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Mirrors > Home > MPE Home > Th. List > ghmgrp | Structured version Visualization version Unicode version |
Description: The image of a group under a group homomorphism is a group. This is a stronger result than that usually found in the literature, since the target of the homomorphism (operator in our model) need not have any of the properties of a group as a prerequisite. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.) |
Ref | Expression |
---|---|
ghmgrp.f | |
ghmgrp.x | |
ghmgrp.y | |
ghmgrp.p | |
ghmgrp.q | |
ghmgrp.1 | |
ghmgrp.3 |
Ref | Expression |
---|---|
ghmgrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ghmgrp.f | . . 3 | |
2 | ghmgrp.x | . . 3 | |
3 | ghmgrp.y | . . 3 | |
4 | ghmgrp.p | . . 3 | |
5 | ghmgrp.q | . . 3 | |
6 | ghmgrp.1 | . . 3 | |
7 | ghmgrp.3 | . . . 4 | |
8 | grpmnd 17429 | . . . 4 | |
9 | 7, 8 | syl 17 | . . 3 |
10 | 1, 2, 3, 4, 5, 6, 9 | mhmmnd 17537 | . 2 |
11 | fof 6115 | . . . . . . . 8 | |
12 | 6, 11 | syl 17 | . . . . . . 7 |
13 | 12 | ad3antrrr 766 | . . . . . 6 |
14 | 7 | ad3antrrr 766 | . . . . . . 7 |
15 | simplr 792 | . . . . . . 7 | |
16 | eqid 2622 | . . . . . . . 8 | |
17 | 2, 16 | grpinvcl 17467 | . . . . . . 7 |
18 | 14, 15, 17 | syl2anc 693 | . . . . . 6 |
19 | 13, 18 | ffvelrnd 6360 | . . . . 5 |
20 | 1 | 3adant1r 1319 | . . . . . . . . 9 |
21 | 7, 17 | sylan 488 | . . . . . . . . 9 |
22 | simpr 477 | . . . . . . . . 9 | |
23 | 20, 21, 22 | mhmlem 17535 | . . . . . . . 8 |
24 | 23 | adantlr 751 | . . . . . . 7 |
25 | 24 | adantr 481 | . . . . . 6 |
26 | eqid 2622 | . . . . . . . . . 10 | |
27 | 2, 4, 26, 16 | grplinv 17468 | . . . . . . . . 9 |
28 | 27 | fveq2d 6195 | . . . . . . . 8 |
29 | 14, 15, 28 | syl2anc 693 | . . . . . . 7 |
30 | 1, 2, 3, 4, 5, 6, 9, 26 | mhmid 17536 | . . . . . . . 8 |
31 | 30 | ad3antrrr 766 | . . . . . . 7 |
32 | 29, 31 | eqtrd 2656 | . . . . . 6 |
33 | simpr 477 | . . . . . . 7 | |
34 | 33 | oveq2d 6666 | . . . . . 6 |
35 | 25, 32, 34 | 3eqtr3rd 2665 | . . . . 5 |
36 | oveq1 6657 | . . . . . . 7 | |
37 | 36 | eqeq1d 2624 | . . . . . 6 |
38 | 37 | rspcev 3309 | . . . . 5 |
39 | 19, 35, 38 | syl2anc 693 | . . . 4 |
40 | foelrni 6244 | . . . . 5 | |
41 | 6, 40 | sylan 488 | . . . 4 |
42 | 39, 41 | r19.29a 3078 | . . 3 |
43 | 42 | ralrimiva 2966 | . 2 |
44 | eqid 2622 | . . 3 | |
45 | 3, 5, 44 | isgrp 17428 | . 2 |
46 | 10, 43, 45 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wrex 2913 wf 5884 wfo 5886 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 c0g 16100 cmnd 17294 cgrp 17422 cminusg 17423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 |
This theorem is referenced by: ghmfghm 18236 ghmabl 18238 |
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