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Mirrors > Home > MPE Home > Th. List > motcl | Structured version Visualization version GIF version |
Description: Closure of motions. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
Ref | Expression |
---|---|
ismot.p | ⊢ 𝑃 = (Base‘𝐺) |
ismot.m | ⊢ − = (dist‘𝐺) |
motgrp.1 | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
motco.2 | ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) |
motcl.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
Ref | Expression |
---|---|
motcl | ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismot.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | ismot.m | . . . 4 ⊢ − = (dist‘𝐺) | |
3 | motgrp.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
4 | motco.2 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐺Ismt𝐺)) | |
5 | 1, 2, 3, 4 | motf1o 26251 | . . 3 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃) |
6 | f1of 6608 | . . 3 ⊢ (𝐹:𝑃–1-1-onto→𝑃 → 𝐹:𝑃⟶𝑃) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝜑 → 𝐹:𝑃⟶𝑃) |
8 | motcl.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | 7, 8 | ffvelrnd 6844 | 1 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⟶wf 6344 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 distcds 16562 Ismtcismt 26245 |
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-pow 5257 ax-pr 5320 ax-un 7450 |
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-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-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-ismt 26246 |
This theorem is referenced by: motcgr3 26258 motrag 26421 |
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