Theorem frgpuptf 18183

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.)

Hypotheses

Ref	Expression
frgpup.b	|- B = ( Base ` H )
frgpup.n	|- N = ( invg ` H )
frgpup.t	|- T = ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) )
frgpup.h	|- ( ph -> H e. Grp )
frgpup.i	|- ( ph -> I e. V )
frgpup.a	|- ( ph -> F : I --> B )

Assertion

Ref	Expression
frgpuptf	|- ( ph -> T : ( I X. 2o ) --> B )

Distinct variable groups:   y, z, F   y, N, z   y, B, z   ph, y, z   y, I, z

Allowed substitution hints:   T( y, z)   H( y, z)   V( y, z)

Proof of Theorem frgpuptf

Step	Hyp	Ref	Expression
1		frgpup.a	. . . . . 6 |- ( ph -> F : I --> B )
2	1	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ y e. I ) -> ( F ` y ) e. B )
3	2	adantrr 753	. . . 4 |- ( ( ph /\ ( y e. I /\ z e. 2o ) ) -> ( F ` y ) e. B )
4		frgpup.h	. . . . . 6 |- ( ph -> H e. Grp )
5	4	adantr 481	. . . . 5 |- ( ( ph /\ ( y e. I /\ z e. 2o ) ) -> H e. Grp )
6		frgpup.b	. . . . . 6 |- B = ( Base ` H )
7		frgpup.n	. . . . . 6 |- N = ( invg ` H )
8	6, 7	grpinvcl 17467	. . . . 5 |- ( ( H e. Grp /\ ( F ` y ) e. B ) -> ( N ` ( F ` y ) ) e. B )
9	5, 3, 8	syl2anc 693	. . . 4 |- ( ( ph /\ ( y e. I /\ z e. 2o ) ) -> ( N ` ( F ` y ) ) e. B )
10	3, 9	ifcld 4131	. . 3 |- ( ( ph /\ ( y e. I /\ z e. 2o ) ) -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) e. B )
11	10	ralrimivva 2971	. 2 |- ( ph -> A. y e. I A. z e. 2o if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) e. B )
12		frgpup.t	. . 3 |- T = ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) )
13	12	fmpt2 7237	. 2 |- ( A. y e. I A. z e. 2o if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) e. B <-> T : ( I X. 2o ) --> B )
14	11, 13	sylib 208	1 |- ( ph -> T : ( I X. 2o ) --> B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915   ifcif 4086   X. cxp 5112   -->wf 5884   ` cfv 5888   |-> cmpt2 6652   2oc2o 7554   Basecbs 15857   Grpcgrp 17422   invgcminusg 17423

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

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-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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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-1st 7168  df-2nd 7169  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426

This theorem is referenced by:  frgpuplem  18185  frgpupf  18186  frgpup1  18188  frgpup2  18189  frgpup3lem  18190