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Mirrors > Home > MPE Home > Th. List > lmiopp | Structured version Visualization version GIF version |
Description: Line mirroring produces points on the opposite side of the mirroring line. Theorem 10.14 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 2-Aug-2020.) |
Ref | Expression |
---|---|
lmiopp.p | ⊢ 𝑃 = (Base‘𝐺) |
lmiopp.m | ⊢ − = (dist‘𝐺) |
lmiopp.i | ⊢ 𝐼 = (Itv‘𝐺) |
lmiopp.l | ⊢ 𝐿 = (LineG‘𝐺) |
lmiopp.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
lmiopp.h | ⊢ (𝜑 → 𝐺DimTarskiG≥2) |
lmiopp.d | ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
lmiopp.o | ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} |
lmiopp.n | ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) |
lmiopp.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
lmiopp.1 | ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) |
Ref | Expression |
---|---|
lmiopp | ⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmiopp.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | lmiopp.m | . 2 ⊢ − = (dist‘𝐺) | |
3 | lmiopp.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | lmiopp.o | . 2 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
5 | lmiopp.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | lmiopp.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | lmiopp.h | . . 3 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
8 | lmiopp.n | . . 3 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) | |
9 | lmiopp.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
10 | lmiopp.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
11 | 1, 2, 3, 6, 7, 8, 9, 10, 5 | lmicl 26572 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) |
12 | eqidd 2822 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘𝐴)) | |
13 | 1, 2, 3, 6, 7, 8, 9, 10, 5, 11 | islmib 26573 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴))))) |
14 | 12, 13 | mpbid 234 | . . 3 ⊢ (𝜑 → ((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴)))) |
15 | 14 | simpld 497 | . 2 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷) |
16 | lmiopp.1 | . 2 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) | |
17 | 1, 2, 3, 6, 7, 8, 9, 10, 5 | lmilmi 26575 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴) |
18 | 17 | eqeq1d 2823 | . . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ 𝐴 = (𝑀‘𝐴))) |
19 | 1, 2, 3, 6, 7, 8, 9, 10, 11 | lmiinv 26578 | . . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ (𝑀‘𝐴) ∈ 𝐷)) |
20 | eqcom 2828 | . . . . . 6 ⊢ (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴) | |
21 | 20 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴)) |
22 | 18, 19, 21 | 3bitr3d 311 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ (𝑀‘𝐴) = 𝐴)) |
23 | 1, 2, 3, 6, 7, 8, 9, 10, 5 | lmiinv 26578 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷)) |
24 | 22, 23 | bitrd 281 | . . 3 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ 𝐴 ∈ 𝐷)) |
25 | 16, 24 | mtbird 327 | . 2 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ 𝐷) |
26 | 1, 2, 3, 6, 7, 5, 11 | midbtwn 26565 | . 2 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ (𝐴𝐼(𝑀‘𝐴))) |
27 | 1, 2, 3, 4, 5, 11, 15, 16, 25, 26 | islnoppd 26526 | 1 ⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ∖ cdif 3933 class class class wbr 5066 {copab 5128 ran crn 5556 ‘cfv 6355 (class class class)co 7156 2c2 11693 Basecbs 16483 distcds 16574 TarskiGcstrkg 26216 DimTarskiG≥cstrkgld 26220 Itvcitv 26222 LineGclng 26223 ⟂Gcperpg 26481 midGcmid 26558 lInvGclmi 26559 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-s2 14210 df-s3 14211 df-trkgc 26234 df-trkgb 26235 df-trkgcb 26236 df-trkgld 26238 df-trkg 26239 df-cgrg 26297 df-leg 26369 df-mir 26439 df-rag 26480 df-perpg 26482 df-mid 26560 df-lmi 26561 |
This theorem is referenced by: trgcopyeulem 26591 |
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