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Mirrors > Home > MPE Home > Th. List > lmicom | Structured version Visualization version GIF version |
Description: The line mirroring function is an involution. Theorem 10.4 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-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) |
lmif.m | β’ π = ((lInvGβπΊ)βπ·) |
lmif.l | β’ πΏ = (LineGβπΊ) |
lmif.d | β’ (π β π· β ran πΏ) |
lmicl.1 | β’ (π β π΄ β π) |
islmib.b | β’ (π β π΅ β π) |
lmicom.1 | β’ (π β (πβπ΄) = π΅) |
Ref | Expression |
---|---|
lmicom | β’ (π β (πβπ΅) = π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismid.p | . . . . 5 β’ π = (BaseβπΊ) | |
2 | ismid.d | . . . . 5 β’ β = (distβπΊ) | |
3 | ismid.i | . . . . 5 β’ πΌ = (ItvβπΊ) | |
4 | ismid.g | . . . . 5 β’ (π β πΊ β TarskiG) | |
5 | ismid.1 | . . . . 5 β’ (π β πΊDimTarskiGβ₯2) | |
6 | lmicl.1 | . . . . 5 β’ (π β π΄ β π) | |
7 | islmib.b | . . . . 5 β’ (π β π΅ β π) | |
8 | 1, 2, 3, 4, 5, 6, 7 | midcom 27766 | . . . 4 β’ (π β (π΄(midGβπΊ)π΅) = (π΅(midGβπΊ)π΄)) |
9 | lmicom.1 | . . . . . . 7 β’ (π β (πβπ΄) = π΅) | |
10 | 9 | eqcomd 2743 | . . . . . 6 β’ (π β π΅ = (πβπ΄)) |
11 | lmif.m | . . . . . . 7 β’ π = ((lInvGβπΊ)βπ·) | |
12 | lmif.l | . . . . . . 7 β’ πΏ = (LineGβπΊ) | |
13 | lmif.d | . . . . . . 7 β’ (π β π· β ran πΏ) | |
14 | 1, 2, 3, 4, 5, 11, 12, 13, 6, 7 | islmib 27771 | . . . . . 6 β’ (π β (π΅ = (πβπ΄) β ((π΄(midGβπΊ)π΅) β π· β§ (π·(βGβπΊ)(π΄πΏπ΅) β¨ π΄ = π΅)))) |
15 | 10, 14 | mpbid 231 | . . . . 5 β’ (π β ((π΄(midGβπΊ)π΅) β π· β§ (π·(βGβπΊ)(π΄πΏπ΅) β¨ π΄ = π΅))) |
16 | 15 | simpld 496 | . . . 4 β’ (π β (π΄(midGβπΊ)π΅) β π·) |
17 | 8, 16 | eqeltrrd 2839 | . . 3 β’ (π β (π΅(midGβπΊ)π΄) β π·) |
18 | 15 | simprd 497 | . . . . . . . . 9 β’ (π β (π·(βGβπΊ)(π΄πΏπ΅) β¨ π΄ = π΅)) |
19 | 18 | orcomd 870 | . . . . . . . 8 β’ (π β (π΄ = π΅ β¨ π·(βGβπΊ)(π΄πΏπ΅))) |
20 | 19 | ord 863 | . . . . . . 7 β’ (π β (Β¬ π΄ = π΅ β π·(βGβπΊ)(π΄πΏπ΅))) |
21 | 4 | adantr 482 | . . . . . . . . . 10 β’ ((π β§ Β¬ π΄ = π΅) β πΊ β TarskiG) |
22 | 6 | adantr 482 | . . . . . . . . . 10 β’ ((π β§ Β¬ π΄ = π΅) β π΄ β π) |
23 | 7 | adantr 482 | . . . . . . . . . 10 β’ ((π β§ Β¬ π΄ = π΅) β π΅ β π) |
24 | simpr 486 | . . . . . . . . . . 11 β’ ((π β§ Β¬ π΄ = π΅) β Β¬ π΄ = π΅) | |
25 | 24 | neqned 2951 | . . . . . . . . . 10 β’ ((π β§ Β¬ π΄ = π΅) β π΄ β π΅) |
26 | 1, 3, 12, 21, 22, 23, 25 | tglinecom 27619 | . . . . . . . . 9 β’ ((π β§ Β¬ π΄ = π΅) β (π΄πΏπ΅) = (π΅πΏπ΄)) |
27 | 26 | breq2d 5122 | . . . . . . . 8 β’ ((π β§ Β¬ π΄ = π΅) β (π·(βGβπΊ)(π΄πΏπ΅) β π·(βGβπΊ)(π΅πΏπ΄))) |
28 | 27 | pm5.74da 803 | . . . . . . 7 β’ (π β ((Β¬ π΄ = π΅ β π·(βGβπΊ)(π΄πΏπ΅)) β (Β¬ π΄ = π΅ β π·(βGβπΊ)(π΅πΏπ΄)))) |
29 | 20, 28 | mpbid 231 | . . . . . 6 β’ (π β (Β¬ π΄ = π΅ β π·(βGβπΊ)(π΅πΏπ΄))) |
30 | 29 | orrd 862 | . . . . 5 β’ (π β (π΄ = π΅ β¨ π·(βGβπΊ)(π΅πΏπ΄))) |
31 | 30 | orcomd 870 | . . . 4 β’ (π β (π·(βGβπΊ)(π΅πΏπ΄) β¨ π΄ = π΅)) |
32 | eqcom 2744 | . . . . 5 β’ (π΄ = π΅ β π΅ = π΄) | |
33 | 32 | orbi2i 912 | . . . 4 β’ ((π·(βGβπΊ)(π΅πΏπ΄) β¨ π΄ = π΅) β (π·(βGβπΊ)(π΅πΏπ΄) β¨ π΅ = π΄)) |
34 | 31, 33 | sylib 217 | . . 3 β’ (π β (π·(βGβπΊ)(π΅πΏπ΄) β¨ π΅ = π΄)) |
35 | 1, 2, 3, 4, 5, 11, 12, 13, 7, 6 | islmib 27771 | . . 3 β’ (π β (π΄ = (πβπ΅) β ((π΅(midGβπΊ)π΄) β π· β§ (π·(βGβπΊ)(π΅πΏπ΄) β¨ π΅ = π΄)))) |
36 | 17, 34, 35 | mpbir2and 712 | . 2 β’ (π β π΄ = (πβπ΅)) |
37 | 36 | eqcomd 2743 | 1 β’ (π β (πβπ΅) = π΄) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 397 β¨ wo 846 = wceq 1542 β wcel 2107 class class class wbr 5110 ran crn 5639 βcfv 6501 (class class class)co 7362 2c2 12215 Basecbs 17090 distcds 17149 TarskiGcstrkg 27411 DimTarskiGβ₯cstrkgld 27415 Itvcitv 27417 LineGclng 27418 βGcperpg 27679 midGcmid 27756 lInvGclmi 27757 |
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 df-lmi 27759 |
This theorem is referenced by: lmilmi 27773 |
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