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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.1 | . . . 4 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) | |
2 | lmiopp.p | . . . . . . . . 9 ⊢ 𝑃 = (Base‘𝐺) | |
3 | lmiopp.m | . . . . . . . . 9 ⊢ − = (dist‘𝐺) | |
4 | lmiopp.i | . . . . . . . . 9 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | lmiopp.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | lmiopp.h | . . . . . . . . 9 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
7 | lmiopp.n | . . . . . . . . 9 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) | |
8 | lmiopp.l | . . . . . . . . 9 ⊢ 𝐿 = (LineG‘𝐺) | |
9 | lmiopp.d | . . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
10 | lmiopp.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | lmilmi 26154 | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴) |
12 | 11 | eqeq1d 2780 | . . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ 𝐴 = (𝑀‘𝐴))) |
13 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | lmicl 26151 | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) |
14 | 2, 3, 4, 5, 6, 7, 8, 9, 13 | lmiinv 26157 | . . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ (𝑀‘𝐴) ∈ 𝐷)) |
15 | eqcom 2785 | . . . . . . . 8 ⊢ (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴) | |
16 | 15 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴)) |
17 | 12, 14, 16 | 3bitr3d 301 | . . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ (𝑀‘𝐴) = 𝐴)) |
18 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | lmiinv 26157 | . . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷)) |
19 | 17, 18 | bitrd 271 | . . . . 5 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ 𝐴 ∈ 𝐷)) |
20 | 1, 19 | mtbird 317 | . . . 4 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ 𝐷) |
21 | 1, 20 | jca 507 | . . 3 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝐷 ∧ ¬ (𝑀‘𝐴) ∈ 𝐷)) |
22 | eqidd 2779 | . . . . . 6 ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘𝐴)) | |
23 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 13 | islmib 26152 | . . . . . 6 ⊢ (𝜑 → ((𝑀‘𝐴) = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴))))) |
24 | 22, 23 | mpbid 224 | . . . . 5 ⊢ (𝜑 → ((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴)))) |
25 | 24 | simpld 490 | . . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷) |
26 | 2, 3, 4, 5, 6, 10, 13 | midbtwn 26144 | . . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ (𝐴𝐼(𝑀‘𝐴))) |
27 | eleq1 2847 | . . . . 5 ⊢ (𝑡 = (𝐴(midG‘𝐺)(𝑀‘𝐴)) → (𝑡 ∈ (𝐴𝐼(𝑀‘𝐴)) ↔ (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ (𝐴𝐼(𝑀‘𝐴)))) | |
28 | 27 | rspcev 3511 | . . . 4 ⊢ (((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ (𝐴𝐼(𝑀‘𝐴))) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼(𝑀‘𝐴))) |
29 | 25, 26, 28 | syl2anc 579 | . . 3 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼(𝑀‘𝐴))) |
30 | 21, 29 | jca 507 | . 2 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ (𝑀‘𝐴) ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼(𝑀‘𝐴)))) |
31 | lmiopp.o | . . 3 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
32 | 2, 3, 4, 31, 10, 13 | islnopp 26104 | . 2 ⊢ (𝜑 → (𝐴𝑂(𝑀‘𝐴) ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ (𝑀‘𝐴) ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼(𝑀‘𝐴))))) |
33 | 30, 32 | mpbird 249 | 1 ⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 ∖ cdif 3789 class class class wbr 4888 {copab 4950 ran crn 5358 ‘cfv 6137 (class class class)co 6924 2c2 11435 Basecbs 16266 distcds 16358 TarskiGcstrkg 25798 DimTarskiG≥cstrkgld 25802 Itvcitv 25804 LineGclng 25805 ⟂Gcperpg 26063 midGcmid 26137 lInvGclmi 26138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-xnn0 11720 df-z 11734 df-uz 11998 df-fz 12649 df-fzo 12790 df-hash 13442 df-word 13606 df-concat 13667 df-s1 13692 df-s2 14005 df-s3 14006 df-trkgc 25816 df-trkgb 25817 df-trkgcb 25818 df-trkgld 25820 df-trkg 25821 df-cgrg 25879 df-leg 25951 df-mir 26021 df-rag 26062 df-perpg 26064 df-mid 26139 df-lmi 26140 |
This theorem is referenced by: trgcopyeulem 26170 |
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