Theorem frgpup3 18191

Description: Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)

Hypotheses

Ref	Expression
frgpup3.g	|- G = (freeGrp ` I )
frgpup3.b	|- B = ( Base ` H )
frgpup3.u	|- U = (varFGrp ` I )

Assertion

Ref	Expression
frgpup3	|- ( ( H e. Grp /\ I e. V /\ F : I --> B ) -> E! m e. ( G GrpHom H ) ( m o. U ) = F )

Distinct variable groups:   B, m   m, F   m, G   m, H   m, I   U, m   m, V

Proof of Theorem frgpup3

Dummy variables g k y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgpup3.b	. . 3 |- B = ( Base ` H )
2		eqid 2622	. . 3 |- ( invg ` H ) = ( invg ` H )
3		eqid 2622	. . 3 |- ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) = ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) )
4		simp1 1061	. . 3 |- ( ( H e. Grp /\ I e. V /\ F : I --> B ) -> H e. Grp )
5		simp2 1062	. . 3 |- ( ( H e. Grp /\ I e. V /\ F : I --> B ) -> I e. V )
6		simp3 1063	. . 3 |- ( ( H e. Grp /\ I e. V /\ F : I --> B ) -> F : I --> B )
7		eqid 2622	. . 3 |- ( _I ` Word ( I X. 2o ) ) = ( _I ` Word ( I X. 2o ) )
8		eqid 2622	. . 3 |- ( ~FG ` I ) = ( ~FG ` I )
9		frgpup3.g	. . 3 |- G = (freeGrp ` I )
10		eqid 2622	. . 3 |- ( Base ` G ) = ( Base ` G )
11		eqid 2622	. . 3 |- ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) = ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. )
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	frgpup1 18188	. 2 |- ( ( H e. Grp /\ I e. V /\ F : I --> B ) -> ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) e. ( G GrpHom H ) )
13	4	adantr 481	. . . . 5 |- ( ( ( H e. Grp /\ I e. V /\ F : I --> B ) /\ k e. I ) -> H e. Grp )
14	5	adantr 481	. . . . 5 |- ( ( ( H e. Grp /\ I e. V /\ F : I --> B ) /\ k e. I ) -> I e. V )
15	6	adantr 481	. . . . 5 |- ( ( ( H e. Grp /\ I e. V /\ F : I --> B ) /\ k e. I ) -> F : I --> B )
16		frgpup3.u	. . . . 5 |- U = (varFGrp ` I )
17		simpr 477	. . . . 5 |- ( ( ( H e. Grp /\ I e. V /\ F : I --> B ) /\ k e. I ) -> k e. I )
18	1, 2, 3, 13, 14, 15, 7, 8, 9, 10, 11, 16, 17	frgpup2 18189	. . . 4 |- ( ( ( H e. Grp /\ I e. V /\ F : I --> B ) /\ k e. I ) -> ( ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) ` ( U ` k ) ) = ( F ` k ) )
19	18	mpteq2dva 4744	. . 3 |- ( ( H e. Grp /\ I e. V /\ F : I --> B ) -> ( k e. I |-> ( ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) ` ( U ` k ) ) ) = ( k e. I |-> ( F ` k ) ) )
20	10, 1	ghmf 17664	. . . . 5 |- ( ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) e. ( G GrpHom H ) -> ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) : ( Base ` G ) --> B )
21	12, 20	syl 17	. . . 4 |- ( ( H e. Grp /\ I e. V /\ F : I --> B ) -> ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) : ( Base ` G ) --> B )
22	8, 16, 9, 10	vrgpf 18181	. . . . 5 |- ( I e. V -> U : I --> ( Base ` G ) )
23	5, 22	syl 17	. . . 4 |- ( ( H e. Grp /\ I e. V /\ F : I --> B ) -> U : I --> ( Base ` G ) )
24		fcompt 6400	. . . 4 |- ( ( ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) : ( Base ` G ) --> B /\ U : I --> ( Base ` G ) ) -> ( ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) o. U ) = ( k e. I |-> ( ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) ` ( U ` k ) ) ) )
25	21, 23, 24	syl2anc 693	. . 3 |- ( ( H e. Grp /\ I e. V /\ F : I --> B ) -> ( ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) o. U ) = ( k e. I |-> ( ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) ` ( U ` k ) ) ) )
26	6	feqmptd 6249	. . 3 |- ( ( H e. Grp /\ I e. V /\ F : I --> B ) -> F = ( k e. I |-> ( F ` k ) ) )
27	19, 25, 26	3eqtr4d 2666	. 2 |- ( ( H e. Grp /\ I e. V /\ F : I --> B ) -> ( ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) o. U ) = F )
28	4	adantr 481	. . . . 5 |- ( ( ( H e. Grp /\ I e. V /\ F : I --> B ) /\ ( m e. ( G GrpHom H ) /\ ( m o. U ) = F ) ) -> H e. Grp )
29	5	adantr 481	. . . . 5 |- ( ( ( H e. Grp /\ I e. V /\ F : I --> B ) /\ ( m e. ( G GrpHom H ) /\ ( m o. U ) = F ) ) -> I e. V )
30	6	adantr 481	. . . . 5 |- ( ( ( H e. Grp /\ I e. V /\ F : I --> B ) /\ ( m e. ( G GrpHom H ) /\ ( m o. U ) = F ) ) -> F : I --> B )
31		simprl 794	. . . . 5 |- ( ( ( H e. Grp /\ I e. V /\ F : I --> B ) /\ ( m e. ( G GrpHom H ) /\ ( m o. U ) = F ) ) -> m e. ( G GrpHom H ) )
32		simprr 796	. . . . 5 |- ( ( ( H e. Grp /\ I e. V /\ F : I --> B ) /\ ( m e. ( G GrpHom H ) /\ ( m o. U ) = F ) ) -> ( m o. U ) = F )
33	1, 2, 3, 28, 29, 30, 7, 8, 9, 10, 11, 16, 31, 32	frgpup3lem 18190	. . . 4 |- ( ( ( H e. Grp /\ I e. V /\ F : I --> B ) /\ ( m e. ( G GrpHom H ) /\ ( m o. U ) = F ) ) -> m = ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) )
34	33	expr 643	. . 3 |- ( ( ( H e. Grp /\ I e. V /\ F : I --> B ) /\ m e. ( G GrpHom H ) ) -> ( ( m o. U ) = F -> m = ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) ) )
35	34	ralrimiva 2966	. 2 |- ( ( H e. Grp /\ I e. V /\ F : I --> B ) -> A. m e. ( G GrpHom H ) ( ( m o. U ) = F -> m = ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) ) )
36		coeq1 5279	. . . 4 |- ( m = ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) -> ( m o. U ) = ( ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) o. U ) )
37	36	eqeq1d 2624	. . 3 |- ( m = ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) -> ( ( m o. U ) = F <-> ( ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) o. U ) = F ) )
38	37	eqreu 3398	. 2 |- ( ( ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) e. ( G GrpHom H ) /\ ( ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) o. U ) = F /\ A. m e. ( G GrpHom H ) ( ( m o. U ) = F -> m = ran ( g e. ( _I ` Word ( I X. 2o ) ) |-> <. [ g ] ( ~FG ` I ) , ( H gsumg ( ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( ( invg ` H ) ` ( F ` y ) ) ) ) o. g ) ) >. ) ) ) -> E! m e. ( G GrpHom H ) ( m o. U ) = F )
39	12, 27, 35, 38	syl3anc 1326	1 |- ( ( H e. Grp /\ I e. V /\ F : I --> B ) -> E! m e. ( G GrpHom H ) ( m o. U ) = F )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E!wreu 2914   (/)c0 3915   ifcif 4086   <.cop 4183   |-> cmpt 4729   _I cid 5023   X. cxp 5112   ran crn 5115   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   2oc2o 7554   [cec 7740  Word cword 13291   Basecbs 15857   gsum_g cgsu 16101   Grpcgrp 17422   invgcminusg 17423   GrpHom cghm 17657   ~_FG cefg 18119  freeGrpcfrgp 18120  var_FGrpcvrgp 18121

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-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  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-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  df-s2 13593  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-0g 16102  df-gsum 16103  df-imas 16168  df-qus 16169  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-frmd 17386  df-vrmd 17387  df-grp 17425  df-minusg 17426  df-ghm 17658  df-efg 18122  df-frgp 18123  df-vrgp 18124

This theorem is referenced by:  0frgp  18192  frgpcyg  19922