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Mirrors > Home > MPE Home > Th. List > mirreu | Structured version Visualization version GIF version |
Description: Any point has a unique antecedent through point inversion. Theorem 7.8 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 3-Jun-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) |
mirmir.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
Ref | Expression |
---|---|
mirreu | ⊢ (𝜑 → ∃!𝑎 ∈ 𝑃 (𝑀‘𝑎) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | mirval.d | . . 3 ⊢ − = (dist‘𝐺) | |
3 | mirval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | mirval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
5 | mirval.s | . . 3 ⊢ 𝑆 = (pInvG‘𝐺) | |
6 | mirval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | mirval.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | mirfv.m | . . 3 ⊢ 𝑀 = (𝑆‘𝐴) | |
9 | mirmir.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mircl 26374 | . 2 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mirmir 26375 | . 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵) |
12 | 6 | ad2antrr 722 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝐺 ∈ TarskiG) |
13 | 7 | ad2antrr 722 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝐴 ∈ 𝑃) |
14 | simplr 765 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝑎 ∈ 𝑃) | |
15 | 1, 2, 3, 4, 5, 12, 13, 8, 14 | mirmir 26375 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → (𝑀‘(𝑀‘𝑎)) = 𝑎) |
16 | simpr 485 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → (𝑀‘𝑎) = 𝐵) | |
17 | 16 | fveq2d 6667 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → (𝑀‘(𝑀‘𝑎)) = (𝑀‘𝐵)) |
18 | 15, 17 | eqtr3d 2855 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝑎 = (𝑀‘𝐵)) |
19 | 18 | ex 413 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → ((𝑀‘𝑎) = 𝐵 → 𝑎 = (𝑀‘𝐵))) |
20 | 19 | ralrimiva 3179 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ((𝑀‘𝑎) = 𝐵 → 𝑎 = (𝑀‘𝐵))) |
21 | fveqeq2 6672 | . . 3 ⊢ (𝑎 = (𝑀‘𝐵) → ((𝑀‘𝑎) = 𝐵 ↔ (𝑀‘(𝑀‘𝐵)) = 𝐵)) | |
22 | 21 | eqreu 3717 | . 2 ⊢ (((𝑀‘𝐵) ∈ 𝑃 ∧ (𝑀‘(𝑀‘𝐵)) = 𝐵 ∧ ∀𝑎 ∈ 𝑃 ((𝑀‘𝑎) = 𝐵 → 𝑎 = (𝑀‘𝐵))) → ∃!𝑎 ∈ 𝑃 (𝑀‘𝑎) = 𝐵) |
23 | 10, 11, 20, 22 | syl3anc 1363 | 1 ⊢ (𝜑 → ∃!𝑎 ∈ 𝑃 (𝑀‘𝑎) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃!wreu 3137 ‘cfv 6348 Basecbs 16471 distcds 16562 TarskiGcstrkg 26143 Itvcitv 26149 LineGclng 26150 pInvGcmir 26365 |
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-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 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-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-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 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 7103 df-ov 7148 df-trkgc 26161 df-trkgb 26162 df-trkgcb 26163 df-trkg 26166 df-mir 26366 |
This theorem is referenced by: (None) |
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