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Mirrors > Home > MPE Home > Th. List > ismhm | Structured version Visualization version Unicode version |
Description: Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.) |
Ref | Expression |
---|---|
ismhm.b | |
ismhm.c | |
ismhm.p | |
ismhm.q | |
ismhm.z | |
ismhm.y |
Ref | Expression |
---|---|
ismhm | MndHom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mhm 17335 | . . 3 MndHom | |
2 | 1 | elmpt2cl 6876 | . 2 MndHom |
3 | fveq2 6191 | . . . . . . . 8 | |
4 | ismhm.c | . . . . . . . 8 | |
5 | 3, 4 | syl6eqr 2674 | . . . . . . 7 |
6 | fveq2 6191 | . . . . . . . 8 | |
7 | ismhm.b | . . . . . . . 8 | |
8 | 6, 7 | syl6eqr 2674 | . . . . . . 7 |
9 | 5, 8 | oveqan12rd 6670 | . . . . . 6 |
10 | 8 | adantr 481 | . . . . . . . 8 |
11 | fveq2 6191 | . . . . . . . . . . . . 13 | |
12 | ismhm.p | . . . . . . . . . . . . 13 | |
13 | 11, 12 | syl6eqr 2674 | . . . . . . . . . . . 12 |
14 | 13 | oveqd 6667 | . . . . . . . . . . 11 |
15 | 14 | fveq2d 6195 | . . . . . . . . . 10 |
16 | fveq2 6191 | . . . . . . . . . . . 12 | |
17 | ismhm.q | . . . . . . . . . . . 12 | |
18 | 16, 17 | syl6eqr 2674 | . . . . . . . . . . 11 |
19 | 18 | oveqd 6667 | . . . . . . . . . 10 |
20 | 15, 19 | eqeqan12d 2638 | . . . . . . . . 9 |
21 | 10, 20 | raleqbidv 3152 | . . . . . . . 8 |
22 | 10, 21 | raleqbidv 3152 | . . . . . . 7 |
23 | fveq2 6191 | . . . . . . . . . 10 | |
24 | ismhm.z | . . . . . . . . . 10 | |
25 | 23, 24 | syl6eqr 2674 | . . . . . . . . 9 |
26 | 25 | fveq2d 6195 | . . . . . . . 8 |
27 | fveq2 6191 | . . . . . . . . 9 | |
28 | ismhm.y | . . . . . . . . 9 | |
29 | 27, 28 | syl6eqr 2674 | . . . . . . . 8 |
30 | 26, 29 | eqeqan12d 2638 | . . . . . . 7 |
31 | 22, 30 | anbi12d 747 | . . . . . 6 |
32 | 9, 31 | rabeqbidv 3195 | . . . . 5 |
33 | ovex 6678 | . . . . . 6 | |
34 | 33 | rabex 4813 | . . . . 5 |
35 | 32, 1, 34 | ovmpt2a 6791 | . . . 4 MndHom |
36 | 35 | eleq2d 2687 | . . 3 MndHom |
37 | fvex 6201 | . . . . . . 7 | |
38 | 4, 37 | eqeltri 2697 | . . . . . 6 |
39 | fvex 6201 | . . . . . . 7 | |
40 | 7, 39 | eqeltri 2697 | . . . . . 6 |
41 | 38, 40 | elmap 7886 | . . . . 5 |
42 | 41 | anbi1i 731 | . . . 4 |
43 | fveq1 6190 | . . . . . . . 8 | |
44 | fveq1 6190 | . . . . . . . . 9 | |
45 | fveq1 6190 | . . . . . . . . 9 | |
46 | 44, 45 | oveq12d 6668 | . . . . . . . 8 |
47 | 43, 46 | eqeq12d 2637 | . . . . . . 7 |
48 | 47 | 2ralbidv 2989 | . . . . . 6 |
49 | fveq1 6190 | . . . . . . 7 | |
50 | 49 | eqeq1d 2624 | . . . . . 6 |
51 | 48, 50 | anbi12d 747 | . . . . 5 |
52 | 51 | elrab 3363 | . . . 4 |
53 | 3anass 1042 | . . . 4 | |
54 | 42, 52, 53 | 3bitr4i 292 | . . 3 |
55 | 36, 54 | syl6bb 276 | . 2 MndHom |
56 | 2, 55 | biadan2 674 | 1 MndHom |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 crab 2916 cvv 3200 wf 5884 cfv 5888 (class class class)co 6650 cmap 7857 cbs 15857 cplusg 15941 c0g 16100 cmnd 17294 MndHom cmhm 17333 |
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-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-rab 2921 df-v 3202 df-sbc 3436 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-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-mhm 17335 |
This theorem is referenced by: mhmf 17340 mhmpropd 17341 mhmlin 17342 mhm0 17343 idmhm 17344 mhmf1o 17345 0mhm 17358 resmhm 17359 resmhm2 17360 resmhm2b 17361 mhmco 17362 prdspjmhm 17367 pwsdiagmhm 17369 pwsco1mhm 17370 pwsco2mhm 17371 frmdup1 17401 mhmfmhm 17538 ghmmhm 17670 frgpmhm 18178 mulgmhm 18233 srglmhm 18535 srgrmhm 18536 dfrhm2 18717 isrhm2d 18728 expmhm 19815 mat1mhm 20290 scmatmhm 20340 mat2pmatmhm 20538 pm2mpmhm 20625 dchrelbas3 24963 xrge0iifmhm 29985 esumcocn 30142 elmrsubrn 31417 deg1mhm 37785 ismhm0 41805 mhmismgmhm 41806 c0mhm 41910 |
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