR
24		leloe 6688	. . . . . . . . . . . 12 |- ((A e. RR /\ 1 e. RR) -> (A <_ 1 <-> (A < 1 \/ A = 1)))
25	23, 24	mpan2 760	. . . . . . . . . . 11 |- (A e. RR -> (A <_ 1 <-> (A < 1 \/ A = 1)))
26	25	3ad2ant1 897	. . . . . . . . . 10 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (A <_ 1 <-> (A < 1 \/ A = 1)))
27	26	adantr 425	. . . . . . . . 9 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ M < N)) -> (A <_ 1 <-> (A < 1 \/ A = 1)))
28		reexpcl 7823	. . . . . . . . . . . 12 |- ((A e. RR /\ N e. NN0) -> (A^N) e. RR)
29	28	3adant2 895	. . . . . . . . . . 11 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (A^N) e. RR)
30		reexpcl 7823	. . . . . . . . . . . 12 |- ((A e. RR /\ M e. NN0) -> (A^M) e. RR)
31	30	3adant3 896	. . . . . . . . . . 11 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (A^M) e. RR)
32		leloe 6688	. . . . . . . . . . 11 |- (((A^N) e. RR /\ (A^M) e. RR) -> ((A^N) <_ (A^M) <-> ((A^N) < (A^M) \/ (A^N) = (A^M))))
33	29, 31, 32	syl11anc 524	. . . . . . . . . 10 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> ((A^N) <_ (A^M) <-> ((A^N) < (A^M) \/ (A^N) = (A^M))))
34	33	adantr 425	. . . . . . . . 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))))
35	22, 27, 34	3imtr4d 602	. . . . . . . 8 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ (0 < A /\ M < N)) -> (A <_ 1 -> (A^N) <_ (A^M)))
36	35	ex 402	. . . . . . 7 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> ((0 < A /\ M < N) -> (A <_ 1 -> (A^N) <_ (A^M))))
37	36	exp3a 405	. . . . . 6 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (0 < A -> (M < N -> (A <_ 1 -> (A^N) <_ (A^M)))))
38	37	com34 40	. . . . 5 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (0 < A -> (A <_ 1 -> (M < N -> (A^N) <_ (A^M)))))
39	38	imp3a 388	. . . 4 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> ((0 < A /\ A <_ 1) -> (M < N -> (A^N) <_ (A^M))))
40	39	imp3a 388	. . 3 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (((0 < A /\ A <_ 1) /\ M < N) -> (A^N) <_ (A^M)))
41	40	imp 377	. 2 |- (((A e. RR /\ M e. NN0 /\ N e. NN0) /\ ((0 < A /\ A <_ 1) /\ M < N)) -> (A^N) <_ (A^M))
42		df-3an 860	. 2 |- ((0 < A /\ A <_ 1 /\ M < N) <-> ((0 < A /\ A <_ 1) /\ M < N))
43	41, 42	sylan2b 501	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 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   class class class wbr 3338  (class class class)co 4884  RRcr 6385  0cc0 6386  1c1 6387   <_ cle 6448  NN0cn0 6450   < clt 6653  ^cexp 7811

This theorem is referenced by:  sin01bndlem2 8734  cos01bndlem2 8736  sin01gt0 8742

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812

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