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Mirrors > Home > MPE Home > Th. List > mirf1o | Structured version Visualization version GIF version |
Description: The point inversion function 𝑀 is a bijection. Theorem 7.11 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 6-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) |
Ref | Expression |
---|---|
mirf1o | ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃) |
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.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
5 | mirval.s | . . . 4 ⊢ 𝑆 = (pInvG‘𝐺) | |
6 | mirval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | mirval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | mirfv.m | . . . 4 ⊢ 𝑀 = (𝑆‘𝐴) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | mirf 26373 | . . 3 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃) |
10 | 9 | ffnd 6508 | . 2 ⊢ (𝜑 → 𝑀 Fn 𝑃) |
11 | 6 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐺 ∈ TarskiG) |
12 | 7 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝐴 ∈ 𝑃) |
13 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → 𝑎 ∈ 𝑃) | |
14 | 1, 2, 3, 4, 5, 11, 12, 8, 13 | mirmir 26375 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → (𝑀‘(𝑀‘𝑎)) = 𝑎) |
15 | 14 | ralrimiva 3179 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 (𝑀‘(𝑀‘𝑎)) = 𝑎) |
16 | nvocnv 7029 | . . 3 ⊢ ((𝑀:𝑃⟶𝑃 ∧ ∀𝑎 ∈ 𝑃 (𝑀‘(𝑀‘𝑎)) = 𝑎) → ◡𝑀 = 𝑀) | |
17 | 9, 15, 16 | syl2anc 584 | . 2 ⊢ (𝜑 → ◡𝑀 = 𝑀) |
18 | nvof1o 7028 | . 2 ⊢ ((𝑀 Fn 𝑃 ∧ ◡𝑀 = 𝑀) → 𝑀:𝑃–1-1-onto→𝑃) | |
19 | 10, 17, 18 | syl2anc 584 | 1 ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ◡ccnv 5547 Fn wfn 6343 ⟶wf 6344 –1-1-onto→wf1o 6347 ‘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: mirmot 26388 |
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