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Mirrors > Home > MPE Home > Th. List > ncolncol | Structured version Visualization version Unicode version |
Description: Deduce non-colinearity from non-colinearity and colinearity. (Contributed by Thierry Arnoux, 27-Aug-2019.) |
Ref | Expression |
---|---|
tglineintmo.p | |
tglineintmo.i | Itv |
tglineintmo.l | LineG |
tglineintmo.g | TarskiG |
tglineinteq.a | |
tglineinteq.b | |
tglineinteq.c | |
tglineinteq.d | |
tglineinteq.e | |
ncolncol.1 | |
ncolncol.2 |
Ref | Expression |
---|---|
ncolncol |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglineinteq.e | . 2 | |
2 | tglineintmo.p | . . 3 | |
3 | tglineintmo.l | . . 3 LineG | |
4 | tglineintmo.i | . . 3 Itv | |
5 | tglineintmo.g | . . . 4 TarskiG | |
6 | 5 | adantr 481 | . . 3 TarskiG |
7 | tglineinteq.a | . . . 4 | |
8 | 7 | adantr 481 | . . 3 |
9 | tglineinteq.b | . . . 4 | |
10 | 9 | adantr 481 | . . 3 |
11 | tglineinteq.c | . . . 4 | |
12 | 11 | adantr 481 | . . 3 |
13 | 5 | ad2antrr 762 | . . . . . 6 TarskiG |
14 | 7 | ad2antrr 762 | . . . . . 6 |
15 | 9 | ad2antrr 762 | . . . . . 6 |
16 | 11 | ad2antrr 762 | . . . . . 6 |
17 | ncolncol.1 | . . . . . . . 8 | |
18 | 2, 3, 4, 5, 7, 9, 17 | tglngne 25445 | . . . . . . 7 |
19 | 18 | ad2antrr 762 | . . . . . 6 |
20 | tglineinteq.d | . . . . . . . . 9 | |
21 | 20 | ad2antrr 762 | . . . . . . . 8 |
22 | ncolncol.2 | . . . . . . . . . 10 | |
23 | 22 | necomd 2849 | . . . . . . . . 9 |
24 | 23 | ad2antrr 762 | . . . . . . . 8 |
25 | simpr 477 | . . . . . . . 8 | |
26 | 2, 4, 3, 13, 15, 21, 16, 24, 25 | lncom 25517 | . . . . . . 7 |
27 | 18 | necomd 2849 | . . . . . . . . 9 |
28 | 2, 4, 3, 5, 9, 7, 20, 27, 17 | lncom 25517 | . . . . . . . . 9 |
29 | 2, 4, 3, 5, 9, 7, 27, 20, 22, 28 | tglineelsb2 25527 | . . . . . . . 8 |
30 | 29 | ad2antrr 762 | . . . . . . 7 |
31 | 26, 30 | eleqtrrd 2704 | . . . . . 6 |
32 | 2, 4, 3, 13, 14, 15, 16, 19, 31 | lncom 25517 | . . . . 5 |
33 | 32 | orcd 407 | . . . 4 |
34 | simpr 477 | . . . . 5 | |
35 | 22 | ad2antrr 762 | . . . . 5 |
36 | 34, 35 | pm2.21ddne 2878 | . . . 4 |
37 | 20 | adantr 481 | . . . . 5 |
38 | simpr 477 | . . . . 5 | |
39 | 2, 3, 4, 6, 10, 12, 37, 38 | colrot2 25455 | . . . 4 |
40 | 33, 36, 39 | mpjaodan 827 | . . 3 |
41 | 2, 3, 4, 6, 8, 10, 12, 40 | colrot1 25454 | . 2 |
42 | 1, 41 | mtand 691 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wo 383 wa 384 wceq 1483 wcel 1990 wne 2794 cfv 5888 (class class class)co 6650 cbs 15857 TarskiGcstrkg 25329 Itvcitv 25335 LineGclng 25336 |
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-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-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-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-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 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-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 df-cgrg 25406 |
This theorem is referenced by: coltr 25542 midexlem 25587 acopy 25724 acopyeu 25725 |
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