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Mirrors > Home > MPE Home > Th. List > ismot | Structured version Visualization version GIF version |
Description: Property of being an isometry mapping to the same space. In geometry, this is also called a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
Ref | Expression |
---|---|
ismot.p | ⊢ 𝑃 = (Base‘𝐺) |
ismot.m | ⊢ − = (dist‘𝐺) |
Ref | Expression |
---|---|
ismot | ⊢ (𝐺 ∈ 𝑉 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismot.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | ismot.m | . . 3 ⊢ − = (dist‘𝐺) | |
3 | 1, 1, 2, 2 | isismt 26328 | . 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉) → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏)))) |
4 | 3 | anidms 570 | 1 ⊢ (𝐺 ∈ 𝑉 → (𝐹 ∈ (𝐺Ismt𝐺) ↔ (𝐹:𝑃–1-1-onto→𝑃 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) − (𝐹‘𝑏)) = (𝑎 − 𝑏)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 Ismtcismt 26326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-ismt 26327 |
This theorem is referenced by: motcgr 26330 idmot 26331 motf1o 26332 motco 26334 cnvmot 26335 motgrp 26337 mirmot 26469 lmimot 26592 |
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