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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 27309 | . . 3 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑃) |
6 | f1of 6781 | . . 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 | ffvelcdmd 7032 | 1 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⟶wf 6489 –1-1-onto→wf1o 6492 ‘cfv 6493 (class class class)co 7351 Basecbs 17043 distcds 17102 Ismtcismt 27303 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-map 8725 df-ismt 27304 |
This theorem is referenced by: motcgr3 27316 motrag 27479 |
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