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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 28599 | . . . 4 β’ (π β (π΄(midGβπΊ)π΅) = (π΅(midGβπΊ)π΄)) |
9 | lmicom.1 | . . . . . . 7 β’ (π β (πβπ΄) = π΅) | |
10 | 9 | eqcomd 2734 | . . . . . 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 28604 | . . . . . 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 494 | . . . 4 β’ (π β (π΄(midGβπΊ)π΅) β π·) |
17 | 8, 16 | eqeltrrd 2830 | . . 3 β’ (π β (π΅(midGβπΊ)π΄) β π·) |
18 | 15 | simprd 495 | . . . . . . . . 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 480 | . . . . . . . . . 10 β’ ((π β§ Β¬ π΄ = π΅) β πΊ β TarskiG) |
22 | 6 | adantr 480 | . . . . . . . . . 10 β’ ((π β§ Β¬ π΄ = π΅) β π΄ β π) |
23 | 7 | adantr 480 | . . . . . . . . . 10 β’ ((π β§ Β¬ π΄ = π΅) β π΅ β π) |
24 | simpr 484 | . . . . . . . . . . 11 β’ ((π β§ Β¬ π΄ = π΅) β Β¬ π΄ = π΅) | |
25 | 24 | neqned 2944 | . . . . . . . . . 10 β’ ((π β§ Β¬ π΄ = π΅) β π΄ β π΅) |
26 | 1, 3, 12, 21, 22, 23, 25 | tglinecom 28452 | . . . . . . . . 9 β’ ((π β§ Β¬ π΄ = π΅) β (π΄πΏπ΅) = (π΅πΏπ΄)) |
27 | 26 | breq2d 5160 | . . . . . . . 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 2735 | . . . . 5 β’ (π΄ = π΅ β π΅ = π΄) | |
33 | 32 | orbi2i 911 | . . . 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 28604 | . . 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 2734 | 1 β’ (π β (πβπ΅) = π΄) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β¨ wo 846 = wceq 1534 β wcel 2099 class class class wbr 5148 ran crn 5679 βcfv 6548 (class class class)co 7420 2c2 12298 Basecbs 17180 distcds 17242 TarskiGcstrkg 28244 DimTarskiGβ₯cstrkgld 28248 Itvcitv 28250 LineGclng 28251 βGcperpg 28512 midGcmid 28589 lInvGclmi 28590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9925 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-xnn0 12576 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-hash 14323 df-word 14498 df-concat 14554 df-s1 14579 df-s2 14832 df-s3 14833 df-trkgc 28265 df-trkgb 28266 df-trkgcb 28267 df-trkgld 28269 df-trkg 28270 df-cgrg 28328 df-leg 28400 df-mir 28470 df-rag 28511 df-perpg 28513 df-mid 28591 df-lmi 28592 |
This theorem is referenced by: lmilmi 28606 |
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