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Mirrors > Home > MPE Home > Th. List > cgrcgra | Structured version Visualization version GIF version |
Description: Triangle congruence implies angle congruence. This is a portion of CPCTC, focusing on a specific angle. (Contributed by Arnoux, 2-Aug-2020.) |
Ref | Expression |
---|---|
cgraid.p | β’ π = (BaseβπΊ) |
cgraid.i | β’ πΌ = (ItvβπΊ) |
cgraid.g | β’ (π β πΊ β TarskiG) |
cgraid.k | β’ πΎ = (hlGβπΊ) |
cgraid.a | β’ (π β π΄ β π) |
cgraid.b | β’ (π β π΅ β π) |
cgraid.c | β’ (π β πΆ β π) |
cgracom.d | β’ (π β π· β π) |
cgracom.e | β’ (π β πΈ β π) |
cgracom.f | β’ (π β πΉ β π) |
cgrcgra.1 | β’ (π β π΄ β π΅) |
cgrcgra.2 | β’ (π β π΅ β πΆ) |
cgrcgra.3 | β’ (π β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπΉββ©) |
Ref | Expression |
---|---|
cgrcgra | β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cgraid.p | . 2 β’ π = (BaseβπΊ) | |
2 | cgraid.i | . 2 β’ πΌ = (ItvβπΊ) | |
3 | cgraid.k | . 2 β’ πΎ = (hlGβπΊ) | |
4 | cgraid.g | . 2 β’ (π β πΊ β TarskiG) | |
5 | cgraid.a | . 2 β’ (π β π΄ β π) | |
6 | cgraid.b | . 2 β’ (π β π΅ β π) | |
7 | cgraid.c | . 2 β’ (π β πΆ β π) | |
8 | cgracom.d | . 2 β’ (π β π· β π) | |
9 | cgracom.e | . 2 β’ (π β πΈ β π) | |
10 | cgracom.f | . 2 β’ (π β πΉ β π) | |
11 | cgrcgra.3 | . 2 β’ (π β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπΉββ©) | |
12 | eqid 2733 | . . . 4 β’ (distβπΊ) = (distβπΊ) | |
13 | eqid 2733 | . . . . 5 β’ (cgrGβπΊ) = (cgrGβπΊ) | |
14 | 1, 12, 2, 13, 4, 5, 6, 7, 8, 9, 10, 11 | cgr3simp1 27761 | . . . 4 β’ (π β (π΄(distβπΊ)π΅) = (π·(distβπΊ)πΈ)) |
15 | cgrcgra.1 | . . . 4 β’ (π β π΄ β π΅) | |
16 | 1, 12, 2, 4, 5, 6, 8, 9, 14, 15 | tgcgrneq 27724 | . . 3 β’ (π β π· β πΈ) |
17 | 1, 2, 3, 8, 5, 9, 4, 16 | hlid 27850 | . 2 β’ (π β π·(πΎβπΈ)π·) |
18 | 1, 12, 2, 13, 4, 5, 6, 7, 8, 9, 10, 11 | cgr3simp2 27762 | . . . . 5 β’ (π β (π΅(distβπΊ)πΆ) = (πΈ(distβπΊ)πΉ)) |
19 | 1, 12, 2, 4, 6, 7, 9, 10, 18 | tgcgrcomlr 27721 | . . . 4 β’ (π β (πΆ(distβπΊ)π΅) = (πΉ(distβπΊ)πΈ)) |
20 | cgrcgra.2 | . . . . 5 β’ (π β π΅ β πΆ) | |
21 | 20 | necomd 2997 | . . . 4 β’ (π β πΆ β π΅) |
22 | 1, 12, 2, 4, 7, 6, 10, 9, 19, 21 | tgcgrneq 27724 | . . 3 β’ (π β πΉ β πΈ) |
23 | 1, 2, 3, 10, 5, 9, 4, 22 | hlid 27850 | . 2 β’ (π β πΉ(πΎβπΈ)πΉ) |
24 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 8, 10, 11, 17, 23 | iscgrad 28052 | 1 β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wne 2941 class class class wbr 5148 βcfv 6541 β¨βcs3 14790 Basecbs 17141 distcds 17203 TarskiGcstrkg 27668 Itvcitv 27674 cgrGccgrg 27751 hlGchlg 27841 cgrAccgra 28048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 df-fzo 13625 df-hash 14288 df-word 14462 df-concat 14518 df-s1 14543 df-s2 14796 df-s3 14797 df-trkgc 27689 df-trkgcb 27691 df-trkg 27694 df-cgrg 27752 df-hlg 27842 df-cgra 28049 |
This theorem is referenced by: acopy 28074 tgsss1 28101 |
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