brbtwn - Metamath Proof Explorer

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Theorem brbtwn 25779

Description: The binary relation form of the betweenness predicate. The statement

should be informally read as "

lies on a line segment between

and

. This exact definition is abstracted away by Tarski's geometry axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)

**Assertion**
Ref	Expression
brbtwn	$/\$ $/\$

Distinct variable groups:

Proof of Theorem brbtwn Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-btwn 25772	. . 3 $/\$ $/\$ $/\$
2	1	breqi 4659	. 2 $/\$ $/\$ $/\$
3		opex 4932	. . . . 5
4		brcnvg 5303	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	mpan2 707	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6	5	3ad2ant1 1082	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		df-br 4654	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		eleq1 2689	. . . . . . . . . 10
9	8	3anbi1d 1403	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
10		fveq1 6190	. . . . . . . . . . . . 13
11	10	oveq2d 6666	. . . . . . . . . . . 12
12	11	oveq1d 6665	. . . . . . . . . . 11
13	12	eqeq2d 2632	. . . . . . . . . 10
14	13	rexralbidv 3058	. . . . . . . . 9
15	9, 14	anbi12d 747	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	15	rexbidv 3052	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17		eleq1 2689	. . . . . . . . . 10
18	17	3anbi2d 1404	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
19		fveq1 6190	. . . . . . . . . . . . 13
20	19	oveq2d 6666	. . . . . . . . . . . 12
21	20	oveq2d 6666	. . . . . . . . . . 11
22	21	eqeq2d 2632	. . . . . . . . . 10
23	22	rexralbidv 3058	. . . . . . . . 9
24	18, 23	anbi12d 747	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	24	rexbidv 3052	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26		eleq1 2689	. . . . . . . . . 10
27	26	3anbi3d 1405	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
28		fveq1 6190	. . . . . . . . . . 11
29	28	eqeq1d 2624	. . . . . . . . . 10
30	29	rexralbidv 3058	. . . . . . . . 9
31	27, 30	anbi12d 747	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	31	rexbidv 3052	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	16, 25, 32	eloprabg 6748	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34		simp1 1061	. . . . . . . . . . . 12 $/\$ $/\$
35		simp1 1061	. . . . . . . . . . . 12 $/\$ $/\$
36		eedimeq 25778	. . . . . . . . . . . 12 $/\$
37	34, 35, 36	syl2anr 495	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
38		oveq2 6658	. . . . . . . . . . . . 13
39	38	raleqdv 3144	. . . . . . . . . . . 12
40	39	rexbidv 3052	. . . . . . . . . . 11
41	37, 40	syl 17	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
42	41	biimpd 219	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
43	42	expimpd 629	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	43	rexlimdvw 3034	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
45		eleenn 25776	. . . . . . . . 9
46	45	3ad2ant1 1082	. . . . . . . 8 $/\$ $/\$
47		fveq2 6191	. . . . . . . . . . . . 13
48	47	eleq2d 2687	. . . . . . . . . . . 12
49	47	eleq2d 2687	. . . . . . . . . . . 12
50	47	eleq2d 2687	. . . . . . . . . . . 12
51	48, 49, 50	3anbi123d 1399	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
52	51, 40	anbi12d 747	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	52	rspcev 3309	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	53	exp32 631	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
55	46, 54	mpcom 38	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
56	44, 55	impbid 202	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
57	33, 56	bitrd 268	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
58	57	3comr 1273	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
59	7, 58	syl5bb 272	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
60	6, 59	bitrd 268	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
61	2, 60	syl5bb 272	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wral 2912

wrex 2913

cvv 3200

cop 4183 class class class wbr 4653

ccnv 5113

cfv 5888 (class class class)co 6650

coprab 6651

cc0 9936

c1 9937

caddc 9939

cmul 9941

cmin 10266

cn 11020

cicc 12178

cfz 12326

cee 25768

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-fz 12327 df-ee 25771 df-btwn 25772

This theorem is referenced by: brbtwn2 25785 axsegcon 25807 ax5seg 25818 axbtwnid 25819 axpasch 25821 axeuclid 25843 axcontlem2 25845 axcontlem4 25847 axcontlem7 25850 axcontlem8 25851

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