Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > isghm | Structured version Visualization version Unicode version |
Description: Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
Ref | Expression |
---|---|
isghm.w | |
isghm.x | |
isghm.a | |
isghm.b |
Ref | Expression |
---|---|
isghm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ghm 17658 | . . 3 | |
2 | 1 | elmpt2cl 6876 | . 2 |
3 | fvex 6201 | . . . . . . . 8 | |
4 | feq2 6027 | . . . . . . . . 9 | |
5 | raleq 3138 | . . . . . . . . . 10 | |
6 | 5 | raleqbi1dv 3146 | . . . . . . . . 9 |
7 | 4, 6 | anbi12d 747 | . . . . . . . 8 |
8 | 3, 7 | sbcie 3470 | . . . . . . 7 |
9 | fveq2 6191 | . . . . . . . . . 10 | |
10 | isghm.w | . . . . . . . . . 10 | |
11 | 9, 10 | syl6eqr 2674 | . . . . . . . . 9 |
12 | 11 | feq2d 6031 | . . . . . . . 8 |
13 | fveq2 6191 | . . . . . . . . . . . . . 14 | |
14 | isghm.a | . . . . . . . . . . . . . 14 | |
15 | 13, 14 | syl6eqr 2674 | . . . . . . . . . . . . 13 |
16 | 15 | oveqd 6667 | . . . . . . . . . . . 12 |
17 | 16 | fveq2d 6195 | . . . . . . . . . . 11 |
18 | 17 | eqeq1d 2624 | . . . . . . . . . 10 |
19 | 11, 18 | raleqbidv 3152 | . . . . . . . . 9 |
20 | 11, 19 | raleqbidv 3152 | . . . . . . . 8 |
21 | 12, 20 | anbi12d 747 | . . . . . . 7 |
22 | 8, 21 | syl5bb 272 | . . . . . 6 |
23 | 22 | abbidv 2741 | . . . . 5 |
24 | fveq2 6191 | . . . . . . . . 9 | |
25 | isghm.x | . . . . . . . . 9 | |
26 | 24, 25 | syl6eqr 2674 | . . . . . . . 8 |
27 | 26 | feq3d 6032 | . . . . . . 7 |
28 | fveq2 6191 | . . . . . . . . . . 11 | |
29 | isghm.b | . . . . . . . . . . 11 | |
30 | 28, 29 | syl6eqr 2674 | . . . . . . . . . 10 |
31 | 30 | oveqd 6667 | . . . . . . . . 9 |
32 | 31 | eqeq2d 2632 | . . . . . . . 8 |
33 | 32 | 2ralbidv 2989 | . . . . . . 7 |
34 | 27, 33 | anbi12d 747 | . . . . . 6 |
35 | 34 | abbidv 2741 | . . . . 5 |
36 | fvex 6201 | . . . . . . . 8 | |
37 | 10, 36 | eqeltri 2697 | . . . . . . 7 |
38 | fvex 6201 | . . . . . . . 8 | |
39 | 25, 38 | eqeltri 2697 | . . . . . . 7 |
40 | mapex 7863 | . . . . . . 7 | |
41 | 37, 39, 40 | mp2an 708 | . . . . . 6 |
42 | simpl 473 | . . . . . . 7 | |
43 | 42 | ss2abi 3674 | . . . . . 6 |
44 | 41, 43 | ssexi 4803 | . . . . 5 |
45 | 23, 35, 1, 44 | ovmpt2 6796 | . . . 4 |
46 | 45 | eleq2d 2687 | . . 3 |
47 | fex 6490 | . . . . . 6 | |
48 | 37, 47 | mpan2 707 | . . . . 5 |
49 | 48 | adantr 481 | . . . 4 |
50 | feq1 6026 | . . . . 5 | |
51 | fveq1 6190 | . . . . . . 7 | |
52 | fveq1 6190 | . . . . . . . 8 | |
53 | fveq1 6190 | . . . . . . . 8 | |
54 | 52, 53 | oveq12d 6668 | . . . . . . 7 |
55 | 51, 54 | eqeq12d 2637 | . . . . . 6 |
56 | 55 | 2ralbidv 2989 | . . . . 5 |
57 | 50, 56 | anbi12d 747 | . . . 4 |
58 | 49, 57 | elab3 3358 | . . 3 |
59 | 46, 58 | syl6bb 276 | . 2 |
60 | 2, 59 | biadan2 674 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wcel 1990 cab 2608 wral 2912 cvv 3200 wsbc 3435 wf 5884 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 cgrp 17422 cghm 17657 |
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 ax-un 6949 |
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-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-pw 4160 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-ov 6653 df-oprab 6654 df-mpt2 6655 df-ghm 17658 |
This theorem is referenced by: isghm3 17661 ghmgrp1 17662 ghmgrp2 17663 ghmf 17664 ghmlin 17665 isghmd 17669 idghm 17675 ghmf1o 17690 islmhm2 19038 expghm 19844 mulgghm2 19845 pi1xfr 22855 pi1coghm 22861 rhmopp 29819 isrnghm 41892 |
Copyright terms: Public domain | W3C validator |