Theorem expword2it 6544

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 6543	. . . . . . . . . . . . . . . 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 6524	. . . . . . . . . . . . . . . . 17 |- (M e. NN0 -> (1^M) = 1)
9	8	adantr 389	. . . . . . . . . . . . . . . 16 |- ((M e. NN0 /\ N e. NN0) -> (1^M) = 1)
10		1expt 6524	. . . . . . . . . . . . . . . . 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 1507	. . . . . . . . . . . . . . 15 |- ((M e. NN0 /\ N e. NN0) -> (1^M) = (1^N))
13	12	3adant1 796	. . . . . . . . . . . . . 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 3959	. . . . . . . . . . . . . 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 3959	. . . . . . . . . . . . . 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 1515	. . . . . . . . . . . 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 564	. . . . . . . . 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 5415	. . . . . . . . . . . 12 |- 1 e. RR
24		leloet 5499	. . . . . . . . . . . 12 |- ((A e. RR /\ 1 e. RR) -> (A <_ 1 <-> (A < 1 \/ A = 1)))
25	23, 24	mpan2 695	. . . . . . . . . . 11 |- (A e. RR -> (A <_ 1 <-> (A < 1 \/ A = 1)))
26	25	3ad2ant1 799	. . . . . . . . . 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 6520	. . . . . . . . . . . . 13 |- ((A e. RR /\ N e. NN0) -> (A^N) e. RR)
29	28	3adant2 797	. . . . . . . . . . . 12 |- ((A e. RR /\ M e. NN0 /\ N e. NN0) -> (A^N) e. RR)
30		reexpclt 6520	. . . . . . . . . . . . 13 |- ((A e. RR /\ M e. NN0) -> (A^M) e. RR)
31	30	3adant3 798	. . . . . . . . . . . 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 5499	. . . . . . . . . . 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 542	. . . . . . . 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 776	. 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 774   = wceq 954   e. wcel 956   class class class wbr 2614  (class class class)co 3954  RRcr 5213  0cc0 5214  1c1 5215   <_ cle 5275  NN0cn0 5277   < clt 5466  ^cexp 6508

This theorem is referenced by:  sin01bndlem2 7418  cos01bndlem2 7420  sin01gt0 7426

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680  df-n 5881  df-n0 6055  df-z 6091  df-seq1 6253  df-exp 6509

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