linethru - Mathbox for Scott Fenton

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Theorem linethru 30970

Description: If

is a line containing two distinct points

and

, then

is the line through

and

. Theorem 6.18 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
linethru	LinesEE $/\$ $/\$ $/\$ Line

Proof of Theorem linethru Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ellines 30969	. . 3 LinesEE $/\$ Line
2		simpll1 1053	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$
3		simpll2 1054	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$
4		simpll3 1055	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$
5		simplr 767	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$
6		liness 30962	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ Line
7	2, 3, 4, 5, 6	syl13anc 1278	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ Line
8		simprll 777	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ Line
9	7, 8	sseldd 3445	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$
10		simprlr 778	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ Line
11	7, 10	sseldd 3445	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$
12		simplll 773	. . . . . . . . . . . . . . . 16 Line $/\$ Line $/\$ $/\$ $/\$ Line
13	12	adantl 472	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ Line
14		simpll1 1053	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$
15		simpll2 1054	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$
16		simpll3 1055	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$
17		simplr 767	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$
18		simprrl 779	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$
19		simprlr 778	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$
20	19	necomd 2691	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$
21		lineelsb2 30965	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ Line Line Line
22	14, 15, 16, 17, 18, 20, 21	syl132anc 1294	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ Line Line Line
23	13, 22	mpd 15	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ Line Line
24		linecom 30967	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ Line Line
25	14, 15, 18, 20, 24	syl13anc 1278	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ Line Line
26	23, 25	eqtrd 2496	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ Line Line
27		neeq2 2699	. . . . . . . . . . . . . . . . 17
28	27	anbi2d 715	. . . . . . . . . . . . . . . 16 Line $/\$ Line $/\$ Line $/\$ Line $/\$
29	28	anbi1d 716	. . . . . . . . . . . . . . 15 Line $/\$ Line $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$
30	29	anbi2d 715	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$
31		oveq2 6328	. . . . . . . . . . . . . . 15 Line Line
32	31	eqeq2d 2472	. . . . . . . . . . . . . 14 Line Line Line Line
33	30, 32	imbi12d 326	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ Line Line $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ Line Line
34	26, 33	mpbiri 241	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ Line Line
35		simp1 1014	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		simp2l 1040	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$
37	35, 36, 10	syl2anc 671	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ Line
38		simp1l1 1107	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$
39		simp1l2 1108	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$
40		simp1l3 1109	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$
41		simp1r 1039	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$
42		simp2rr 1084	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$
43		simp3 1016	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$
44	43	necomd 2691	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$
45		lineelsb2 30965	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$ Line Line Line
46	38, 39, 40, 41, 42, 44, 45	syl132anc 1294	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ Line Line Line
47	37, 46	mpd 15	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ Line Line
48		linecom 30967	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ Line Line
49	38, 39, 42, 44, 48	syl13anc 1278	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ Line Line
50	47, 49	eqtrd 2496	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ Line Line
51	36	simplld 766	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ Line
52	51, 50	eleqtrd 2542	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ Line
53		simp2rl 1083	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$
54		simp2lr 1082	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$
55	54	necomd 2691	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$
56		lineelsb2 30965	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ Line Line Line
57	38, 42, 39, 43, 53, 55, 56	syl132anc 1294	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ Line Line Line
58	52, 57	mpd 15	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ Line Line
59		linecom 30967	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ Line Line
60	38, 42, 53, 55, 59	syl13anc 1278	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ Line Line
61	50, 58, 60	3eqtrd 2500	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ Line Line
62	61	3expa 1215	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ $/\$ Line Line
63	62	expcom 441	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ Line Line
64	34, 63	pm2.61ine 2719	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ $/\$ Line Line
65	64	expr 624	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ $/\$ Line Line
66	9, 11, 65	mp2and 690	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ Line $/\$ Line $/\$ Line Line
67	66	ex 440	. . . . . . . 8 $/\$ $/\$ $/\$ Line $/\$ Line $/\$ Line Line
68		eleq2 2529	. . . . . . . . . . 11 Line Line
69		eleq2 2529	. . . . . . . . . . 11 Line Line
70	68, 69	anbi12d 722	. . . . . . . . . 10 Line $/\$ Line $/\$ Line
71	70	anbi1d 716	. . . . . . . . 9 Line $/\$ $/\$ Line $/\$ Line $/\$
72		eqeq1 2466	. . . . . . . . 9 Line Line Line Line
73	71, 72	imbi12d 326	. . . . . . . 8 Line $/\$ $/\$ Line Line $/\$ Line $/\$ Line Line
74	67, 73	syl5ibrcom 230	. . . . . . 7 $/\$ $/\$ $/\$ Line $/\$ $/\$ Line
75	74	expimpd 612	. . . . . 6 $/\$ $/\$ $/\$ Line $/\$ $/\$ Line
76	75	3expa 1215	. . . . 5 $/\$ $/\$ $/\$ Line $/\$ $/\$ Line
77	76	rexlimdva 2891	. . . 4 $/\$ $/\$ Line $/\$ $/\$ Line
78	77	rexlimivv 2896	. . 3 $/\$ Line $/\$ $/\$ Line
79	1, 78	sylbi 200	. 2 LinesEE $/\$ $/\$ Line
80	79	3impib 1213	1 LinesEE $/\$ $/\$ $/\$ Line

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 375 $/\$ w3a 991

wceq 1455

wcel 1898

wne 2633

wrex 2750

wss 3416

cfv 5605 (class class class)co 6320

cn 10642

cee 24974 Linecline2 30951 LinesEEclines2 30953

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4531 ax-sep 4541 ax-nul 4550 ax-pow 4598 ax-pr 4656 ax-un 6615 ax-inf2 8177 ax-cnex 9626 ax-resscn 9627 ax-1cn 9628 ax-icn 9629 ax-addcl 9630 ax-addrcl 9631 ax-mulcl 9632 ax-mulrcl 9633 ax-mulcom 9634 ax-addass 9635 ax-mulass 9636 ax-distr 9637 ax-i2m1 9638 ax-1ne0 9639 ax-1rid 9640 ax-rnegex 9641 ax-rrecex 9642 ax-cnre 9643 ax-pre-lttri 9644 ax-pre-lttrn 9645 ax-pre-ltadd 9646 ax-pre-mulgt0 9647 ax-pre-sup 9648

This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1458 df-fal 1461 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-nel 2636 df-ral 2754 df-rex 2755 df-reu 2756 df-rmo 2757 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-pss 3432 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-tp 3985 df-op 3987 df-uni 4213 df-int 4249 df-iun 4294 df-br 4419 df-opab 4478 df-mpt 4479 df-tr 4514 df-eprel 4767 df-id 4771 df-po 4777 df-so 4778 df-fr 4815 df-se 4816 df-we 4817 df-xp 4862 df-rel 4863 df-cnv 4864 df-co 4865 df-dm 4866 df-rn 4867 df-res 4868 df-ima 4869 df-pred 5403 df-ord 5449 df-on 5450 df-lim 5451 df-suc 5452 df-iota 5569 df-fun 5607 df-fn 5608 df-f 5609 df-f1 5610 df-fo 5611 df-f1o 5612 df-fv 5613 df-isom 5614 df-riota 6282 df-ov 6323 df-oprab 6324 df-mpt2 6325 df-om 6725 df-1st 6825 df-2nd 6826 df-wrecs 7059 df-recs 7121 df-rdg 7159 df-1o 7213 df-oadd 7217 df-er 7394 df-ec 7396 df-map 7505 df-en 7601 df-dom 7602 df-sdom 7603 df-fin 7604 df-sup 7987 df-oi 8056 df-card 8404 df-pnf 9708 df-mnf 9709 df-xr 9710 df-ltxr 9711 df-le 9712 df-sub 9893 df-neg 9894 df-div 10303 df-nn 10643 df-2 10701 df-3 10702 df-n0 10904 df-z 10972 df-uz 11194 df-rp 11337 df-ico 11675 df-icc 11676 df-fz 11820 df-fzo 11953 df-seq 12252 df-exp 12311 df-hash 12554 df-cj 13217 df-re 13218 df-im 13219 df-sqrt 13353 df-abs 13354 df-clim 13607 df-sum 13808 df-ee 24977 df-btwn 24978 df-cgr 24979 df-ofs 30800 df-colinear 30856 df-ifs 30857 df-cgr3 30858 df-fs 30859 df-line2 30954 df-lines2 30956

This theorem is referenced by: hilbert1.2 30972 lineintmo 30974

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