Theorem axbtwnid 25819

Description: Points are indivisible. That is, if A lies between B and B, then A = B. Axiom A6 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 3-Jun-2013.)

Assertion

Ref	Expression
axbtwnid	|- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( A Btwn <. B , B >. -> A = B ) )

Proof of Theorem axbtwnid

Dummy variables t i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1062	. . 3 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> A e. ( EE ` N ) )
2		simp3 1063	. . 3 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> B e. ( EE ` N ) )
3		brbtwn 25779	. . 3 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( A Btwn <. B , B >. <-> E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( A ` i ) = ( ( ( 1 - t ) x. ( B ` i ) ) + ( t x. ( B ` i ) ) ) ) )
4	1, 2, 2, 3	syl3anc 1326	. 2 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( A Btwn <. B , B >. <-> E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( A ` i ) = ( ( ( 1 - t ) x. ( B ` i ) ) + ( t x. ( B ` i ) ) ) ) )
5		0re 10040	. . . . . . 7 |- 0 e. RR
6		1re 10039	. . . . . . 7 |- 1 e. RR
7	5, 6	elicc2i 12239	. . . . . 6 |- ( t e. ( 0 [,] 1 ) <-> ( t e. RR /\ 0 <_ t /\ t <_ 1 ) )
8	7	simp1bi 1076	. . . . 5 |- ( t e. ( 0 [,] 1 ) -> t e. RR )
9	8	recnd 10068	. . . 4 |- ( t e. ( 0 [,] 1 ) -> t e. CC )
10		eqeefv 25783	. . . . . . . 8 |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( A = B <-> A. i e. ( 1 ... N ) ( A ` i ) = ( B ` i ) ) )
11	10	3adant1 1079	. . . . . . 7 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( A = B <-> A. i e. ( 1 ... N ) ( A ` i ) = ( B ` i ) ) )
12	11	adantr 481	. . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) -> ( A = B <-> A. i e. ( 1 ... N ) ( A ` i ) = ( B ` i ) ) )
13		ax-1cn 9994	. . . . . . . . . . . 12 |- 1 e. CC
14		npcan 10290	. . . . . . . . . . . 12 |- ( ( 1 e. CC /\ t e. CC ) -> ( ( 1 - t ) + t ) = 1 )
15	13, 14	mpan 706	. . . . . . . . . . 11 |- ( t e. CC -> ( ( 1 - t ) + t ) = 1 )
16	15	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) /\ i e. ( 1 ... N ) ) -> ( ( 1 - t ) + t ) = 1 )
17	16	oveq1d 6665	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - t ) + t ) x. ( B ` i ) ) = ( 1 x. ( B ` i ) ) )
18		subcl 10280	. . . . . . . . . . . 12 |- ( ( 1 e. CC /\ t e. CC ) -> ( 1 - t ) e. CC )
19	13, 18	mpan 706	. . . . . . . . . . 11 |- ( t e. CC -> ( 1 - t ) e. CC )
20	19	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) /\ i e. ( 1 ... N ) ) -> ( 1 - t ) e. CC )
21		simplr 792	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) /\ i e. ( 1 ... N ) ) -> t e. CC )
22		simpll3 1102	. . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) /\ i e. ( 1 ... N ) ) -> B e. ( EE ` N ) )
23		fveecn 25782	. . . . . . . . . . 11 |- ( ( B e. ( EE ` N ) /\ i e. ( 1 ... N ) ) -> ( B ` i ) e. CC )
24	22, 23	sylancom 701	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) /\ i e. ( 1 ... N ) ) -> ( B ` i ) e. CC )
25	20, 21, 24	adddird 10065	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) /\ i e. ( 1 ... N ) ) -> ( ( ( 1 - t ) + t ) x. ( B ` i ) ) = ( ( ( 1 - t ) x. ( B ` i ) ) + ( t x. ( B ` i ) ) ) )
26	24	mulid2d 10058	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) /\ i e. ( 1 ... N ) ) -> ( 1 x. ( B ` i ) ) = ( B ` i ) )
27	17, 25, 26	3eqtr3rd 2665	. . . . . . . 8 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) /\ i e. ( 1 ... N ) ) -> ( B ` i ) = ( ( ( 1 - t ) x. ( B ` i ) ) + ( t x. ( B ` i ) ) ) )
28	27	eqeq2d 2632	. . . . . . 7 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) /\ i e. ( 1 ... N ) ) -> ( ( A ` i ) = ( B ` i ) <-> ( A ` i ) = ( ( ( 1 - t ) x. ( B ` i ) ) + ( t x. ( B ` i ) ) ) ) )
29	28	ralbidva 2985	. . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) -> ( A. i e. ( 1 ... N ) ( A ` i ) = ( B ` i ) <-> A. i e. ( 1 ... N ) ( A ` i ) = ( ( ( 1 - t ) x. ( B ` i ) ) + ( t x. ( B ` i ) ) ) ) )
30	12, 29	bitrd 268	. . . . 5 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) -> ( A = B <-> A. i e. ( 1 ... N ) ( A ` i ) = ( ( ( 1 - t ) x. ( B ` i ) ) + ( t x. ( B ` i ) ) ) ) )
31	30	biimprd 238	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. CC ) -> ( A. i e. ( 1 ... N ) ( A ` i ) = ( ( ( 1 - t ) x. ( B ` i ) ) + ( t x. ( B ` i ) ) ) -> A = B ) )
32	9, 31	sylan2 491	. . 3 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ t e. ( 0 [,] 1 ) ) -> ( A. i e. ( 1 ... N ) ( A ` i ) = ( ( ( 1 - t ) x. ( B ` i ) ) + ( t x. ( B ` i ) ) ) -> A = B ) )
33	32	rexlimdva 3031	. 2 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( E. t e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( A ` i ) = ( ( ( 1 - t ) x. ( B ` i ) ) + ( t x. ( B ` i ) ) ) -> A = B ) )
34	4, 33	sylbid 230	1 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( A Btwn <. B , B >. -> A = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   <.cop 4183   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   [,]cicc 12178   ...cfz 12326   EEcee 25768   Btwn cbtwn 25769

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-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-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-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-z 11378  df-uz 11688  df-icc 12182  df-fz 12327  df-ee 25771  df-btwn 25772

This theorem is referenced by:  eengtrkg  25865  btwncomim  32120  btwnswapid  32124  btwnintr  32126  btwnexch3  32127  ifscgr  32151  idinside  32191  btwnconn1lem12  32205  outsideofrflx  32234