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Theorem linethru 26052
 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.
StepHypRef Expression
1 ellines 26051 . . 3 LinesEE Line
2 simpll1 996 . . . . . . . . . . . 12 Line Line
3 simpll2 997 . . . . . . . . . . . 12 Line Line
4 simpll3 998 . . . . . . . . . . . 12 Line Line
5 simplr 732 . . . . . . . . . . . 12 Line Line
6 liness 26044 . . . . . . . . . . . 12 Line
72, 3, 4, 5, 6syl13anc 1186 . . . . . . . . . . 11 Line Line Line
8 simprll 739 . . . . . . . . . . 11 Line Line Line
97, 8sseldd 3341 . . . . . . . . . 10 Line Line
10 simprlr 740 . . . . . . . . . . 11 Line Line Line
117, 10sseldd 3341 . . . . . . . . . 10 Line Line
12 simplll 735 . . . . . . . . . . . . . . . 16 Line Line Line
1312adantl 453 . . . . . . . . . . . . . . 15 Line Line Line
14 simpll1 996 . . . . . . . . . . . . . . . 16 Line Line
15 simpll2 997 . . . . . . . . . . . . . . . 16 Line Line
16 simpll3 998 . . . . . . . . . . . . . . . 16 Line Line
17 simplr 732 . . . . . . . . . . . . . . . 16 Line Line
18 simprrl 741 . . . . . . . . . . . . . . . 16 Line Line
19 simprlr 740 . . . . . . . . . . . . . . . . 17 Line Line
2019necomd 2681 . . . . . . . . . . . . . . . 16 Line Line
21 lineelsb2 26047 . . . . . . . . . . . . . . . 16 Line Line Line
2214, 15, 16, 17, 18, 20, 21syl132anc 1202 . . . . . . . . . . . . . . 15 Line Line Line Line Line
2313, 22mpd 15 . . . . . . . . . . . . . 14 Line Line Line Line
24 linecom 26049 . . . . . . . . . . . . . . 15 Line Line
2514, 15, 18, 20, 24syl13anc 1186 . . . . . . . . . . . . . 14 Line Line Line Line
2623, 25eqtrd 2467 . . . . . . . . . . . . 13 Line Line Line Line
27 neeq2 2607 . . . . . . . . . . . . . . . . 17
2827anbi2d 685 . . . . . . . . . . . . . . . 16 Line Line Line Line
2928anbi1d 686 . . . . . . . . . . . . . . 15 Line Line Line Line
3029anbi2d 685 . . . . . . . . . . . . . 14 Line Line Line Line
31 oveq2 6081 . . . . . . . . . . . . . . 15 Line Line
3231eqeq2d 2446 . . . . . . . . . . . . . 14 Line Line Line Line
3330, 32imbi12d 312 . . . . . . . . . . . . 13 Line Line Line Line Line Line Line Line
3426, 33mpbiri 225 . . . . . . . . . . . 12 Line Line Line Line
35 simp1 957 . . . . . . . . . . . . . . . . . 18 Line Line
36 simp2l 983 . . . . . . . . . . . . . . . . . 18 Line Line Line Line
3735, 36, 10syl2anc 643 . . . . . . . . . . . . . . . . 17 Line Line Line
38 simp1l1 1050 . . . . . . . . . . . . . . . . . 18 Line Line
39 simp1l2 1051 . . . . . . . . . . . . . . . . . 18 Line Line
40 simp1l3 1052 . . . . . . . . . . . . . . . . . 18 Line Line
41 simp1r 982 . . . . . . . . . . . . . . . . . 18 Line Line
42 simp2rr 1027 . . . . . . . . . . . . . . . . . 18 Line Line
43 simp3 959 . . . . . . . . . . . . . . . . . . 19 Line Line
4443necomd 2681 . . . . . . . . . . . . . . . . . 18 Line Line
45 lineelsb2 26047 . . . . . . . . . . . . . . . . . 18 Line Line Line
4638, 39, 40, 41, 42, 44, 45syl132anc 1202 . . . . . . . . . . . . . . . . 17 Line Line Line Line Line
4737, 46mpd 15 . . . . . . . . . . . . . . . 16 Line Line Line Line
48 linecom 26049 . . . . . . . . . . . . . . . . 17 Line Line
4938, 39, 42, 44, 48syl13anc 1186 . . . . . . . . . . . . . . . 16 Line Line Line Line
5047, 49eqtrd 2467 . . . . . . . . . . . . . . 15 Line Line Line Line
51 simpll 731 . . . . . . . . . . . . . . . . . 18 Line Line Line
5236, 51syl 16 . . . . . . . . . . . . . . . . 17 Line Line Line
5352, 50eleqtrd 2511 . . . . . . . . . . . . . . . 16 Line Line Line
54 simp2rl 1026 . . . . . . . . . . . . . . . . 17 Line Line
55 simp2lr 1025 . . . . . . . . . . . . . . . . . 18 Line Line
5655necomd 2681 . . . . . . . . . . . . . . . . 17 Line Line
57 lineelsb2 26047 . . . . . . . . . . . . . . . . 17 Line Line Line
5838, 42, 39, 43, 54, 56, 57syl132anc 1202 . . . . . . . . . . . . . . . 16 Line Line Line Line Line
5953, 58mpd 15 . . . . . . . . . . . . . . 15 Line Line Line Line
60 linecom 26049 . . . . . . . . . . . . . . . 16 Line Line
6138, 42, 54, 56, 60syl13anc 1186 . . . . . . . . . . . . . . 15 Line Line Line Line
6250, 59, 613eqtrd 2471 . . . . . . . . . . . . . 14 Line Line Line Line
63623expa 1153 . . . . . . . . . . . . 13 Line Line Line Line
6463expcom 425 . . . . . . . . . . . 12 Line Line Line Line
6534, 64pm2.61ine 2674 . . . . . . . . . . 11 Line Line Line Line
6665expr 599 . . . . . . . . . 10 Line Line Line Line
679, 11, 66mp2and 661 . . . . . . . . 9 Line Line Line Line
6867ex 424 . . . . . . . 8 Line Line Line Line
69 eleq2 2496 . . . . . . . . . . 11 Line Line
70 eleq2 2496 . . . . . . . . . . 11 Line Line
7169, 70anbi12d 692 . . . . . . . . . 10 Line Line Line
7271anbi1d 686 . . . . . . . . 9 Line Line Line
73 eqeq1 2441 . . . . . . . . 9 Line Line Line Line
7472, 73imbi12d 312 . . . . . . . 8 Line Line Line Line Line Line
7568, 74syl5ibrcom 214 . . . . . . 7 Line Line
7675expimpd 587 . . . . . 6 Line Line
77763expa 1153 . . . . 5 Line Line
7877rexlimdva 2822 . . . 4 Line Line
7978rexlimivv 2827 . . 3 Line Line
801, 79sylbi 188 . 2 LinesEE Line
81803impib 1151 1 LinesEE Line
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725   wne 2598  wrex 2698   wss 3312  cfv 5446  (class class class)co 6073  cn 9990  cee 25792  Linecline2 26033  LinesEEclines2 26035 This theorem is referenced by:  hilbert1.2  26054  lineintmo  26056 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-ec 6899  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7469  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-ico 10912  df-icc 10913  df-fz 11034  df-fzo 11126  df-seq 11314  df-exp 11373  df-hash 11609  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-clim 12272  df-sum 12470  df-ee 25795  df-btwn 25796  df-cgr 25797  df-ofs 25882  df-ifs 25938  df-cgr3 25939  df-colinear 25940  df-fs 25941  df-line2 26036  df-lines2 26038
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