Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > frmdup1 | Structured version Visualization version Unicode version |
Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) |
Ref | Expression |
---|---|
frmdup.m | freeMnd |
frmdup.b | |
frmdup.e | Word g |
frmdup.g | |
frmdup.i | |
frmdup.a |
Ref | Expression |
---|---|
frmdup1 | MndHom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frmdup.i | . . . 4 | |
2 | frmdup.m | . . . . 5 freeMnd | |
3 | 2 | frmdmnd 17396 | . . . 4 |
4 | 1, 3 | syl 17 | . . 3 |
5 | frmdup.g | . . 3 | |
6 | 4, 5 | jca 554 | . 2 |
7 | 5 | adantr 481 | . . . . . 6 Word |
8 | simpr 477 | . . . . . . 7 Word Word | |
9 | frmdup.a | . . . . . . . 8 | |
10 | 9 | adantr 481 | . . . . . . 7 Word |
11 | wrdco 13577 | . . . . . . 7 Word Word | |
12 | 8, 10, 11 | syl2anc 693 | . . . . . 6 Word Word |
13 | frmdup.b | . . . . . . 7 | |
14 | 13 | gsumwcl 17377 | . . . . . 6 Word g |
15 | 7, 12, 14 | syl2anc 693 | . . . . 5 Word g |
16 | frmdup.e | . . . . 5 Word g | |
17 | 15, 16 | fmptd 6385 | . . . 4 Word |
18 | eqid 2622 | . . . . . . 7 | |
19 | 2, 18 | frmdbas 17389 | . . . . . 6 Word |
20 | 1, 19 | syl 17 | . . . . 5 Word |
21 | 20 | feq2d 6031 | . . . 4 Word |
22 | 17, 21 | mpbird 247 | . . 3 |
23 | 2, 18 | frmdelbas 17390 | . . . . . . . . 9 Word |
24 | 23 | ad2antrl 764 | . . . . . . . 8 Word |
25 | 2, 18 | frmdelbas 17390 | . . . . . . . . 9 Word |
26 | 25 | ad2antll 765 | . . . . . . . 8 Word |
27 | 9 | adantr 481 | . . . . . . . 8 |
28 | ccatco 13581 | . . . . . . . 8 Word Word ++ ++ | |
29 | 24, 26, 27, 28 | syl3anc 1326 | . . . . . . 7 ++ ++ |
30 | 29 | oveq2d 6666 | . . . . . 6 g ++ g ++ |
31 | 5 | adantr 481 | . . . . . . 7 |
32 | wrdco 13577 | . . . . . . . 8 Word Word | |
33 | 24, 27, 32 | syl2anc 693 | . . . . . . 7 Word |
34 | wrdco 13577 | . . . . . . . 8 Word Word | |
35 | 26, 27, 34 | syl2anc 693 | . . . . . . 7 Word |
36 | eqid 2622 | . . . . . . . 8 | |
37 | 13, 36 | gsumccat 17378 | . . . . . . 7 Word Word g ++ g g |
38 | 31, 33, 35, 37 | syl3anc 1326 | . . . . . 6 g ++ g g |
39 | 30, 38 | eqtrd 2656 | . . . . 5 g ++ g g |
40 | eqid 2622 | . . . . . . . . 9 | |
41 | 2, 18, 40 | frmdadd 17392 | . . . . . . . 8 ++ |
42 | 41 | adantl 482 | . . . . . . 7 ++ |
43 | 42 | fveq2d 6195 | . . . . . 6 ++ |
44 | ccatcl 13359 | . . . . . . . 8 Word Word ++ Word | |
45 | 24, 26, 44 | syl2anc 693 | . . . . . . 7 ++ Word |
46 | coeq2 5280 | . . . . . . . . 9 ++ ++ | |
47 | 46 | oveq2d 6666 | . . . . . . . 8 ++ g g ++ |
48 | ovex 6678 | . . . . . . . 8 g | |
49 | 47, 16, 48 | fvmpt3i 6287 | . . . . . . 7 ++ Word ++ g ++ |
50 | 45, 49 | syl 17 | . . . . . 6 ++ g ++ |
51 | 43, 50 | eqtrd 2656 | . . . . 5 g ++ |
52 | coeq2 5280 | . . . . . . . . 9 | |
53 | 52 | oveq2d 6666 | . . . . . . . 8 g g |
54 | 53, 16, 48 | fvmpt3i 6287 | . . . . . . 7 Word g |
55 | coeq2 5280 | . . . . . . . . 9 | |
56 | 55 | oveq2d 6666 | . . . . . . . 8 g g |
57 | 56, 16, 48 | fvmpt3i 6287 | . . . . . . 7 Word g |
58 | 54, 57 | oveqan12d 6669 | . . . . . 6 Word Word g g |
59 | 24, 26, 58 | syl2anc 693 | . . . . 5 g g |
60 | 39, 51, 59 | 3eqtr4d 2666 | . . . 4 |
61 | 60 | ralrimivva 2971 | . . 3 |
62 | wrd0 13330 | . . . 4 Word | |
63 | coeq2 5280 | . . . . . . . 8 | |
64 | co02 5649 | . . . . . . . 8 | |
65 | 63, 64 | syl6eq 2672 | . . . . . . 7 |
66 | 65 | oveq2d 6666 | . . . . . 6 g g |
67 | eqid 2622 | . . . . . . 7 | |
68 | 67 | gsum0 17278 | . . . . . 6 g |
69 | 66, 68 | syl6eq 2672 | . . . . 5 g |
70 | 69, 16, 48 | fvmpt3i 6287 | . . . 4 Word |
71 | 62, 70 | mp1i 13 | . . 3 |
72 | 22, 61, 71 | 3jca 1242 | . 2 |
73 | 2 | frmd0 17397 | . . 3 |
74 | 18, 13, 40, 36, 73, 67 | ismhm 17337 | . 2 MndHom |
75 | 6, 72, 74 | sylanbrc 698 | 1 MndHom |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 c0 3915 cmpt 4729 ccom 5118 wf 5884 cfv 5888 (class class class)co 6650 Word cword 13291 ++ cconcat 13293 cbs 15857 cplusg 15941 c0g 16100 g cgsu 16101 cmnd 17294 MndHom cmhm 17333 freeMndcfrmd 17384 |
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 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-nel 2898 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-word 13299 df-concat 13301 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-frmd 17386 |
This theorem is referenced by: frmdup3 17404 |
Copyright terms: Public domain | W3C validator |