![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > frgpupval | Structured version Visualization version GIF 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, 2-Oct-2015.) |
Ref | Expression |
---|---|
frgpup.b | β’ π΅ = (Baseβπ») |
frgpup.n | β’ π = (invgβπ») |
frgpup.t | β’ π = (π¦ β πΌ, π§ β 2o β¦ if(π§ = β , (πΉβπ¦), (πβ(πΉβπ¦)))) |
frgpup.h | β’ (π β π» β Grp) |
frgpup.i | β’ (π β πΌ β π) |
frgpup.a | β’ (π β πΉ:πΌβΆπ΅) |
frgpup.w | β’ π = ( I βWord (πΌ Γ 2o)) |
frgpup.r | β’ βΌ = ( ~FG βπΌ) |
frgpup.g | β’ πΊ = (freeGrpβπΌ) |
frgpup.x | β’ π = (BaseβπΊ) |
frgpup.e | β’ πΈ = ran (π β π β¦ β¨[π] βΌ , (π» Ξ£g (π β π))β©) |
Ref | Expression |
---|---|
frgpupval | β’ ((π β§ π΄ β π) β (πΈβ[π΄] βΌ ) = (π» Ξ£g (π β π΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgpup.e | . 2 β’ πΈ = ran (π β π β¦ β¨[π] βΌ , (π» Ξ£g (π β π))β©) | |
2 | ovexd 7396 | . 2 β’ ((π β§ π β π) β (π» Ξ£g (π β π)) β V) | |
3 | frgpup.w | . . . 4 β’ π = ( I βWord (πΌ Γ 2o)) | |
4 | frgpup.r | . . . 4 β’ βΌ = ( ~FG βπΌ) | |
5 | 3, 4 | efger 19508 | . . 3 β’ βΌ Er π |
6 | 5 | a1i 11 | . 2 β’ (π β βΌ Er π) |
7 | 3 | fvexi 6860 | . . 3 β’ π β V |
8 | 7 | a1i 11 | . 2 β’ (π β π β V) |
9 | coeq2 5818 | . . 3 β’ (π = π΄ β (π β π) = (π β π΄)) | |
10 | 9 | oveq2d 7377 | . 2 β’ (π = π΄ β (π» Ξ£g (π β π)) = (π» Ξ£g (π β π΄))) |
11 | frgpup.b | . . . 4 β’ π΅ = (Baseβπ») | |
12 | frgpup.n | . . . 4 β’ π = (invgβπ») | |
13 | frgpup.t | . . . 4 β’ π = (π¦ β πΌ, π§ β 2o β¦ if(π§ = β , (πΉβπ¦), (πβ(πΉβπ¦)))) | |
14 | frgpup.h | . . . 4 β’ (π β π» β Grp) | |
15 | frgpup.i | . . . 4 β’ (π β πΌ β π) | |
16 | frgpup.a | . . . 4 β’ (π β πΉ:πΌβΆπ΅) | |
17 | frgpup.g | . . . 4 β’ πΊ = (freeGrpβπΌ) | |
18 | frgpup.x | . . . 4 β’ π = (BaseβπΊ) | |
19 | 11, 12, 13, 14, 15, 16, 3, 4, 17, 18, 1 | frgpupf 19563 | . . 3 β’ (π β πΈ:πβΆπ΅) |
20 | 19 | ffund 6676 | . 2 β’ (π β Fun πΈ) |
21 | 1, 2, 6, 8, 10, 20 | qliftval 8751 | 1 β’ ((π β§ π΄ β π) β (πΈβ[π΄] βΌ ) = (π» Ξ£g (π β π΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3447 β c0 4286 ifcif 4490 β¨cop 4596 β¦ cmpt 5192 I cid 5534 Γ cxp 5635 ran crn 5638 β ccom 5641 βΆwf 6496 βcfv 6500 (class class class)co 7361 β cmpo 7363 2oc2o 8410 Er wer 8651 [cec 8652 Word cword 14411 Basecbs 17091 Ξ£g cgsu 17330 Grpcgrp 18756 invgcminusg 18757 ~FG cefg 19496 freeGrpcfrgp 19497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-ot 4599 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-ec 8656 df-qs 8660 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-inf 9387 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-fzo 13577 df-seq 13916 df-hash 14240 df-word 14412 df-concat 14468 df-s1 14493 df-substr 14538 df-pfx 14568 df-splice 14647 df-s2 14746 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-0g 17331 df-gsum 17332 df-imas 17398 df-qus 17399 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-frmd 18667 df-grp 18759 df-minusg 18760 df-efg 19499 df-frgp 19500 |
This theorem is referenced by: frgpup1 19565 frgpup2 19566 frgpup3lem 19567 |
Copyright terms: Public domain | W3C validator |