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Mirrors > Home > MPE Home > Th. List > lmimid | Structured version Visualization version Unicode version |
Description: If we have a right angle, then the mirror point is the point inversion. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
Ref | Expression |
---|---|
ismid.p |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
ismid.d |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
ismid.i |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
ismid.g |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
ismid.1 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
lmif.m |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
lmif.l |
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
lmif.d |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
lmicl.1 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
lmimid.s |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
lmimid.r |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
lmimid.a |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
lmimid.b |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
lmimid.c |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
lmimid.d |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
lmimid |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmimid.s |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 1 | a1i 11 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | 2 | fveq1d 5872 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | ismid.p |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | ismid.d |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | ismid.i |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | ismid.g |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | ismid.1 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | lmimid.c |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | lmif.l |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | eqid 2453 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | lmif.d |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | lmimid.b |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 4, 10, 6, 7, 12, 13 | tglnpt 24606 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 4, 5, 6, 10, 11, 7, 14, 1, 9 | mircl 24718 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 | 4, 5, 6, 7, 8, 9, 15, 11, 14 | ismidb 24832 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 3, 16 | mpbid 214 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 17, 13 | eqeltrd 2531 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | df-ne 2626 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 7 | adantr 467 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 12 | adantr 467 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 9 | adantr 467 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 15 | adantr 467 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | simpr 463 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 4, 6, 10, 20, 22, 23, 24 | tgelrnln 24687 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 13 | adantr 467 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 14 | adantr 467 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 4, 5, 6, 7, 8, 9, 15 | midbtwn 24833 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 17, 28 | eqeltrrd 2532 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 29 | adantr 467 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 4, 6, 10, 20, 22, 23, 27, 24, 30 | btwnlng1 24676 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 26, 31 | elind 3620 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | lmimid.a |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
34 | 33 | adantr 467 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 4, 6, 10, 20, 22, 23, 24 | tglinerflx1 24690 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | lmimid.d |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
37 | 36 | adantr 467 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | 4, 5, 6, 10, 11, 7, 14, 1, 9 | mirinv 24723 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | eqcom 2460 |
. . . . . . . . . . . . . 14
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
40 | 38, 39 | syl6bb 265 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 40 | biimpar 488 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | 41 | eqcomd 2459 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 42 | ex 436 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
44 | 43 | necon3d 2647 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 44 | imp 431 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | lmimid.r |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
47 | 46 | adantr 467 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 4, 5, 6, 10, 20, 21, 25, 32, 34, 35, 37, 45, 47 | ragperp 24774 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 48 | ex 436 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | 19, 49 | syl5bir 222 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
51 | 50 | orrd 380 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 51 | orcomd 390 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | lmif.m |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
54 | 4, 5, 6, 7, 8, 53, 10, 12, 9, 15 | islmib 24841 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
55 | 18, 52, 54 | mpbir2and 934 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
56 | 55 | eqcomd 2459 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-cnex 9600 ax-resscn 9601 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-addrcl 9605 ax-mulcl 9606 ax-mulrcl 9607 ax-mulcom 9608 ax-addass 9609 ax-mulass 9610 ax-distr 9611 ax-i2m1 9612 ax-1ne0 9613 ax-1rid 9614 ax-rnegex 9615 ax-rrecex 9616 ax-cnre 9617 ax-pre-lttri 9618 ax-pre-lttrn 9619 ax-pre-ltadd 9620 ax-pre-mulgt0 9621 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-nel 2627 df-ral 2744 df-rex 2745 df-reu 2746 df-rmo 2747 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-int 4238 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-om 6698 df-1st 6798 df-2nd 6799 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-1o 7187 df-oadd 7191 df-er 7368 df-map 7479 df-pm 7480 df-en 7575 df-dom 7576 df-sdom 7577 df-fin 7578 df-card 8378 df-cda 8603 df-pnf 9682 df-mnf 9683 df-xr 9684 df-ltxr 9685 df-le 9686 df-sub 9867 df-neg 9868 df-nn 10617 df-2 10675 df-3 10676 df-n0 10877 df-z 10945 df-uz 11167 df-fz 11792 df-fzo 11923 df-hash 12523 df-word 12671 df-concat 12673 df-s1 12674 df-s2 12951 df-s3 12952 df-trkgc 24508 df-trkgb 24509 df-trkgcb 24510 df-trkgld 24512 df-trkg 24513 df-cgrg 24568 df-leg 24640 df-mir 24710 df-rag 24751 df-perpg 24753 df-mid 24828 df-lmi 24829 |
This theorem is referenced by: hypcgrlem1 24853 |
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