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| Mirrors > Home > MPE Home > Th. List > mirinv | Structured version Visualization version GIF version | ||
| Description: The only invariant point of a point inversion Theorem 7.3 of [Schwabhauser] p. 49, Theorem 7.10 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 30-Jul-2019.) | 
| Ref | Expression | 
|---|---|
| mirval.p | ⊢ 𝑃 = (Base‘𝐺) | 
| mirval.d | ⊢ − = (dist‘𝐺) | 
| mirval.i | ⊢ 𝐼 = (Itv‘𝐺) | 
| mirval.l | ⊢ 𝐿 = (LineG‘𝐺) | 
| mirval.s | ⊢ 𝑆 = (pInvG‘𝐺) | 
| mirval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) | 
| mirval.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) | 
| mirfv.m | ⊢ 𝑀 = (𝑆‘𝐴) | 
| mirinv.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) | 
| Ref | Expression | 
|---|---|
| mirinv | ⊢ (𝜑 → ((𝑀‘𝐵) = 𝐵 ↔ 𝐴 = 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | mirval.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
| 2 | mirval.d | . . . 4 ⊢ − = (dist‘𝐺) | |
| 3 | mirval.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
| 4 | mirval.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 5 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐺 ∈ TarskiG) | 
| 6 | mirinv.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 7 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐵 ∈ 𝑃) | 
| 8 | mirval.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 9 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 ∈ 𝑃) | 
| 10 | mirval.l | . . . . . 6 ⊢ 𝐿 = (LineG‘𝐺) | |
| 11 | mirval.s | . . . . . 6 ⊢ 𝑆 = (pInvG‘𝐺) | |
| 12 | mirfv.m | . . . . . 6 ⊢ 𝑀 = (𝑆‘𝐴) | |
| 13 | 1, 2, 3, 10, 11, 5, 9, 12, 7 | mirbtwn 28667 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)) | 
| 14 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → (𝑀‘𝐵) = 𝐵) | |
| 15 | 14 | oveq1d 7447 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → ((𝑀‘𝐵)𝐼𝐵) = (𝐵𝐼𝐵)) | 
| 16 | 13, 15 | eleqtrd 2842 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵)) | 
| 17 | 1, 2, 3, 5, 7, 9, 16 | axtgbtwnid 28475 | . . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐵 = 𝐴) | 
| 18 | 17 | eqcomd 2742 | . 2 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 = 𝐵) | 
| 19 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG) | 
| 20 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃) | 
| 21 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃) | 
| 22 | eqidd 2737 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐴 − 𝐵)) | |
| 23 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
| 24 | 1, 2, 3, 19, 21, 21 | tgbtwntriv1 28500 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ (𝐵𝐼𝐵)) | 
| 25 | 23, 24 | eqeltrd 2840 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵)) | 
| 26 | 1, 2, 3, 10, 11, 19, 20, 12, 21, 21, 22, 25 | ismir 28668 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 = (𝑀‘𝐵)) | 
| 27 | 26 | eqcomd 2742 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) = 𝐵) | 
| 28 | 18, 27 | impbida 800 | 1 ⊢ (𝜑 → ((𝑀‘𝐵) = 𝐵 ↔ 𝐴 = 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 distcds 17307 TarskiGcstrkg 28436 Itvcitv 28442 LineGclng 28443 pInvGcmir 28661 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-trkgc 28457 df-trkgb 28458 df-trkgcb 28459 df-trkg 28462 df-mir 28662 | 
| This theorem is referenced by: mirne 28676 mircinv 28677 mirln2 28686 miduniq 28694 miduniq2 28696 krippenlem 28699 ragflat2 28712 footexALT 28727 footexlem1 28728 footexlem2 28729 colperpexlem2 28740 colperpexlem3 28741 opphllem6 28761 lmimid 28803 hypcgrlem2 28809 | 
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