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Mirrors > Home > MPE Home > Th. List > lmireu | Structured version Visualization version GIF version |
Description: Any point has a unique antecedent through line mirroring. Theorem 10.6 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 |
---|---|
lmireu | ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 (𝑀‘𝑏) = 𝐴) |
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 | lmicl 28016 | . 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | lmilmi 28019 | . 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴) |
12 | 4 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝐺 ∈ TarskiG) |
13 | 5 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝐺DimTarskiG≥2) |
14 | 8 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝐷 ∈ ran 𝐿) |
15 | simplr 768 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝑏 ∈ 𝑃) | |
16 | 1, 2, 3, 12, 13, 6, 7, 14, 15 | lmilmi 28019 | . . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → (𝑀‘(𝑀‘𝑏)) = 𝑏) |
17 | simpr 486 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → (𝑀‘𝑏) = 𝐴) | |
18 | 17 | fveq2d 6891 | . . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → (𝑀‘(𝑀‘𝑏)) = (𝑀‘𝐴)) |
19 | 16, 18 | eqtr3d 2775 | . . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝑃) ∧ (𝑀‘𝑏) = 𝐴) → 𝑏 = (𝑀‘𝐴)) |
20 | 19 | ex 414 | . . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → ((𝑀‘𝑏) = 𝐴 → 𝑏 = (𝑀‘𝐴))) |
21 | 20 | ralrimiva 3147 | . 2 ⊢ (𝜑 → ∀𝑏 ∈ 𝑃 ((𝑀‘𝑏) = 𝐴 → 𝑏 = (𝑀‘𝐴))) |
22 | fveqeq2 6896 | . . 3 ⊢ (𝑏 = (𝑀‘𝐴) → ((𝑀‘𝑏) = 𝐴 ↔ (𝑀‘(𝑀‘𝐴)) = 𝐴)) | |
23 | 22 | eqreu 3723 | . 2 ⊢ (((𝑀‘𝐴) ∈ 𝑃 ∧ (𝑀‘(𝑀‘𝐴)) = 𝐴 ∧ ∀𝑏 ∈ 𝑃 ((𝑀‘𝑏) = 𝐴 → 𝑏 = (𝑀‘𝐴))) → ∃!𝑏 ∈ 𝑃 (𝑀‘𝑏) = 𝐴) |
24 | 10, 11, 21, 23 | syl3anc 1372 | 1 ⊢ (𝜑 → ∃!𝑏 ∈ 𝑃 (𝑀‘𝑏) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃!wreu 3375 class class class wbr 5146 ran crn 5675 ‘cfv 6539 2c2 12262 Basecbs 17139 distcds 17201 TarskiGcstrkg 27657 DimTarskiG≥cstrkgld 27661 Itvcitv 27663 LineGclng 27664 lInvGclmi 28003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-oadd 8464 df-er 8698 df-map 8817 df-pm 8818 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-dju 9891 df-card 9929 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-3 12271 df-n0 12468 df-xnn0 12540 df-z 12554 df-uz 12818 df-fz 13480 df-fzo 13623 df-hash 14286 df-word 14460 df-concat 14516 df-s1 14541 df-s2 14794 df-s3 14795 df-trkgc 27678 df-trkgb 27679 df-trkgcb 27680 df-trkgld 27682 df-trkg 27683 df-cgrg 27741 df-leg 27813 df-mir 27883 df-rag 27924 df-perpg 27926 df-mid 28004 df-lmi 28005 |
This theorem is referenced by: lmieq 28021 |
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