Theorem btwnconn1lem7 32200

Description: Lemma for btwnconn1 32208. Under our assumptions, C and d are distinct. (Contributed by Scott Fenton, 8-Oct-2013.)

Assertion

Ref

Expression

btwnconn1lem7

|- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) ) ) -> C =/= d )

Proof of Theorem btwnconn1lem7

Step	Hyp	Ref	Expression
1		simp1l3 1156	. . . . 5 |- ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) -> C =/= c )
2	1	adantr 481	. . . 4 |- ( ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) ) -> C =/= c )
3		simp2rr 1131	. . . . 5 |- ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) -> <. C , d >.Cgr <. C , D >. )
4	3	adantr 481	. . . 4 |- ( ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) ) -> <. C , d >.Cgr <. C , D >. )
5		simp2lr 1129	. . . . 5 |- ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) -> <. D , c >.Cgr <. C , D >. )
6	5	adantr 481	. . . 4 |- ( ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) ) -> <. D , c >.Cgr <. C , D >. )
7	2, 4, 6	3jca 1242	. . 3 |- ( ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) ) -> ( C =/= c /\ <. C , d >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) )
8		simp11 1091	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> N e. NN )
9		simp21 1094	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
10		simp22 1095	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
11		simp23 1096	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> c e. ( EE ` N ) )
12		simp31 1097	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> d e. ( EE ` N ) )
13		simpr1 1067	. . . . . 6 |- ( ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) /\ ( C =/= c /\ <. C , d >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) ) -> C =/= c )
14		opeq2 4403	. . . . . . . . . . . 12 |- ( C = d -> <. C , C >. = <. C , d >. )
15	14	breq1d 4663	. . . . . . . . . . 11 |- ( C = d -> ( <. C , C >.Cgr <. C , D >. <-> <. C , d >.Cgr <. C , D >. ) )
16	15	3anbi2d 1404	. . . . . . . . . 10 |- ( C = d -> ( ( C =/= c /\ <. C , C >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) <-> ( C =/= c /\ <. C , d >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) ) )
17	16	biimparc 504	. . . . . . . . 9 |- ( ( ( C =/= c /\ <. C , d >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) /\ C = d ) -> ( C =/= c /\ <. C , C >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) )
18		simp2 1062	. . . . . . . . . . . . 13 |- ( ( C =/= c /\ <. C , C >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) -> <. C , C >.Cgr <. C , D >. )
19		simp1 1061	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> N e. NN )
20		simp2l 1087	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> C e. ( EE ` N ) )
21		simp2r 1088	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> D e. ( EE ` N ) )
22		cgrid2 32110	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( <. C , C >.Cgr <. C , D >. -> C = D ) )
23	19, 20, 20, 21, 22	syl13anc 1328	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> ( <. C , C >.Cgr <. C , D >. -> C = D ) )
24	18, 23	syl5 34	. . . . . . . . . . . 12 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> ( ( C =/= c /\ <. C , C >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) -> C = D ) )
25	24	imp 445	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) /\ ( C =/= c /\ <. C , C >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) ) -> C = D )
26		opeq1 4402	. . . . . . . . . . . . . . . 16 |- ( C = D -> <. C , c >. = <. D , c >. )
27		opeq2 4403	. . . . . . . . . . . . . . . 16 |- ( C = D -> <. C , C >. = <. C , D >. )
28	26, 27	breq12d 4666	. . . . . . . . . . . . . . 15 |- ( C = D -> ( <. C , c >.Cgr <. C , C >. <-> <. D , c >.Cgr <. C , D >. ) )
29	28	biimparc 504	. . . . . . . . . . . . . 14 |- ( ( <. D , c >.Cgr <. C , D >. /\ C = D ) -> <. C , c >.Cgr <. C , C >. )
30		simp3l 1089	. . . . . . . . . . . . . . 15 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> c e. ( EE ` N ) )
31		axcgrid 25796	. . . . . . . . . . . . . . 15 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ c e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( <. C , c >.Cgr <. C , C >. -> C = c ) )
32	19, 20, 30, 20, 31	syl13anc 1328	. . . . . . . . . . . . . 14 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> ( <. C , c >.Cgr <. C , C >. -> C = c ) )
33	29, 32	syl5 34	. . . . . . . . . . . . 13 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> ( ( <. D , c >.Cgr <. C , D >. /\ C = D ) -> C = c ) )
34	33	expdimp 453	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) /\ <. D , c >.Cgr <. C , D >. ) -> ( C = D -> C = c ) )
35	34	3ad2antr3 1228	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) /\ ( C =/= c /\ <. C , C >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) ) -> ( C = D -> C = c ) )
36	25, 35	mpd 15	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) /\ ( C =/= c /\ <. C , C >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) ) -> C = c )
37	36	ex 450	. . . . . . . . 9 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> ( ( C =/= c /\ <. C , C >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) -> C = c ) )
38	17, 37	syl5 34	. . . . . . . 8 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> ( ( ( C =/= c /\ <. C , d >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) /\ C = d ) -> C = c ) )
39	38	expdimp 453	. . . . . . 7 |- ( ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) /\ ( C =/= c /\ <. C , d >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) ) -> ( C = d -> C = c ) )
40	39	necon3d 2815	. . . . . 6 |- ( ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) /\ ( C =/= c /\ <. C , d >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) ) -> ( C =/= c -> C =/= d ) )
41	13, 40	mpd 15	. . . . 5 |- ( ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) /\ ( C =/= c /\ <. C , d >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) ) -> C =/= d )
42	41	ex 450	. . . 4 |- ( ( N e. NN /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) /\ ( c e. ( EE ` N ) /\ d e. ( EE ` N ) ) ) -> ( ( C =/= c /\ <. C , d >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) -> C =/= d ) )
43	8, 9, 10, 11, 12, 42	syl122anc 1335	. . 3 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( C =/= c /\ <. C , d >.Cgr <. C , D >. /\ <. D , c >.Cgr <. C , D >. ) -> C =/= d ) )
44	7, 43	syl5 34	. 2 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) ) -> C =/= d ) )
45	44	imp 445	1 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( d e. ( EE ` N ) /\ b e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) /\ ( ( ( ( A =/= B /\ B =/= C /\ C =/= c ) /\ ( B Btwn <. A , C >. /\ B Btwn <. A , D >. ) ) /\ ( ( D Btwn <. A , c >. /\ <. D , c >.Cgr <. C , D >. ) /\ ( C Btwn <. A , d >. /\ <. C , d >.Cgr <. C , D >. ) ) /\ ( ( c Btwn <. A , b >. /\ <. c , b >.Cgr <. C , B >. ) /\ ( d Btwn <. A , b >. /\ <. d , b >.Cgr <. D , B >. ) ) ) /\ ( E Btwn <. C , c >. /\ E Btwn <. D , d >. ) ) ) -> C =/= d )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770

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

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-ee 25771  df-cgr 25773

This theorem is referenced by:  btwnconn1lem8  32201  btwnconn1lem12  32205