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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 26570 | . . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴)) |
9 | lmicom.1 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐵) | |
10 | 9 | eqcomd 2829 | . . . . . 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 26575 | . . . . . 6 ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)))) |
15 | 10, 14 | mpbid 234 | . . . . 5 ⊢ (𝜑 → ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))) |
16 | 15 | simpld 497 | . . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝐷) |
17 | 8, 16 | eqeltrrd 2916 | . . 3 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐴) ∈ 𝐷) |
18 | 15 | simprd 498 | . . . . . . . . 9 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) |
19 | 18 | orcomd 867 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵))) |
20 | 19 | ord 860 | . . . . . . 7 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵))) |
21 | 4 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG) |
22 | 6 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃) |
23 | 7 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃) |
24 | simpr 487 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵) | |
25 | 24 | neqned 3025 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ≠ 𝐵) |
26 | 1, 3, 12, 21, 22, 23, 25 | tglinecom 26423 | . . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐴𝐿𝐵) = (𝐵𝐿𝐴)) |
27 | 26 | breq2d 5080 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))) |
28 | 27 | pm5.74da 802 | . . . . . . 7 ⊢ (𝜑 → ((¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)) ↔ (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴)))) |
29 | 20, 28 | mpbid 234 | . . . . . 6 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))) |
30 | 29 | orrd 859 | . . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))) |
31 | 30 | orcomd 867 | . . . 4 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵)) |
32 | eqcom 2830 | . . . . 5 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
33 | 32 | orbi2i 909 | . . . 4 ⊢ ((𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵) ↔ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴)) |
34 | 31, 33 | sylib 220 | . . 3 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴)) |
35 | 1, 2, 3, 4, 5, 11, 12, 13, 7, 6 | islmib 26575 | . . 3 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐵) ↔ ((𝐵(midG‘𝐺)𝐴) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴)))) |
36 | 17, 34, 35 | mpbir2and 711 | . 2 ⊢ (𝜑 → 𝐴 = (𝑀‘𝐵)) |
37 | 36 | eqcomd 2829 | 1 ⊢ (𝜑 → (𝑀‘𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ran crn 5558 ‘cfv 6357 (class class class)co 7158 2c2 11695 Basecbs 16485 distcds 16576 TarskiGcstrkg 26218 DimTarskiG≥cstrkgld 26222 Itvcitv 26224 LineGclng 26225 ⟂Gcperpg 26483 midGcmid 26560 lInvGclmi 26561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-s2 14212 df-s3 14213 df-trkgc 26236 df-trkgb 26237 df-trkgcb 26238 df-trkgld 26240 df-trkg 26241 df-cgrg 26299 df-leg 26371 df-mir 26441 df-rag 26482 df-perpg 26484 df-mid 26562 df-lmi 26563 |
This theorem is referenced by: lmilmi 26577 |
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