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Mirrors > Home > MPE Home > Th. List > idmot | Structured version Visualization version GIF version |
Description: The identity is a motion. (Contributed by Thierry Arnoux, 15-Dec-2019.) |
Ref | Expression |
---|---|
ismot.p | β’ π = (BaseβπΊ) |
ismot.m | β’ β = (distβπΊ) |
motgrp.1 | β’ (π β πΊ β π) |
Ref | Expression |
---|---|
idmot | β’ (π β ( I βΎ π) β (πΊIsmtπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | motgrp.1 | . 2 β’ (π β πΊ β π) | |
2 | f1oi 6862 | . . 3 β’ ( I βΎ π):πβ1-1-ontoβπ | |
3 | 2 | a1i 11 | . 2 β’ (π β ( I βΎ π):πβ1-1-ontoβπ) |
4 | fvresi 7164 | . . . . 5 β’ (π β π β (( I βΎ π)βπ) = π) | |
5 | 4 | ad2antrl 725 | . . . 4 β’ ((π β§ (π β π β§ π β π)) β (( I βΎ π)βπ) = π) |
6 | fvresi 7164 | . . . . 5 β’ (π β π β (( I βΎ π)βπ) = π) | |
7 | 6 | ad2antll 726 | . . . 4 β’ ((π β§ (π β π β§ π β π)) β (( I βΎ π)βπ) = π) |
8 | 5, 7 | oveq12d 7420 | . . 3 β’ ((π β§ (π β π β§ π β π)) β ((( I βΎ π)βπ) β (( I βΎ π)βπ)) = (π β π)) |
9 | 8 | ralrimivva 3192 | . 2 β’ (π β βπ β π βπ β π ((( I βΎ π)βπ) β (( I βΎ π)βπ)) = (π β π)) |
10 | ismot.p | . . . 4 β’ π = (BaseβπΊ) | |
11 | ismot.m | . . . 4 β’ β = (distβπΊ) | |
12 | 10, 11 | ismot 28279 | . . 3 β’ (πΊ β π β (( I βΎ π) β (πΊIsmtπΊ) β (( I βΎ π):πβ1-1-ontoβπ β§ βπ β π βπ β π ((( I βΎ π)βπ) β (( I βΎ π)βπ)) = (π β π)))) |
13 | 12 | biimpar 477 | . 2 β’ ((πΊ β π β§ (( I βΎ π):πβ1-1-ontoβπ β§ βπ β π βπ β π ((( I βΎ π)βπ) β (( I βΎ π)βπ)) = (π β π))) β ( I βΎ π) β (πΊIsmtπΊ)) |
14 | 1, 3, 9, 13 | syl12anc 834 | 1 β’ (π β ( I βΎ π) β (πΊIsmtπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3053 I cid 5564 βΎ cres 5669 β1-1-ontoβwf1o 6533 βcfv 6534 (class class class)co 7402 Basecbs 17149 distcds 17211 Ismtcismt 28276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-map 8819 df-ismt 28277 |
This theorem is referenced by: motgrp 28287 |
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