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Theorem cxple2 24443
Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
Assertion
Ref Expression
cxple2 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )
Proof of Theorem cxple2
StepHypRef Expression
1 simpl1l 1112 . . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> A e. RR )
2 simpr 477 . . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> 0 < A )
31, 2elrpd 11869 . . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> A e. RR+ )
43adantr 481 . . . 4 |- ( ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) /\ 0 < B ) -> A e. RR+ )
5 simp2l 1087 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> B e. RR )
65ad2antrr 762 . . . . 5 |- ( ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) /\ 0 < B ) -> B e. RR )
7 simpr 477 . . . . 5 |- ( ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) /\ 0 < B ) -> 0 < B )
86, 7elrpd 11869 . . . 4 |- ( ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) /\ 0 < B ) -> B e. RR+ )
9 simp3 1063 . . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> C e. RR+ )
109ad2antrr 762 . . . 4 |- ( ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) /\ 0 < B ) -> C e. RR+ )
11 simp3 1063 . . . . . . . 8 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> C e. RR+ )
1211rpred 11872 . . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> C e. RR )
13 relogcl 24322 . . . . . . . 8 |- ( A e. RR+ -> ( log ` A ) e. RR )
14133ad2ant1 1082 . . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( log ` A ) e. RR )
1512, 14remulcld 10070 . . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( C x. ( log ` A ) ) e. RR )
16 relogcl 24322 . . . . . . . 8 |- ( B e. RR+ -> ( log ` B ) e. RR )
17163ad2ant2 1083 . . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( log ` B ) e. RR )
1812, 17remulcld 10070 . . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( C x. ( log ` B ) ) e. RR )
19 efle 14848 . . . . . 6 |- ( ( ( C x. ( log ` A ) ) e. RR /\ ( C x. ( log ` B ) ) e. RR ) -> ( ( C x. ( log ` A ) ) <_ ( C x. ( log ` B ) ) <-> ( exp ` ( C x. ( log ` A ) ) ) <_ ( exp ` ( C x. ( log ` B ) ) ) ) )
2015, 18, 19syl2anc 693 . . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( ( C x. ( log ` A ) ) <_ ( C x. ( log ` B ) ) <-> ( exp ` ( C x. ( log ` A ) ) ) <_ ( exp ` ( C x. ( log ` B ) ) ) ) )
21 efle 14848 . . . . . . 7 |- ( ( ( log ` A ) e. RR /\ ( log ` B ) e. RR ) -> ( ( log ` A ) <_ ( log ` B ) <-> ( exp ` ( log ` A ) ) <_ ( exp ` ( log ` B ) ) ) )
2214, 17, 21syl2anc 693 . . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( ( log ` A ) <_ ( log ` B ) <-> ( exp ` ( log ` A ) ) <_ ( exp ` ( log ` B ) ) ) )
2314, 17, 11lemul2d 11916 . . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( ( log ` A ) <_ ( log ` B ) <-> ( C x. ( log ` A ) ) <_ ( C x. ( log ` B ) ) ) )
24 reeflog 24327 . . . . . . . 8 |- ( A e. RR+ -> ( exp ` ( log ` A ) ) = A )
25243ad2ant1 1082 . . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( exp ` ( log ` A ) ) = A )
26 reeflog 24327 . . . . . . . 8 |- ( B e. RR+ -> ( exp ` ( log ` B ) ) = B )
27263ad2ant2 1083 . . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( exp ` ( log ` B ) ) = B )
2825, 27breq12d 4666 . . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( ( exp ` ( log ` A ) ) <_ ( exp ` ( log ` B ) ) <-> A <_ B ) )
2922, 23, 283bitr3rd 299 . . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A <_ B <-> ( C x. ( log ` A ) ) <_ ( C x. ( log ` B ) ) ) )
30 rpre 11839 . . . . . . . . 9 |- ( A e. RR+ -> A e. RR )
31303ad2ant1 1082 . . . . . . . 8 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> A e. RR )
3231recnd 10068 . . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> A e. CC )
33 rpne0 11848 . . . . . . . 8 |- ( A e. RR+ -> A =/= 0 )
34333ad2ant1 1082 . . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> A =/= 0 )
3512recnd 10068 . . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> C e. CC )
36 cxpef 24411 . . . . . . 7 |- ( ( A e. CC /\ A =/= 0 /\ C e. CC ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
3732, 34, 35, 36syl3anc 1326 . . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
38 rpre 11839 . . . . . . . . 9 |- ( B e. RR+ -> B e. RR )
39383ad2ant2 1083 . . . . . . . 8 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> B e. RR )
4039recnd 10068 . . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> B e. CC )
41 rpne0 11848 . . . . . . . 8 |- ( B e. RR+ -> B =/= 0 )
42413ad2ant2 1083 . . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> B =/= 0 )
43 cxpef 24411 . . . . . . 7 |- ( ( B e. CC /\ B =/= 0 /\ C e. CC ) -> ( B ^c C ) = ( exp ` ( C x. ( log ` B ) ) ) )
4440, 42, 35, 43syl3anc 1326 . . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( B ^c C ) = ( exp ` ( C x. ( log ` B ) ) ) )
4537, 44breq12d 4666 . . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( ( A ^c C ) <_ ( B ^c C ) <-> ( exp ` ( C x. ( log ` A ) ) ) <_ ( exp ` ( C x. ( log ` B ) ) ) ) )
4620, 29, 453bitr4d 300 . . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )
474, 8, 10, 46syl3anc 1326 . . 3 |- ( ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) /\ 0 < B ) -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )
48 0re 10040 . . . . . . . 8 |- 0 e. RR
49 simp1l 1085 . . . . . . . 8 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> A e. RR )
50 ltnle 10117 . . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A <-> -. A <_ 0 ) )
5148, 49, 50sylancr 695 . . . . . . 7 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> ( 0 < A <-> -. A <_ 0 ) )
5251biimpa 501 . . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> -. A <_ 0 )
539rpred 11872 . . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> C e. RR )
5453adantr 481 . . . . . . . . 9 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> C e. RR )
55 rpcxpcl 24422 . . . . . . . . 9 |- ( ( A e. RR+ /\ C e. RR ) -> ( A ^c C ) e. RR+ )
563, 54, 55syl2anc 693 . . . . . . . 8 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> ( A ^c C ) e. RR+ )
57 rpgt0 11844 . . . . . . . . 9 |- ( ( A ^c C ) e. RR+ -> 0 < ( A ^c C ) )
58 rpre 11839 . . . . . . . . . 10 |- ( ( A ^c C ) e. RR+ -> ( A ^c C ) e. RR )
59 ltnle 10117 . . . . . . . . . 10 |- ( ( 0 e. RR /\ ( A ^c C ) e. RR ) -> ( 0 < ( A ^c C ) <-> -. ( A ^c C ) <_ 0 ) )
6048, 58, 59sylancr 695 . . . . . . . . 9 |- ( ( A ^c C ) e. RR+ -> ( 0 < ( A ^c C ) <-> -. ( A ^c C ) <_ 0 ) )
6157, 60mpbid 222 . . . . . . . 8 |- ( ( A ^c C ) e. RR+ -> -. ( A ^c C ) <_ 0 )
6256, 61syl 17 . . . . . . 7 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> -. ( A ^c C ) <_ 0 )
6353recnd 10068 . . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> C e. CC )
649rpne0d 11877 . . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> C =/= 0 )
65 0cxp 24412 . . . . . . . . . 10 |- ( ( C e. CC /\ C =/= 0 ) -> ( 0 ^c C ) = 0 )
6663, 64, 65syl2anc 693 . . . . . . . . 9 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> ( 0 ^c C ) = 0 )
6766adantr 481 . . . . . . . 8 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> ( 0 ^c C ) = 0 )
6867breq2d 4665 . . . . . . 7 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> ( ( A ^c C ) <_ ( 0 ^c C ) <-> ( A ^c C ) <_ 0 ) )
6962, 68mtbird 315 . . . . . 6 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> -. ( A ^c C ) <_ ( 0 ^c C ) )
7052, 692falsed 366 . . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> ( A <_ 0 <-> ( A ^c C ) <_ ( 0 ^c C ) ) )
71 breq2 4657 . . . . . 6 |- ( 0 = B -> ( A <_ 0 <-> A <_ B ) )
72 oveq1 6657 . . . . . . 7 |- ( 0 = B -> ( 0 ^c C ) = ( B ^c C ) )
7372breq2d 4665 . . . . . 6 |- ( 0 = B -> ( ( A ^c C ) <_ ( 0 ^c C ) <-> ( A ^c C ) <_ ( B ^c C ) ) )
7471, 73bibi12d 335 . . . . 5 |- ( 0 = B -> ( ( A <_ 0 <-> ( A ^c C ) <_ ( 0 ^c C ) ) <-> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) ) )
7570, 74syl5ibcom 235 . . . 4 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> ( 0 = B -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) ) )
7675imp 445 . . 3 |- ( ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) /\ 0 = B ) -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )
77 simp2r 1088 . . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> 0 <_ B )
78 leloe 10124 . . . . . 6 |- ( ( 0 e. RR /\ B e. RR ) -> ( 0 <_ B <-> ( 0 < B \/ 0 = B ) ) )
7948, 5, 78sylancr 695 . . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> ( 0 <_ B <-> ( 0 < B \/ 0 = B ) ) )
8077, 79mpbid 222 . . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> ( 0 < B \/ 0 = B ) )
8180adantr 481 . . 3 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> ( 0 < B \/ 0 = B ) )
8247, 76, 81mpjaodan 827 . 2 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 < A ) -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )
83 simpr 477 . . . 4 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 = A ) -> 0 = A )
84 simpl2r 1115 . . . 4 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 = A ) -> 0 <_ B )
8583, 84eqbrtrrd 4677 . . 3 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 = A ) -> A <_ B )
8666adantr 481 . . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 = A ) -> ( 0 ^c C ) = 0 )
8783oveq1d 6665 . . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 = A ) -> ( 0 ^c C ) = ( A ^c C ) )
8886, 87eqtr3d 2658 . . . 4 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 = A ) -> 0 = ( A ^c C ) )
89 simpl2l 1114 . . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 = A ) -> B e. RR )
9053adantr 481 . . . . 5 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 = A ) -> C e. RR )
91 cxpge0 24429 . . . . 5 |- ( ( B e. RR /\ 0 <_ B /\ C e. RR ) -> 0 <_ ( B ^c C ) )
9289, 84, 90, 91syl3anc 1326 . . . 4 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 = A ) -> 0 <_ ( B ^c C ) )
9388, 92eqbrtrrd 4677 . . 3 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 = A ) -> ( A ^c C ) <_ ( B ^c C ) )
9485, 932thd 255 . 2 |- ( ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) /\ 0 = A ) -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )
95 simp1r 1086 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> 0 <_ A )
96 leloe 10124 . . . 4 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) )
9748, 49, 96sylancr 695 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) )
9895, 97mpbid 222 . 2 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> ( 0 < A \/ 0 = A ) )
9982, 94, 98mpjaodan 827 1 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. RR+ ) -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   < clt 10074   <_ cle 10075   RR+crp 11832   expce 14792   logclog 24301   ^c ccxp 24302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304
This theorem is referenced by:  cxplt2  24444  cxple2a  24445  cxple2d  24473
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