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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 28794 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) | 
| 12 | eqidd 2738 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘𝐴)) | |
| 13 | 1, 2, 3, 6, 7, 8, 9, 10, 5, 11 | islmib 28795 | . . . 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 232 | . . 3 ⊢ (𝜑 → ((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴)))) | 
| 15 | 14 | simpld 494 | . 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 28797 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴) | 
| 18 | 17 | eqeq1d 2739 | . . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ 𝐴 = (𝑀‘𝐴))) | 
| 19 | 1, 2, 3, 6, 7, 8, 9, 10, 11 | lmiinv 28800 | . . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ (𝑀‘𝐴) ∈ 𝐷)) | 
| 20 | eqcom 2744 | . . . . . 6 ⊢ (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴) | |
| 21 | 20 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴)) | 
| 22 | 18, 19, 21 | 3bitr3d 309 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ (𝑀‘𝐴) = 𝐴)) | 
| 23 | 1, 2, 3, 6, 7, 8, 9, 10, 5 | lmiinv 28800 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷)) | 
| 24 | 22, 23 | bitrd 279 | . . 3 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ 𝐴 ∈ 𝐷)) | 
| 25 | 16, 24 | mtbird 325 | . 2 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ 𝐷) | 
| 26 | 1, 2, 3, 6, 7, 5, 11 | midbtwn 28787 | . 2 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ (𝐴𝐼(𝑀‘𝐴))) | 
| 27 | 1, 2, 3, 4, 5, 11, 15, 16, 25, 26 | islnoppd 28748 | 1 ⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ∖ cdif 3948 class class class wbr 5143 {copab 5205 ran crn 5686 ‘cfv 6561 (class class class)co 7431 2c2 12321 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 DimTarskiG≥cstrkgld 28439 Itvcitv 28441 LineGclng 28442 ⟂Gcperpg 28703 midGcmid 28780 lInvGclmi 28781 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkgld 28460 df-trkg 28461 df-cgrg 28519 df-leg 28591 df-mir 28661 df-rag 28702 df-perpg 28704 df-mid 28782 df-lmi 28783 | 
| This theorem is referenced by: trgcopyeulem 28813 | 
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