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Mirrors > Home > MPE Home > Th. List > lmiinv | Structured version Visualization version GIF version |
Description: The invariants of the line mirroring lie on the mirror line. Theorem 10.8 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) |
Ref | Expression |
---|---|
lmiinv | ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismid.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | ismid.d | . . 3 ⊢ − = (dist‘𝐺) | |
3 | ismid.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | ismid.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | ismid.1 | . . 3 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
6 | lmif.m | . . 3 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) | |
7 | lmif.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
8 | lmif.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
9 | lmicl.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 9 | islmib 28706 | . 2 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐴) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐴) ∨ 𝐴 = 𝐴)))) |
11 | eqcom 2732 | . . 3 ⊢ (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴) | |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴)) |
13 | eqidd 2726 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐴) | |
14 | 13 | olcd 872 | . . . 4 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐴) ∨ 𝐴 = 𝐴)) |
15 | 14 | biantrud 530 | . . 3 ⊢ (𝜑 → ((𝐴(midG‘𝐺)𝐴) ∈ 𝐷 ↔ ((𝐴(midG‘𝐺)𝐴) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐴) ∨ 𝐴 = 𝐴)))) |
16 | 1, 2, 3, 4, 5, 9, 9 | midid 28700 | . . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐴) = 𝐴) |
17 | 16 | eleq1d 2810 | . . 3 ⊢ (𝜑 → ((𝐴(midG‘𝐺)𝐴) ∈ 𝐷 ↔ 𝐴 ∈ 𝐷)) |
18 | 15, 17 | bitr3d 280 | . 2 ⊢ (𝜑 → (((𝐴(midG‘𝐺)𝐴) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐴) ∨ 𝐴 = 𝐴)) ↔ 𝐴 ∈ 𝐷)) |
19 | 10, 12, 18 | 3bitr3d 308 | 1 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ran crn 5682 ‘cfv 6553 (class class class)co 7423 2c2 12314 Basecbs 17208 distcds 17270 TarskiGcstrkg 28346 DimTarskiG≥cstrkgld 28350 Itvcitv 28352 LineGclng 28353 ⟂Gcperpg 28614 midGcmid 28691 lInvGclmi 28692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-oadd 8499 df-er 8733 df-map 8856 df-pm 8857 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-dju 9940 df-card 9978 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-nn 12260 df-2 12322 df-3 12323 df-n0 12520 df-xnn0 12592 df-z 12606 df-uz 12870 df-fz 13534 df-fzo 13677 df-hash 14343 df-word 14518 df-concat 14574 df-s1 14599 df-s2 14852 df-s3 14853 df-trkgc 28367 df-trkgb 28368 df-trkgcb 28369 df-trkgld 28371 df-trkg 28372 df-cgrg 28430 df-leg 28502 df-mir 28572 df-rag 28613 df-perpg 28615 df-mid 28693 df-lmi 28694 |
This theorem is referenced by: lmicinv 28712 lmiisolem 28715 lmiopp 28721 |
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