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Mirrors > Home > MPE Home > Th. List > mirmid | Structured version Visualization version GIF version |
Description: Point inversion preserves midpoints. (Contributed by Thierry Arnoux, 12-Dec-2019.) |
Ref | Expression |
---|---|
ismid.p | β’ π = (BaseβπΊ) |
ismid.d | β’ β = (distβπΊ) |
ismid.i | β’ πΌ = (ItvβπΊ) |
ismid.g | β’ (π β πΊ β TarskiG) |
ismid.1 | β’ (π β πΊDimTarskiGβ₯2) |
midcl.1 | β’ (π β π΄ β π) |
midcl.2 | β’ (π β π΅ β π) |
mirmid.s | β’ π = ((pInvGβπΊ)βπ) |
mirmid.x | β’ (π β π β π) |
Ref | Expression |
---|---|
mirmid | β’ (π β ((πβπ΄)(midGβπΊ)(πβπ΅)) = (πβ(π΄(midGβπΊ)π΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2738 | . . . . 5 β’ (π β (π΄(midGβπΊ)π΅) = (π΄(midGβπΊ)π΅)) | |
2 | ismid.p | . . . . . 6 β’ π = (BaseβπΊ) | |
3 | ismid.d | . . . . . 6 β’ β = (distβπΊ) | |
4 | ismid.i | . . . . . 6 β’ πΌ = (ItvβπΊ) | |
5 | ismid.g | . . . . . 6 β’ (π β πΊ β TarskiG) | |
6 | ismid.1 | . . . . . 6 β’ (π β πΊDimTarskiGβ₯2) | |
7 | midcl.1 | . . . . . 6 β’ (π β π΄ β π) | |
8 | midcl.2 | . . . . . 6 β’ (π β π΅ β π) | |
9 | eqid 2737 | . . . . . 6 β’ (pInvGβπΊ) = (pInvGβπΊ) | |
10 | 2, 3, 4, 5, 6, 7, 8 | midcl 27761 | . . . . . 6 β’ (π β (π΄(midGβπΊ)π΅) β π) |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | ismidb 27762 | . . . . 5 β’ (π β (π΅ = (((pInvGβπΊ)β(π΄(midGβπΊ)π΅))βπ΄) β (π΄(midGβπΊ)π΅) = (π΄(midGβπΊ)π΅))) |
12 | 1, 11 | mpbird 257 | . . . 4 β’ (π β π΅ = (((pInvGβπΊ)β(π΄(midGβπΊ)π΅))βπ΄)) |
13 | 12 | fveq2d 6851 | . . 3 β’ (π β (πβπ΅) = (πβ(((pInvGβπΊ)β(π΄(midGβπΊ)π΅))βπ΄))) |
14 | eqid 2737 | . . . 4 β’ (LineGβπΊ) = (LineGβπΊ) | |
15 | mirmid.x | . . . 4 β’ (π β π β π) | |
16 | mirmid.s | . . . 4 β’ π = ((pInvGβπΊ)βπ) | |
17 | 2, 3, 4, 14, 9, 5, 15, 16, 7, 10 | mirmir2 27658 | . . 3 β’ (π β (πβ(((pInvGβπΊ)β(π΄(midGβπΊ)π΅))βπ΄)) = (((pInvGβπΊ)β(πβ(π΄(midGβπΊ)π΅)))β(πβπ΄))) |
18 | 13, 17 | eqtrd 2777 | . 2 β’ (π β (πβπ΅) = (((pInvGβπΊ)β(πβ(π΄(midGβπΊ)π΅)))β(πβπ΄))) |
19 | 2, 3, 4, 14, 9, 5, 15, 16, 7 | mircl 27645 | . . 3 β’ (π β (πβπ΄) β π) |
20 | 2, 3, 4, 14, 9, 5, 15, 16, 8 | mircl 27645 | . . 3 β’ (π β (πβπ΅) β π) |
21 | 2, 3, 4, 14, 9, 5, 15, 16, 10 | mircl 27645 | . . 3 β’ (π β (πβ(π΄(midGβπΊ)π΅)) β π) |
22 | 2, 3, 4, 5, 6, 19, 20, 9, 21 | ismidb 27762 | . 2 β’ (π β ((πβπ΅) = (((pInvGβπΊ)β(πβ(π΄(midGβπΊ)π΅)))β(πβπ΄)) β ((πβπ΄)(midGβπΊ)(πβπ΅)) = (πβ(π΄(midGβπΊ)π΅)))) |
23 | 18, 22 | mpbid 231 | 1 β’ (π β ((πβπ΄)(midGβπΊ)(πβπ΅)) = (πβ(π΄(midGβπΊ)π΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 class class class wbr 5110 βcfv 6501 (class class class)co 7362 2c2 12215 Basecbs 17090 distcds 17149 TarskiGcstrkg 27411 DimTarskiGβ₯cstrkgld 27415 Itvcitv 27417 LineGclng 27418 pInvGcmir 27636 midGcmid 27756 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-fz 13432 df-fzo 13575 df-hash 14238 df-word 14410 df-concat 14466 df-s1 14491 df-s2 14744 df-s3 14745 df-trkgc 27432 df-trkgb 27433 df-trkgcb 27434 df-trkgld 27436 df-trkg 27437 df-cgrg 27495 df-leg 27567 df-mir 27637 df-rag 27678 df-perpg 27680 df-mid 27758 |
This theorem is referenced by: lmiisolem 27780 |
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