Theorem ax5seglem5 25813

Description: Lemma for ax5seg 25818. If B is between A and C, and A is distinct from B, then A is distinct from C. (Contributed by Scott Fenton, 11-Jun-2013.)

Assertion

Ref	Expression
ax5seglem5	|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A =/= B /\ T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) =/= 0 )

Distinct variable groups:   A, i, j   B, i, j   C, i, j   T, i   i, N, j

Allowed substitution hint:   T( j)

Proof of Theorem ax5seglem5

Step	Hyp	Ref	Expression
1		fveq1 6190	. . . . . . . . . . . . . . 15 |- ( A = C -> ( A ` i ) = ( C ` i ) )
2	1	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( A = C -> ( T x. ( A ` i ) ) = ( T x. ( C ` i ) ) )
3	2	oveq2d 6666	. . . . . . . . . . . . 13 |- ( A = C -> ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( A ` i ) ) ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) )
4	3	eqeq2d 2632	. . . . . . . . . . . 12 |- ( A = C -> ( ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( A ` i ) ) ) <-> ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) )
5	4	ralbidv 2986	. . . . . . . . . . 11 |- ( A = C -> ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( A ` i ) ) ) <-> A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) )
6	5	biimparc 504	. . . . . . . . . 10 |- ( ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) /\ A = C ) -> A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( A ` i ) ) ) )
7		simplr1 1103	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ T e. ( 0 [,] 1 ) ) -> A e. ( EE ` N ) )
8		simplr2 1104	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ T e. ( 0 [,] 1 ) ) -> B e. ( EE ` N ) )
9		eqeefv 25783	. . . . . . . . . . . 12 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( A = B <-> A. i e. ( 1 ... N ) ( A ` i ) = ( B ` i ) ) )
10	7, 8, 9	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ T e. ( 0 [,] 1 ) ) -> ( A = B <-> A. i e. ( 1 ... N ) ( A ` i ) = ( B ` i ) ) )
11		fveecn 25782	. . . . . . . . . . . . . . 15 |- ( ( A e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( A ` i ) e. CC )
12	7, 11	sylan 488	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ T e. ( 0 [,] 1 ) ) /\ i e. ( 1 ... N ) ) -> ( A ` i ) e. CC )
13		0re 10040	. . . . . . . . . . . . . . . . . 18 |- 0 e. RR
14		1re 10039	. . . . . . . . . . . . . . . . . 18 |- 1 e. RR
15	13, 14	elicc2i 12239	. . . . . . . . . . . . . . . . 17 |- ( T e. ( 0 [,] 1 ) <-> ( T e. RR /\ 0 <_ T /\ T <_ 1 ) )
16	15	simp1bi 1076	. . . . . . . . . . . . . . . 16 |- ( T e. ( 0 [,] 1 ) -> T e. RR )
17	16	recnd 10068	. . . . . . . . . . . . . . 15 |- ( T e. ( 0 [,] 1 ) -> T e. CC )
18	17	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ T e. ( 0 [,] 1 ) ) /\ i e. ( 1 ... N ) ) -> T e. CC )
19		ax-1cn 9994	. . . . . . . . . . . . . . . . . . 19 |- 1 e. CC
20		npcan 10290	. . . . . . . . . . . . . . . . . . 19 |- ( ( 1 e. CC /\ T e. CC ) -> ( ( 1 - T ) + T ) = 1 )
21	19, 20	mpan 706	. . . . . . . . . . . . . . . . . 18 |- ( T e. CC -> ( ( 1 - T ) + T ) = 1 )
22	21	oveq1d 6665	. . . . . . . . . . . . . . . . 17 |- ( T e. CC -> ( ( ( 1 - T ) + T ) x. ( A ` i ) ) = ( 1 x. ( A ` i ) ) )
23		mulid2 10038	. . . . . . . . . . . . . . . . 17 |- ( ( A ` i ) e. CC -> ( 1 x. ( A ` i ) ) = ( A ` i ) )
24	22, 23	sylan9eqr 2678	. . . . . . . . . . . . . . . 16 |- ( ( ( A ` i ) e. CC /\ T e. CC ) -> ( ( ( 1 - T ) + T ) x. ( A ` i ) ) = ( A ` i ) )
25		subcl 10280	. . . . . . . . . . . . . . . . . . 19 |- ( ( 1 e. CC /\ T e. CC ) -> ( 1 - T ) e. CC )
26	19, 25	mpan 706	. . . . . . . . . . . . . . . . . 18 |- ( T e. CC -> ( 1 - T ) e. CC )
27	26	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( A ` i ) e. CC /\ T e. CC ) -> ( 1 - T ) e. CC )
28		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( A ` i ) e. CC /\ T e. CC ) -> T e. CC )
29		simpl 473	. . . . . . . . . . . . . . . . 17 |- ( ( ( A ` i ) e. CC /\ T e. CC ) -> ( A ` i ) e. CC )
30	27, 28, 29	adddird 10065	. . . . . . . . . . . . . . . 16 |- ( ( ( A ` i ) e. CC /\ T e. CC ) -> ( ( ( 1 - T ) + T ) x. ( A ` i ) ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( A ` i ) ) ) )
31	24, 30	eqtr3d 2658	. . . . . . . . . . . . . . 15 |- ( ( ( A ` i ) e. CC /\ T e. CC ) -> ( A ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( A ` i ) ) ) )
32	31	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( ( ( A ` i ) e. CC /\ T e. CC ) -> ( ( A ` i ) = ( B ` i ) <-> ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( A ` i ) ) ) = ( B ` i ) ) )
33	12, 18, 32	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ T e. ( 0 [,] 1 ) ) /\ i e. ( 1 ... N ) ) -> ( ( A ` i ) = ( B ` i ) <-> ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( A ` i ) ) ) = ( B ` i ) ) )
34		eqcom 2629	. . . . . . . . . . . . 13 |- ( ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( A ` i ) ) ) = ( B ` i ) <-> ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( A ` i ) ) ) )
35	33, 34	syl6bb 276	. . . . . . . . . . . 12 |- ( ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ T e. ( 0 [,] 1 ) ) /\ i e. ( 1 ... N ) ) -> ( ( A ` i ) = ( B ` i ) <-> ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( A ` i ) ) ) ) )
36	35	ralbidva 2985	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ T e. ( 0 [,] 1 ) ) -> ( A. i e. ( 1 ... N ) ( A ` i ) = ( B ` i ) <-> A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( A ` i ) ) ) ) )
37	10, 36	bitrd 268	. . . . . . . . . 10 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ T e. ( 0 [,] 1 ) ) -> ( A = B <-> A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( A ` i ) ) ) ) )
38	6, 37	syl5ibr 236	. . . . . . . . 9 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ T e. ( 0 [,] 1 ) ) -> ( ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) /\ A = C ) -> A = B ) )
39	38	expd 452	. . . . . . . 8 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ T e. ( 0 [,] 1 ) ) -> ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) -> ( A = C -> A = B ) ) )
40	39	impr 649	. . . . . . 7 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> ( A = C -> A = B ) )
41	40	necon3d 2815	. . . . . 6 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> ( A =/= B -> A =/= C ) )
42	41	ex 450	. . . . 5 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) -> ( A =/= B -> A =/= C ) ) )
43	42	com23 86	. . . 4 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A =/= B -> ( ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) -> A =/= C ) ) )
44	43	exp4a 633	. . 3 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) -> ( A =/= B -> ( T e. ( 0 [,] 1 ) -> ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) -> A =/= C ) ) ) )
45	44	3imp2 1282	. 2 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A =/= B /\ T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> A =/= C )
46		simplr1 1103	. . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A =/= B /\ T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> A e. ( EE ` N ) )
47		simplr3 1105	. . . 4 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A =/= B /\ T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> C e. ( EE ` N ) )
48		eqeelen 25784	. . . 4 |- ( ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) -> ( A = C <-> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) = 0 ) )
49	46, 47, 48	syl2anc 693	. . 3 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A =/= B /\ T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> ( A = C <-> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) = 0 ) )
50	49	necon3bid 2838	. 2 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A =/= B /\ T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> ( A =/= C <-> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) =/= 0 ) )
51	45, 50	mpbid 222	1 |- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A =/= B /\ T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) =/= 0 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   [,]cicc 12178   ...cfz 12326   ^cexp 12860   sum_csu 14416   EEcee 25768

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-icc 12182  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

This theorem is referenced by:  ax5seglem6  25814