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Mirrors > Home > MPE Home > Th. List > frgpupf | 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 |
---|---|
frgpupf | β’ (π β πΈ:πβΆπ΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgpup.e | . . . 4 β’ πΈ = ran (π β π β¦ β¨[π] βΌ , (π» Ξ£g (π β π))β©) | |
2 | frgpup.h | . . . . . 6 β’ (π β π» β Grp) | |
3 | 2 | grpmndd 18910 | . . . . 5 β’ (π β π» β Mnd) |
4 | frgpup.w | . . . . . . . 8 β’ π = ( I βWord (πΌ Γ 2o)) | |
5 | fviss 6980 | . . . . . . . 8 β’ ( I βWord (πΌ Γ 2o)) β Word (πΌ Γ 2o) | |
6 | 4, 5 | eqsstri 4016 | . . . . . . 7 β’ π β Word (πΌ Γ 2o) |
7 | 6 | sseli 3978 | . . . . . 6 β’ (π β π β π β Word (πΌ Γ 2o)) |
8 | frgpup.b | . . . . . . 7 β’ π΅ = (Baseβπ») | |
9 | frgpup.n | . . . . . . 7 β’ π = (invgβπ») | |
10 | frgpup.t | . . . . . . 7 β’ π = (π¦ β πΌ, π§ β 2o β¦ if(π§ = β , (πΉβπ¦), (πβ(πΉβπ¦)))) | |
11 | frgpup.i | . . . . . . 7 β’ (π β πΌ β π) | |
12 | frgpup.a | . . . . . . 7 β’ (π β πΉ:πΌβΆπ΅) | |
13 | 8, 9, 10, 2, 11, 12 | frgpuptf 19732 | . . . . . 6 β’ (π β π:(πΌ Γ 2o)βΆπ΅) |
14 | wrdco 14822 | . . . . . 6 β’ ((π β Word (πΌ Γ 2o) β§ π:(πΌ Γ 2o)βΆπ΅) β (π β π) β Word π΅) | |
15 | 7, 13, 14 | syl2anr 595 | . . . . 5 β’ ((π β§ π β π) β (π β π) β Word π΅) |
16 | 8 | gsumwcl 18798 | . . . . 5 β’ ((π» β Mnd β§ (π β π) β Word π΅) β (π» Ξ£g (π β π)) β π΅) |
17 | 3, 15, 16 | syl2an2r 683 | . . . 4 β’ ((π β§ π β π) β (π» Ξ£g (π β π)) β π΅) |
18 | frgpup.r | . . . . . 6 β’ βΌ = ( ~FG βπΌ) | |
19 | 4, 18 | efger 19680 | . . . . 5 β’ βΌ Er π |
20 | 19 | a1i 11 | . . . 4 β’ (π β βΌ Er π) |
21 | 4 | fvexi 6916 | . . . . 5 β’ π β V |
22 | 21 | a1i 11 | . . . 4 β’ (π β π β V) |
23 | coeq2 5865 | . . . . 5 β’ (π = β β (π β π) = (π β β)) | |
24 | 23 | oveq2d 7442 | . . . 4 β’ (π = β β (π» Ξ£g (π β π)) = (π» Ξ£g (π β β))) |
25 | 8, 9, 10, 2, 11, 12, 4, 18 | frgpuplem 19734 | . . . 4 β’ ((π β§ π βΌ β) β (π» Ξ£g (π β π)) = (π» Ξ£g (π β β))) |
26 | 1, 17, 20, 22, 24, 25 | qliftfund 8828 | . . 3 β’ (π β Fun πΈ) |
27 | 1, 17, 20, 22 | qliftf 8830 | . . 3 β’ (π β (Fun πΈ β πΈ:(π / βΌ )βΆπ΅)) |
28 | 26, 27 | mpbid 231 | . 2 β’ (π β πΈ:(π / βΌ )βΆπ΅) |
29 | frgpup.x | . . . 4 β’ π = (BaseβπΊ) | |
30 | frgpup.g | . . . . . . 7 β’ πΊ = (freeGrpβπΌ) | |
31 | eqid 2728 | . . . . . . 7 β’ (freeMndβ(πΌ Γ 2o)) = (freeMndβ(πΌ Γ 2o)) | |
32 | 30, 31, 18 | frgpval 19720 | . . . . . 6 β’ (πΌ β π β πΊ = ((freeMndβ(πΌ Γ 2o)) /s βΌ )) |
33 | 11, 32 | syl 17 | . . . . 5 β’ (π β πΊ = ((freeMndβ(πΌ Γ 2o)) /s βΌ )) |
34 | 2on 8507 | . . . . . . . . 9 β’ 2o β On | |
35 | xpexg 7758 | . . . . . . . . 9 β’ ((πΌ β π β§ 2o β On) β (πΌ Γ 2o) β V) | |
36 | 11, 34, 35 | sylancl 584 | . . . . . . . 8 β’ (π β (πΌ Γ 2o) β V) |
37 | wrdexg 14514 | . . . . . . . 8 β’ ((πΌ Γ 2o) β V β Word (πΌ Γ 2o) β V) | |
38 | fvi 6979 | . . . . . . . 8 β’ (Word (πΌ Γ 2o) β V β ( I βWord (πΌ Γ 2o)) = Word (πΌ Γ 2o)) | |
39 | 36, 37, 38 | 3syl 18 | . . . . . . 7 β’ (π β ( I βWord (πΌ Γ 2o)) = Word (πΌ Γ 2o)) |
40 | 4, 39 | eqtrid 2780 | . . . . . 6 β’ (π β π = Word (πΌ Γ 2o)) |
41 | eqid 2728 | . . . . . . . 8 β’ (Baseβ(freeMndβ(πΌ Γ 2o))) = (Baseβ(freeMndβ(πΌ Γ 2o))) | |
42 | 31, 41 | frmdbas 18811 | . . . . . . 7 β’ ((πΌ Γ 2o) β V β (Baseβ(freeMndβ(πΌ Γ 2o))) = Word (πΌ Γ 2o)) |
43 | 36, 42 | syl 17 | . . . . . 6 β’ (π β (Baseβ(freeMndβ(πΌ Γ 2o))) = Word (πΌ Γ 2o)) |
44 | 40, 43 | eqtr4d 2771 | . . . . 5 β’ (π β π = (Baseβ(freeMndβ(πΌ Γ 2o)))) |
45 | 18 | fvexi 6916 | . . . . . 6 β’ βΌ β V |
46 | 45 | a1i 11 | . . . . 5 β’ (π β βΌ β V) |
47 | fvexd 6917 | . . . . 5 β’ (π β (freeMndβ(πΌ Γ 2o)) β V) | |
48 | 33, 44, 46, 47 | qusbas 17534 | . . . 4 β’ (π β (π / βΌ ) = (BaseβπΊ)) |
49 | 29, 48 | eqtr4id 2787 | . . 3 β’ (π β π = (π / βΌ )) |
50 | 49 | feq2d 6713 | . 2 β’ (π β (πΈ:πβΆπ΅ β πΈ:(π / βΌ )βΆπ΅)) |
51 | 28, 50 | mpbird 256 | 1 β’ (π β πΈ:πβΆπ΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3473 β c0 4326 ifcif 4532 β¨cop 4638 β¦ cmpt 5235 I cid 5579 Γ cxp 5680 ran crn 5683 β ccom 5686 Oncon0 6374 Fun wfun 6547 βΆwf 6549 βcfv 6553 (class class class)co 7426 β cmpo 7428 2oc2o 8487 Er wer 8728 [cec 8729 / cqs 8730 Word cword 14504 Basecbs 17187 Ξ£g cgsu 17429 /s cqus 17494 Mndcmnd 18701 freeMndcfrmd 18806 Grpcgrp 18897 invgcminusg 18898 ~FG cefg 19668 freeGrpcfrgp 19669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-ec 8733 df-qs 8737 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-word 14505 df-concat 14561 df-s1 14586 df-substr 14631 df-pfx 14661 df-splice 14740 df-s2 14839 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-0g 17430 df-gsum 17431 df-imas 17497 df-qus 17498 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-frmd 18808 df-grp 18900 df-minusg 18901 df-efg 19671 df-frgp 19672 |
This theorem is referenced by: frgpupval 19736 frgpup1 19737 |
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