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Mirrors > Home > MPE Home > Th. List > mirtrcgr | Structured version Visualization version GIF version |
Description: Point inversion of one point of a triangle around another point preserves triangle congruence. (Contributed by Thierry Arnoux, 4-Oct-2020.) |
Ref | Expression |
---|---|
mirval.p | β’ π = (BaseβπΊ) |
mirval.d | β’ β = (distβπΊ) |
mirval.i | β’ πΌ = (ItvβπΊ) |
mirval.l | β’ πΏ = (LineGβπΊ) |
mirval.s | β’ π = (pInvGβπΊ) |
mirval.g | β’ (π β πΊ β TarskiG) |
mirtrcgr.e | β’ βΌ = (cgrGβπΊ) |
mirtrcgr.m | β’ π = (πβπ΅) |
mirtrcgr.n | β’ π = (πβπ) |
mirtrcgr.a | β’ (π β π΄ β π) |
mirtrcgr.b | β’ (π β π΅ β π) |
mirtrcgr.x | β’ (π β π β π) |
mirtrcgr.y | β’ (π β π β π) |
mirtrcgr.c | β’ (π β πΆ β π) |
mirtrcgr.z | β’ (π β π β π) |
mirtrcgr.1 | β’ (π β π΄ β π΅) |
mirtrcgr.2 | β’ (π β β¨βπ΄π΅πΆββ© βΌ β¨βπππββ©) |
Ref | Expression |
---|---|
mirtrcgr | β’ (π β β¨β(πβπ΄)π΅πΆββ© βΌ β¨β(πβπ)ππββ©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . 2 β’ π = (BaseβπΊ) | |
2 | mirval.d | . 2 β’ β = (distβπΊ) | |
3 | mirtrcgr.e | . 2 β’ βΌ = (cgrGβπΊ) | |
4 | mirval.g | . 2 β’ (π β πΊ β TarskiG) | |
5 | mirval.i | . . 3 β’ πΌ = (ItvβπΊ) | |
6 | mirval.l | . . 3 β’ πΏ = (LineGβπΊ) | |
7 | mirval.s | . . 3 β’ π = (pInvGβπΊ) | |
8 | mirtrcgr.b | . . 3 β’ (π β π΅ β π) | |
9 | mirtrcgr.m | . . 3 β’ π = (πβπ΅) | |
10 | mirtrcgr.a | . . 3 β’ (π β π΄ β π) | |
11 | 1, 2, 5, 6, 7, 4, 8, 9, 10 | mircl 28180 | . 2 β’ (π β (πβπ΄) β π) |
12 | mirtrcgr.c | . 2 β’ (π β πΆ β π) | |
13 | mirtrcgr.y | . . 3 β’ (π β π β π) | |
14 | mirtrcgr.n | . . 3 β’ π = (πβπ) | |
15 | mirtrcgr.x | . . 3 β’ (π β π β π) | |
16 | 1, 2, 5, 6, 7, 4, 13, 14, 15 | mircl 28180 | . 2 β’ (π β (πβπ) β π) |
17 | mirtrcgr.z | . 2 β’ (π β π β π) | |
18 | mirtrcgr.2 | . . . . . 6 β’ (π β β¨βπ΄π΅πΆββ© βΌ β¨βπππββ©) | |
19 | 1, 2, 5, 3, 4, 10, 8, 12, 15, 13, 17, 18 | cgr3simp1 28039 | . . . . 5 β’ (π β (π΄ β π΅) = (π β π)) |
20 | 1, 2, 5, 4, 10, 8, 15, 13, 19 | tgcgrcomlr 27999 | . . . 4 β’ (π β (π΅ β π΄) = (π β π)) |
21 | 1, 2, 5, 6, 7, 4, 8, 9, 10 | mircgr 28176 | . . . 4 β’ (π β (π΅ β (πβπ΄)) = (π΅ β π΄)) |
22 | 1, 2, 5, 6, 7, 4, 13, 14, 15 | mircgr 28176 | . . . 4 β’ (π β (π β (πβπ)) = (π β π)) |
23 | 20, 21, 22 | 3eqtr4d 2781 | . . 3 β’ (π β (π΅ β (πβπ΄)) = (π β (πβπ))) |
24 | 1, 2, 5, 4, 8, 11, 13, 16, 23 | tgcgrcomlr 27999 | . 2 β’ (π β ((πβπ΄) β π΅) = ((πβπ) β π)) |
25 | 1, 2, 5, 3, 4, 10, 8, 12, 15, 13, 17, 18 | cgr3simp2 28040 | . 2 β’ (π β (π΅ β πΆ) = (π β π)) |
26 | 1, 2, 5, 6, 7, 4, 8, 9, 10 | mirbtwn 28177 | . . . . . 6 β’ (π β π΅ β ((πβπ΄)πΌπ΄)) |
27 | 1, 6, 5, 4, 11, 10, 8, 26 | btwncolg1 28074 | . . . . 5 β’ (π β (π΅ β ((πβπ΄)πΏπ΄) β¨ (πβπ΄) = π΄)) |
28 | 1, 6, 5, 4, 11, 10, 8, 27 | colcom 28077 | . . . 4 β’ (π β (π΅ β (π΄πΏ(πβπ΄)) β¨ π΄ = (πβπ΄))) |
29 | 1, 2, 5, 6, 7, 4, 3, 9, 14, 10, 8, 15, 13, 19 | mircgrextend 28201 | . . . . . 6 β’ (π β (π΄ β (πβπ΄)) = (π β (πβπ))) |
30 | 1, 2, 5, 4, 10, 11, 15, 16, 29 | tgcgrcomlr 27999 | . . . . 5 β’ (π β ((πβπ΄) β π΄) = ((πβπ) β π)) |
31 | 1, 2, 3, 4, 10, 8, 11, 15, 13, 16, 19, 23, 30 | trgcgr 28035 | . . . 4 β’ (π β β¨βπ΄π΅(πβπ΄)ββ© βΌ β¨βππ(πβπ)ββ©) |
32 | 1, 2, 5, 3, 4, 10, 8, 12, 15, 13, 17, 18 | cgr3simp3 28041 | . . . . 5 β’ (π β (πΆ β π΄) = (π β π)) |
33 | 1, 2, 5, 4, 12, 10, 17, 15, 32 | tgcgrcomlr 27999 | . . . 4 β’ (π β (π΄ β πΆ) = (π β π)) |
34 | mirtrcgr.1 | . . . 4 β’ (π β π΄ β π΅) | |
35 | 1, 6, 5, 4, 10, 8, 11, 3, 15, 13, 2, 12, 16, 17, 28, 31, 33, 25, 34 | tgfscgr 28087 | . . 3 β’ (π β ((πβπ΄) β πΆ) = ((πβπ) β π)) |
36 | 1, 2, 5, 4, 11, 12, 16, 17, 35 | tgcgrcomlr 27999 | . 2 β’ (π β (πΆ β (πβπ΄)) = (π β (πβπ))) |
37 | 1, 2, 3, 4, 11, 8, 12, 16, 13, 17, 24, 25, 36 | trgcgr 28035 | 1 β’ (π β β¨β(πβπ΄)π΅πΆββ© βΌ β¨β(πβπ)ππββ©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 β wne 2939 class class class wbr 5148 βcfv 6543 (class class class)co 7412 β¨βcs3 14798 Basecbs 17149 distcds 17211 TarskiGcstrkg 27946 Itvcitv 27952 LineGclng 27953 cgrGccgrg 28029 pInvGcmir 28171 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-oadd 8474 df-er 8707 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-dju 9900 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-xnn0 12550 df-z 12564 df-uz 12828 df-fz 13490 df-fzo 13633 df-hash 14296 df-word 14470 df-concat 14526 df-s1 14551 df-s2 14804 df-s3 14805 df-trkgc 27967 df-trkgb 27968 df-trkgcb 27969 df-trkg 27972 df-cgrg 28030 df-mir 28172 |
This theorem is referenced by: sacgr 28350 |
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