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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 26500 | . 2 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐴) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐴) ∨ 𝐴 = 𝐴)))) |
11 | eqcom 2825 | . . 3 ⊢ (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴) | |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴)) |
13 | eqidd 2819 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐴) | |
14 | 13 | olcd 870 | . . . 4 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐴) ∨ 𝐴 = 𝐴)) |
15 | 14 | biantrud 532 | . . 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 26494 | . . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐴) = 𝐴) |
17 | 16 | eleq1d 2894 | . . 3 ⊢ (𝜑 → ((𝐴(midG‘𝐺)𝐴) ∈ 𝐷 ↔ 𝐴 ∈ 𝐷)) |
18 | 15, 17 | bitr3d 282 | . 2 ⊢ (𝜑 → (((𝐴(midG‘𝐺)𝐴) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐴) ∨ 𝐴 = 𝐴)) ↔ 𝐴 ∈ 𝐷)) |
19 | 10, 12, 18 | 3bitr3d 310 | 1 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ran crn 5549 ‘cfv 6348 (class class class)co 7145 2c2 11680 Basecbs 16471 distcds 16562 TarskiGcstrkg 26143 DimTarskiG≥cstrkgld 26147 Itvcitv 26149 LineGclng 26150 ⟂Gcperpg 26408 midGcmid 26485 lInvGclmi 26486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-s2 14198 df-s3 14199 df-trkgc 26161 df-trkgb 26162 df-trkgcb 26163 df-trkgld 26165 df-trkg 26166 df-cgrg 26224 df-leg 26296 df-mir 26366 df-rag 26407 df-perpg 26409 df-mid 26487 df-lmi 26488 |
This theorem is referenced by: lmicinv 26506 lmiisolem 26509 lmiopp 26515 |
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