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Mirrors > Home > MPE Home > Th. List > cgrancol | Structured version Visualization version GIF version |
Description: Angle congruence preserves non-colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.) |
Ref | Expression |
---|---|
cgracol.p | β’ π = (BaseβπΊ) |
cgracol.i | β’ πΌ = (ItvβπΊ) |
cgracol.m | β’ β = (distβπΊ) |
cgracol.g | β’ (π β πΊ β TarskiG) |
cgracol.a | β’ (π β π΄ β π) |
cgracol.b | β’ (π β π΅ β π) |
cgracol.c | β’ (π β πΆ β π) |
cgracol.d | β’ (π β π· β π) |
cgracol.e | β’ (π β πΈ β π) |
cgracol.f | β’ (π β πΉ β π) |
cgracol.1 | β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) |
cgrancol.l | β’ πΏ = (LineGβπΊ) |
cgrancol.2 | β’ (π β Β¬ (πΆ β (π΄πΏπ΅) β¨ π΄ = π΅)) |
Ref | Expression |
---|---|
cgrancol | β’ (π β Β¬ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cgrancol.2 | . 2 β’ (π β Β¬ (πΆ β (π΄πΏπ΅) β¨ π΄ = π΅)) | |
2 | cgracol.p | . . 3 β’ π = (BaseβπΊ) | |
3 | cgracol.i | . . 3 β’ πΌ = (ItvβπΊ) | |
4 | cgracol.m | . . 3 β’ β = (distβπΊ) | |
5 | cgracol.g | . . . 4 β’ (π β πΊ β TarskiG) | |
6 | 5 | adantr 479 | . . 3 β’ ((π β§ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) β πΊ β TarskiG) |
7 | cgracol.d | . . . 4 β’ (π β π· β π) | |
8 | 7 | adantr 479 | . . 3 β’ ((π β§ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) β π· β π) |
9 | cgracol.e | . . . 4 β’ (π β πΈ β π) | |
10 | 9 | adantr 479 | . . 3 β’ ((π β§ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) β πΈ β π) |
11 | cgracol.f | . . . 4 β’ (π β πΉ β π) | |
12 | 11 | adantr 479 | . . 3 β’ ((π β§ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) β πΉ β π) |
13 | cgracol.a | . . . 4 β’ (π β π΄ β π) | |
14 | 13 | adantr 479 | . . 3 β’ ((π β§ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) β π΄ β π) |
15 | cgracol.b | . . . 4 β’ (π β π΅ β π) | |
16 | 15 | adantr 479 | . . 3 β’ ((π β§ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) β π΅ β π) |
17 | cgracol.c | . . . 4 β’ (π β πΆ β π) | |
18 | 17 | adantr 479 | . . 3 β’ ((π β§ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) β πΆ β π) |
19 | eqid 2728 | . . . 4 β’ (hlGβπΊ) = (hlGβπΊ) | |
20 | cgracol.1 | . . . . 5 β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) | |
21 | 20 | adantr 479 | . . . 4 β’ ((π β§ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) |
22 | 2, 3, 6, 19, 14, 16, 18, 8, 10, 12, 21 | cgracom 28654 | . . 3 β’ ((π β§ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) β β¨βπ·πΈπΉββ©(cgrAβπΊ)β¨βπ΄π΅πΆββ©) |
23 | cgrancol.l | . . 3 β’ πΏ = (LineGβπΊ) | |
24 | simpr 483 | . . 3 β’ ((π β§ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) β (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) | |
25 | 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 22, 23, 24 | cgracol 28660 | . 2 β’ ((π β§ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) β (πΆ β (π΄πΏπ΅) β¨ π΄ = π΅)) |
26 | 1, 25 | mtand 814 | 1 β’ (π β Β¬ (πΉ β (π·πΏπΈ) β¨ π· = πΈ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β¨ wo 845 = wceq 1533 β wcel 2098 class class class wbr 5152 βcfv 6553 (class class class)co 7426 β¨βcs3 14835 Basecbs 17189 distcds 17251 TarskiGcstrkg 28259 Itvcitv 28265 LineGclng 28266 hlGchlg 28432 cgrAccgra 28639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-oadd 8499 df-er 8733 df-map 8855 df-pm 8856 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-dju 9934 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-n0 12513 df-xnn0 12585 df-z 12599 df-uz 12863 df-fz 13527 df-fzo 13670 df-hash 14332 df-word 14507 df-concat 14563 df-s1 14588 df-s2 14841 df-s3 14842 df-trkgc 28280 df-trkgb 28281 df-trkgcb 28282 df-trkg 28285 df-cgrg 28343 df-leg 28415 df-hlg 28433 df-cgra 28640 |
This theorem is referenced by: acopyeu 28666 tgasa1 28690 |
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