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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 | ⊢ 𝑃 = (Base‘𝐺) |
ismid.d | ⊢ − = (dist‘𝐺) |
ismid.i | ⊢ 𝐼 = (Itv‘𝐺) |
ismid.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
ismid.1 | ⊢ (𝜑 → 𝐺DimTarskiG≥2) |
lmif.m | ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) |
lmif.l | ⊢ 𝐿 = (LineG‘𝐺) |
lmif.d | ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
lmicl.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
islmib.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
lmicom.1 | ⊢ (𝜑 → (𝑀‘𝐴) = 𝐵) |
Ref | Expression |
---|---|
lmicom | ⊢ (𝜑 → (𝑀‘𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismid.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
2 | ismid.d | . . . . 5 ⊢ − = (dist‘𝐺) | |
3 | ismid.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | ismid.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | ismid.1 | . . . . 5 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
6 | lmicl.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | islmib.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
8 | 1, 2, 3, 4, 5, 6, 7 | midcom 25674 | . . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴)) |
9 | lmicom.1 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐵) | |
10 | 9 | eqcomd 2628 | . . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑀‘𝐴)) |
11 | lmif.m | . . . . . . 7 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) | |
12 | lmif.l | . . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺) | |
13 | lmif.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
14 | 1, 2, 3, 4, 5, 11, 12, 13, 6, 7 | islmib 25679 | . . . . . 6 ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)))) |
15 | 10, 14 | mpbid 222 | . . . . 5 ⊢ (𝜑 → ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))) |
16 | 15 | simpld 475 | . . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝐷) |
17 | 8, 16 | eqeltrrd 2702 | . . 3 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐴) ∈ 𝐷) |
18 | 15 | simprd 479 | . . . . . . . . 9 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) |
19 | 18 | orcomd 403 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵))) |
20 | 19 | ord 392 | . . . . . . 7 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵))) |
21 | 4 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG) |
22 | 6 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃) |
23 | 7 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃) |
24 | simpr 477 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵) | |
25 | 24 | neqned 2801 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ≠ 𝐵) |
26 | 1, 3, 12, 21, 22, 23, 25 | tglinecom 25530 | . . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐴𝐿𝐵) = (𝐵𝐿𝐴)) |
27 | 26 | breq2d 4665 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))) |
28 | 27 | pm5.74da 723 | . . . . . . 7 ⊢ (𝜑 → ((¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)) ↔ (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴)))) |
29 | 20, 28 | mpbid 222 | . . . . . 6 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))) |
30 | 29 | orrd 393 | . . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))) |
31 | 30 | orcomd 403 | . . . 4 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵)) |
32 | eqcom 2629 | . . . . 5 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
33 | 32 | orbi2i 541 | . . . 4 ⊢ ((𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵) ↔ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴)) |
34 | 31, 33 | sylib 208 | . . 3 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴)) |
35 | 1, 2, 3, 4, 5, 11, 12, 13, 7, 6 | islmib 25679 | . . 3 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐵) ↔ ((𝐵(midG‘𝐺)𝐴) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴)))) |
36 | 17, 34, 35 | mpbir2and 957 | . 2 ⊢ (𝜑 → 𝐴 = (𝑀‘𝐵)) |
37 | 36 | eqcomd 2628 | 1 ⊢ (𝜑 → (𝑀‘𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ran crn 5115 ‘cfv 5888 (class class class)co 6650 2c2 11070 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 DimTarskiG≥cstrkgld 25333 Itvcitv 25335 LineGclng 25336 ⟂Gcperpg 25590 midGcmid 25664 lInvGclmi 25665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkgld 25351 df-trkg 25352 df-cgrg 25406 df-leg 25478 df-mir 25548 df-rag 25589 df-perpg 25591 df-mid 25666 df-lmi 25667 |
This theorem is referenced by: lmilmi 25681 |
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