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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 25556 | . 2 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | mirmir 25557 | . 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵) |
12 | 6 | ad2antrr 762 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝐺 ∈ TarskiG) |
13 | 7 | ad2antrr 762 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝐴 ∈ 𝑃) |
14 | simplr 792 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝑎 ∈ 𝑃) | |
15 | 1, 2, 3, 4, 5, 12, 13, 8, 14 | mirmir 25557 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → (𝑀‘(𝑀‘𝑎)) = 𝑎) |
16 | simpr 477 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → (𝑀‘𝑎) = 𝐵) | |
17 | 16 | fveq2d 6195 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → (𝑀‘(𝑀‘𝑎)) = (𝑀‘𝐵)) |
18 | 15, 17 | eqtr3d 2658 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝑎 = (𝑀‘𝐵)) |
19 | 18 | ex 450 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → ((𝑀‘𝑎) = 𝐵 → 𝑎 = (𝑀‘𝐵))) |
20 | 19 | ralrimiva 2966 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ((𝑀‘𝑎) = 𝐵 → 𝑎 = (𝑀‘𝐵))) |
21 | fveq2 6191 | . . . 4 ⊢ (𝑎 = (𝑀‘𝐵) → (𝑀‘𝑎) = (𝑀‘(𝑀‘𝐵))) | |
22 | 21 | eqeq1d 2624 | . . 3 ⊢ (𝑎 = (𝑀‘𝐵) → ((𝑀‘𝑎) = 𝐵 ↔ (𝑀‘(𝑀‘𝐵)) = 𝐵)) |
23 | 22 | eqreu 3398 | . 2 ⊢ (((𝑀‘𝐵) ∈ 𝑃 ∧ (𝑀‘(𝑀‘𝐵)) = 𝐵 ∧ ∀𝑎 ∈ 𝑃 ((𝑀‘𝑎) = 𝐵 → 𝑎 = (𝑀‘𝐵))) → ∃!𝑎 ∈ 𝑃 (𝑀‘𝑎) = 𝐵) |
24 | 10, 11, 20, 23 | syl3anc 1326 | 1 ⊢ (𝜑 → ∃!𝑎 ∈ 𝑃 (𝑀‘𝑎) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃!wreu 2914 ‘cfv 5888 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 LineGclng 25336 pInvGcmir 25547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 df-mir 25548 |
This theorem is referenced by: (None) |
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