Theorem frmdup3 17404

Description: Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
frmdup3.m	|- M = (freeMnd ` I )
frmdup3.b	|- B = ( Base ` G )
frmdup3.u	|- U = (varFMnd ` I )

Assertion

Ref	Expression
frmdup3	|- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> E! m e. ( M MndHom G ) ( m o. U ) = A )

Distinct variable groups:   A, m   B, m   m, G   m, I   m, M   U, m   m, V

Proof of Theorem frmdup3

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frmdup3.m	. . 3 |- M = (freeMnd ` I )
2		frmdup3.b	. . 3 |- B = ( Base ` G )
3		eqid 2622	. . 3 |- ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) = ( x e. Word I |-> ( G gsumg ( A o. x ) ) )
4		simp1 1061	. . 3 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> G e. Mnd )
5		simp2 1062	. . 3 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> I e. V )
6		simp3 1063	. . 3 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> A : I --> B )
7	1, 2, 3, 4, 5, 6	frmdup1 17401	. 2 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) e. ( M MndHom G ) )
8	4	adantr 481	. . . . 5 |- ( ( ( G e. Mnd /\ I e. V /\ A : I --> B ) /\ y e. I ) -> G e. Mnd )
9	5	adantr 481	. . . . 5 |- ( ( ( G e. Mnd /\ I e. V /\ A : I --> B ) /\ y e. I ) -> I e. V )
10	6	adantr 481	. . . . 5 |- ( ( ( G e. Mnd /\ I e. V /\ A : I --> B ) /\ y e. I ) -> A : I --> B )
11		frmdup3.u	. . . . 5 |- U = (varFMnd ` I )
12		simpr 477	. . . . 5 |- ( ( ( G e. Mnd /\ I e. V /\ A : I --> B ) /\ y e. I ) -> y e. I )
13	1, 2, 3, 8, 9, 10, 11, 12	frmdup2 17402	. . . 4 |- ( ( ( G e. Mnd /\ I e. V /\ A : I --> B ) /\ y e. I ) -> ( ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) ` ( U ` y ) ) = ( A ` y ) )
14	13	mpteq2dva 4744	. . 3 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> ( y e. I |-> ( ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) ` ( U ` y ) ) ) = ( y e. I |-> ( A ` y ) ) )
15		eqid 2622	. . . . . 6 |- ( Base ` M ) = ( Base ` M )
16	15, 2	mhmf 17340	. . . . 5 |- ( ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) e. ( M MndHom G ) -> ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) : ( Base ` M ) --> B )
17	7, 16	syl 17	. . . 4 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) : ( Base ` M ) --> B )
18	11	vrmdf 17395	. . . . . 6 |- ( I e. V -> U : I -->Word I )
19	18	3ad2ant2 1083	. . . . 5 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> U : I -->Word I )
20	1, 15	frmdbas 17389	. . . . . . 7 |- ( I e. V -> ( Base ` M ) = Word I )
21	20	3ad2ant2 1083	. . . . . 6 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> ( Base ` M ) = Word I )
22	21	feq3d 6032	. . . . 5 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> ( U : I --> ( Base ` M ) <-> U : I -->Word I ) )
23	19, 22	mpbird 247	. . . 4 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> U : I --> ( Base ` M ) )
24		fcompt 6400	. . . 4 |- ( ( ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) : ( Base ` M ) --> B /\ U : I --> ( Base ` M ) ) -> ( ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) o. U ) = ( y e. I |-> ( ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) ` ( U ` y ) ) ) )
25	17, 23, 24	syl2anc 693	. . 3 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> ( ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) o. U ) = ( y e. I |-> ( ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) ` ( U ` y ) ) ) )
26	6	feqmptd 6249	. . 3 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> A = ( y e. I |-> ( A ` y ) ) )
27	14, 25, 26	3eqtr4d 2666	. 2 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> ( ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) o. U ) = A )
28	1, 2, 11	frmdup3lem 17403	. . . 4 |- ( ( ( G e. Mnd /\ I e. V /\ A : I --> B ) /\ ( m e. ( M MndHom G ) /\ ( m o. U ) = A ) ) -> m = ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) )
29	28	expr 643	. . 3 |- ( ( ( G e. Mnd /\ I e. V /\ A : I --> B ) /\ m e. ( M MndHom G ) ) -> ( ( m o. U ) = A -> m = ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) ) )
30	29	ralrimiva 2966	. 2 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> A. m e. ( M MndHom G ) ( ( m o. U ) = A -> m = ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) ) )
31		coeq1 5279	. . . 4 |- ( m = ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) -> ( m o. U ) = ( ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) o. U ) )
32	31	eqeq1d 2624	. . 3 |- ( m = ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) -> ( ( m o. U ) = A <-> ( ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) o. U ) = A ) )
33	32	eqreu 3398	. 2 |- ( ( ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) e. ( M MndHom G ) /\ ( ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) o. U ) = A /\ A. m e. ( M MndHom G ) ( ( m o. U ) = A -> m = ( x e. Word I |-> ( G gsumg ( A o. x ) ) ) ) ) -> E! m e. ( M MndHom G ) ( m o. U ) = A )
34	7, 27, 30, 33	syl3anc 1326	1 |- ( ( G e. Mnd /\ I e. V /\ A : I --> B ) -> E! m e. ( M MndHom G ) ( m o. U ) = A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E!wreu 2914   |-> cmpt 4729   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650  Word cword 13291   Basecbs 15857   gsum_g cgsu 16101   Mndcmnd 17294   MndHom cmhm 17333  freeMndcfrmd 17384  var_FMndcvrmd 17385

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-xnn0 11364  df-z 11378  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-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  df-vrmd 17387

This theorem is referenced by: (None)