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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 26260 | . . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴)) |
9 | lmicom.1 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐵) | |
10 | 9 | eqcomd 2778 | . . . . . 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 26265 | . . . . . 6 ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)))) |
15 | 10, 14 | mpbid 224 | . . . . 5 ⊢ (𝜑 → ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))) |
16 | 15 | simpld 487 | . . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝐷) |
17 | 8, 16 | eqeltrrd 2861 | . . 3 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐴) ∈ 𝐷) |
18 | 15 | simprd 488 | . . . . . . . . 9 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) |
19 | 18 | orcomd 857 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵))) |
20 | 19 | ord 850 | . . . . . . 7 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵))) |
21 | 4 | adantr 473 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG) |
22 | 6 | adantr 473 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃) |
23 | 7 | adantr 473 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃) |
24 | simpr 477 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵) | |
25 | 24 | neqned 2968 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ≠ 𝐵) |
26 | 1, 3, 12, 21, 22, 23, 25 | tglinecom 26113 | . . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐴𝐿𝐵) = (𝐵𝐿𝐴)) |
27 | 26 | breq2d 4935 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))) |
28 | 27 | pm5.74da 791 | . . . . . . 7 ⊢ (𝜑 → ((¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)) ↔ (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴)))) |
29 | 20, 28 | mpbid 224 | . . . . . 6 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))) |
30 | 29 | orrd 849 | . . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))) |
31 | 30 | orcomd 857 | . . . 4 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵)) |
32 | eqcom 2779 | . . . . 5 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
33 | 32 | orbi2i 896 | . . . 4 ⊢ ((𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵) ↔ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴)) |
34 | 31, 33 | sylib 210 | . . 3 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴)) |
35 | 1, 2, 3, 4, 5, 11, 12, 13, 7, 6 | islmib 26265 | . . 3 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐵) ↔ ((𝐵(midG‘𝐺)𝐴) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴)))) |
36 | 17, 34, 35 | mpbir2and 700 | . 2 ⊢ (𝜑 → 𝐴 = (𝑀‘𝐵)) |
37 | 36 | eqcomd 2778 | 1 ⊢ (𝜑 → (𝑀‘𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2048 class class class wbr 4923 ran crn 5401 ‘cfv 6182 (class class class)co 6970 2c2 11488 Basecbs 16329 distcds 16420 TarskiGcstrkg 25908 DimTarskiG≥cstrkgld 25912 Itvcitv 25914 LineGclng 25915 ⟂Gcperpg 26173 midGcmid 26250 lInvGclmi 26251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-map 8200 df-pm 8201 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-dju 9116 df-card 9154 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-n0 11701 df-xnn0 11773 df-z 11787 df-uz 12052 df-fz 12702 df-fzo 12843 df-hash 13499 df-word 13663 df-concat 13724 df-s1 13749 df-s2 14062 df-s3 14063 df-trkgc 25926 df-trkgb 25927 df-trkgcb 25928 df-trkgld 25930 df-trkg 25931 df-cgrg 25989 df-leg 26061 df-mir 26131 df-rag 26172 df-perpg 26174 df-mid 26252 df-lmi 26253 |
This theorem is referenced by: lmilmi 26267 |
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