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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 6871 | . . 3 β’ ( I βΎ π):πβ1-1-ontoβπ | |
3 | 2 | a1i 11 | . 2 β’ (π β ( I βΎ π):πβ1-1-ontoβπ) |
4 | fvresi 7176 | . . . . 5 β’ (π β π β (( I βΎ π)βπ) = π) | |
5 | 4 | ad2antrl 727 | . . . 4 β’ ((π β§ (π β π β§ π β π)) β (( I βΎ π)βπ) = π) |
6 | fvresi 7176 | . . . . 5 β’ (π β π β (( I βΎ π)βπ) = π) | |
7 | 6 | ad2antll 728 | . . . 4 β’ ((π β§ (π β π β§ π β π)) β (( I βΎ π)βπ) = π) |
8 | 5, 7 | oveq12d 7432 | . . 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 3196 | . 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 28332 | . . 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 836 | 1 β’ (π β ( I βΎ π) β (πΊIsmtπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βwral 3057 I cid 5569 βΎ cres 5674 β1-1-ontoβwf1o 6541 βcfv 6542 (class class class)co 7414 Basecbs 17173 distcds 17235 Ismtcismt 28329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8840 df-ismt 28330 |
This theorem is referenced by: motgrp 28340 |
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