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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 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐺 ∈ TarskiG) |
6 | mirinv.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐵 ∈ 𝑃) |
8 | mirval.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | 8 | adantr 483 | . . . 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 26436 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)) |
14 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → (𝑀‘𝐵) = 𝐵) | |
15 | 14 | oveq1d 7163 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → ((𝑀‘𝐵)𝐼𝐵) = (𝐵𝐼𝐵)) |
16 | 13, 15 | eleqtrd 2913 | . . . 4 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵)) |
17 | 1, 2, 3, 5, 7, 9, 16 | axtgbtwnid 26244 | . . 3 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐵 = 𝐴) |
18 | 17 | eqcomd 2825 | . 2 ⊢ ((𝜑 ∧ (𝑀‘𝐵) = 𝐵) → 𝐴 = 𝐵) |
19 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG) |
20 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃) |
21 | 6 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃) |
22 | eqidd 2820 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐴 − 𝐵)) | |
23 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
24 | 1, 2, 3, 19, 21, 21 | tgbtwntriv1 26269 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ (𝐵𝐼𝐵)) |
25 | 23, 24 | eqeltrd 2911 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵)) |
26 | 1, 2, 3, 10, 11, 19, 20, 12, 21, 21, 22, 25 | ismir 26437 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 = (𝑀‘𝐵)) |
27 | 26 | eqcomd 2825 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) = 𝐵) |
28 | 18, 27 | impbida 799 | 1 ⊢ (𝜑 → ((𝑀‘𝐵) = 𝐵 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 distcds 16566 TarskiGcstrkg 26208 Itvcitv 26214 LineGclng 26215 pInvGcmir 26430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 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-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 7106 df-ov 7151 df-trkgc 26226 df-trkgb 26227 df-trkgcb 26228 df-trkg 26231 df-mir 26431 |
This theorem is referenced by: mirne 26445 mircinv 26446 mirln2 26455 miduniq 26463 miduniq2 26465 krippenlem 26468 ragflat2 26481 footexALT 26496 footexlem1 26497 footexlem2 26498 colperpexlem2 26509 colperpexlem3 26510 opphllem6 26530 lmimid 26572 hypcgrlem2 26578 |
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