Theorem efgtf 18135

Description: Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)

Hypotheses

Ref	Expression
efgval.w	|- W = ( _I ` Word ( I X. 2o ) )
efgval.r	|- .~ = ( ~FG ` I )
efgval2.m	|- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
efgval2.t	|- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )

Assertion

Ref	Expression
efgtf	|- ( X e. W -> ( ( T ` X ) = ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) /\ ( T ` X ) : ( ( 0 ... ( # ` X ) ) X. ( I X. 2o ) ) --> W ) )

Distinct variable groups:   a, b, y, z   v, n, w, y, z, a   M, a   n, b, v, w, M   T, a, b   X, a, b   W, a, b, n, v, w, y, z   .~ , a, b, y, z   I, a, b, n, v, w, y, z

Allowed substitution hints:   .~ ( w, v, n)   T( y, z, w, v, n)   M( y, z)   X( y, z, w, v, n)

Proof of Theorem efgtf

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . . . . . . 10 |- W = ( _I ` Word ( I X. 2o ) )
2		fviss 6256	. . . . . . . . . 10 |- ( _I ` Word ( I X. 2o ) ) C_ Word ( I X. 2o )
3	1, 2	eqsstri 3635	. . . . . . . . 9 |- W C_ Word ( I X. 2o )
4		simpl 473	. . . . . . . . 9 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> X e. W )
5	3, 4	sseldi 3601	. . . . . . . 8 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> X e. Word ( I X. 2o ) )
6		simprr 796	. . . . . . . . 9 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> b e. ( I X. 2o ) )
7		efgval2.m	. . . . . . . . . . . 12 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
8	7	efgmf 18126	. . . . . . . . . . 11 |- M : ( I X. 2o ) --> ( I X. 2o )
9	8	ffvelrni 6358	. . . . . . . . . 10 |- ( b e. ( I X. 2o ) -> ( M ` b ) e. ( I X. 2o ) )
10	9	ad2antll 765	. . . . . . . . 9 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> ( M ` b ) e. ( I X. 2o ) )
11	6, 10	s2cld 13616	. . . . . . . 8 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> <" b ( M ` b ) "> e. Word ( I X. 2o ) )
12		splcl 13503	. . . . . . . 8 |- ( ( X e. Word ( I X. 2o ) /\ <" b ( M ` b ) "> e. Word ( I X. 2o ) ) -> ( X splice <. a , a , <" b ( M ` b ) "> >. ) e. Word ( I X. 2o ) )
13	5, 11, 12	syl2anc 693	. . . . . . 7 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> ( X splice <. a , a , <" b ( M ` b ) "> >. ) e. Word ( I X. 2o ) )
14	1	efgrcl 18128	. . . . . . . . 9 |- ( X e. W -> ( I e. _V /\ W = Word ( I X. 2o ) ) )
15	14	simprd 479	. . . . . . . 8 |- ( X e. W -> W = Word ( I X. 2o ) )
16	15	adantr 481	. . . . . . 7 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> W = Word ( I X. 2o ) )
17	13, 16	eleqtrrd 2704	. . . . . 6 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> ( X splice <. a , a , <" b ( M ` b ) "> >. ) e. W )
18	17	ralrimivva 2971	. . . . 5 |- ( X e. W -> A. a e. ( 0 ... ( # ` X ) ) A. b e. ( I X. 2o ) ( X splice <. a , a , <" b ( M ` b ) "> >. ) e. W )
19		eqid 2622	. . . . . 6 |- ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) = ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) )
20	19	fmpt2 7237	. . . . 5 |- ( A. a e. ( 0 ... ( # ` X ) ) A. b e. ( I X. 2o ) ( X splice <. a , a , <" b ( M ` b ) "> >. ) e. W <-> ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) : ( ( 0 ... ( # ` X ) ) X. ( I X. 2o ) ) --> W )
21	18, 20	sylib 208	. . . 4 |- ( X e. W -> ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) : ( ( 0 ... ( # ` X ) ) X. ( I X. 2o ) ) --> W )
22		ovex 6678	. . . . 5 |- ( 0 ... ( # ` X ) ) e. _V
23	14	simpld 475	. . . . . 6 |- ( X e. W -> I e. _V )
24		2on 7568	. . . . . 6 |- 2o e. On
25		xpexg 6960	. . . . . 6 |- ( ( I e. _V /\ 2o e. On ) -> ( I X. 2o ) e. _V )
26	23, 24, 25	sylancl 694	. . . . 5 |- ( X e. W -> ( I X. 2o ) e. _V )
27		xpexg 6960	. . . . 5 |- ( ( ( 0 ... ( # ` X ) ) e. _V /\ ( I X. 2o ) e. _V ) -> ( ( 0 ... ( # ` X ) ) X. ( I X. 2o ) ) e. _V )
28	22, 26, 27	sylancr 695	. . . 4 |- ( X e. W -> ( ( 0 ... ( # ` X ) ) X. ( I X. 2o ) ) e. _V )
29		fvex 6201	. . . . . 6 |- ( _I ` Word ( I X. 2o ) ) e. _V
30	1, 29	eqeltri 2697	. . . . 5 |- W e. _V
31	30	a1i 11	. . . 4 |- ( X e. W -> W e. _V )
32		fex2 7121	. . . 4 |- ( ( ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) : ( ( 0 ... ( # ` X ) ) X. ( I X. 2o ) ) --> W /\ ( ( 0 ... ( # ` X ) ) X. ( I X. 2o ) ) e. _V /\ W e. _V ) -> ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) e. _V )
33	21, 28, 31, 32	syl3anc 1326	. . 3 |- ( X e. W -> ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) e. _V )
34		fveq2 6191	. . . . . 6 |- ( u = X -> ( # ` u ) = ( # ` X ) )
35	34	oveq2d 6666	. . . . 5 |- ( u = X -> ( 0 ... ( # ` u ) ) = ( 0 ... ( # ` X ) ) )
36		eqidd 2623	. . . . 5 |- ( u = X -> ( I X. 2o ) = ( I X. 2o ) )
37		oveq1 6657	. . . . 5 |- ( u = X -> ( u splice <. a , a , <" b ( M ` b ) "> >. ) = ( X splice <. a , a , <" b ( M ` b ) "> >. ) )
38	35, 36, 37	mpt2eq123dv 6717	. . . 4 |- ( u = X -> ( a e. ( 0 ... ( # ` u ) ) , b e. ( I X. 2o ) |-> ( u splice <. a , a , <" b ( M ` b ) "> >. ) ) = ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) )
39		efgval2.t	. . . . 5 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
40		oteq1 4411	. . . . . . . . . 10 |- ( n = a -> <. n , n , <" w ( M ` w ) "> >. = <. a , n , <" w ( M ` w ) "> >. )
41		oteq2 4412	. . . . . . . . . 10 |- ( n = a -> <. a , n , <" w ( M ` w ) "> >. = <. a , a , <" w ( M ` w ) "> >. )
42	40, 41	eqtrd 2656	. . . . . . . . 9 |- ( n = a -> <. n , n , <" w ( M ` w ) "> >. = <. a , a , <" w ( M ` w ) "> >. )
43	42	oveq2d 6666	. . . . . . . 8 |- ( n = a -> ( v splice <. n , n , <" w ( M ` w ) "> >. ) = ( v splice <. a , a , <" w ( M ` w ) "> >. ) )
44		id 22	. . . . . . . . . . 11 |- ( w = b -> w = b )
45		fveq2 6191	. . . . . . . . . . 11 |- ( w = b -> ( M ` w ) = ( M ` b ) )
46	44, 45	s2eqd 13608	. . . . . . . . . 10 |- ( w = b -> <" w ( M ` w ) "> = <" b ( M ` b ) "> )
47	46	oteq3d 4416	. . . . . . . . 9 |- ( w = b -> <. a , a , <" w ( M ` w ) "> >. = <. a , a , <" b ( M ` b ) "> >. )
48	47	oveq2d 6666	. . . . . . . 8 |- ( w = b -> ( v splice <. a , a , <" w ( M ` w ) "> >. ) = ( v splice <. a , a , <" b ( M ` b ) "> >. ) )
49	43, 48	cbvmpt2v 6735	. . . . . . 7 |- ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) = ( a e. ( 0 ... ( # ` v ) ) , b e. ( I X. 2o ) |-> ( v splice <. a , a , <" b ( M ` b ) "> >. ) )
50		fveq2 6191	. . . . . . . . 9 |- ( v = u -> ( # ` v ) = ( # ` u ) )
51	50	oveq2d 6666	. . . . . . . 8 |- ( v = u -> ( 0 ... ( # ` v ) ) = ( 0 ... ( # ` u ) ) )
52		eqidd 2623	. . . . . . . 8 |- ( v = u -> ( I X. 2o ) = ( I X. 2o ) )
53		oveq1 6657	. . . . . . . 8 |- ( v = u -> ( v splice <. a , a , <" b ( M ` b ) "> >. ) = ( u splice <. a , a , <" b ( M ` b ) "> >. ) )
54	51, 52, 53	mpt2eq123dv 6717	. . . . . . 7 |- ( v = u -> ( a e. ( 0 ... ( # ` v ) ) , b e. ( I X. 2o ) |-> ( v splice <. a , a , <" b ( M ` b ) "> >. ) ) = ( a e. ( 0 ... ( # ` u ) ) , b e. ( I X. 2o ) |-> ( u splice <. a , a , <" b ( M ` b ) "> >. ) ) )
55	49, 54	syl5eq 2668	. . . . . 6 |- ( v = u -> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) = ( a e. ( 0 ... ( # ` u ) ) , b e. ( I X. 2o ) |-> ( u splice <. a , a , <" b ( M ` b ) "> >. ) ) )
56	55	cbvmptv 4750	. . . . 5 |- ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) ) = ( u e. W |-> ( a e. ( 0 ... ( # ` u ) ) , b e. ( I X. 2o ) |-> ( u splice <. a , a , <" b ( M ` b ) "> >. ) ) )
57	39, 56	eqtri 2644	. . . 4 |- T = ( u e. W |-> ( a e. ( 0 ... ( # ` u ) ) , b e. ( I X. 2o ) |-> ( u splice <. a , a , <" b ( M ` b ) "> >. ) ) )
58	38, 57	fvmptg 6280	. . 3 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) e. _V ) -> ( T ` X ) = ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) )
59	33, 58	mpdan 702	. 2 |- ( X e. W -> ( T ` X ) = ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) )
60	59	feq1d 6030	. . 3 |- ( X e. W -> ( ( T ` X ) : ( ( 0 ... ( # ` X ) ) X. ( I X. 2o ) ) --> W <-> ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) : ( ( 0 ... ( # ` X ) ) X. ( I X. 2o ) ) --> W ) )
61	21, 60	mpbird 247	. 2 |- ( X e. W -> ( T ` X ) : ( ( 0 ... ( # ` X ) ) X. ( I X. 2o ) ) --> W )
62	59, 61	jca 554	1 |- ( X e. W -> ( ( T ` X ) = ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) /\ ( T ` X ) : ( ( 0 ... ( # ` X ) ) X. ( I X. 2o ) ) --> W ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   <.cop 4183   <.cotp 4185   |-> cmpt 4729   _I cid 5023   X. cxp 5112   Oncon0 5723   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   0cc0 9936   ...cfz 12326   #chash 13117  Word cword 13291   splice csplice 13296   <"cs2 13586   ~_FG cefg 18119

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-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-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-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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-s2 13593

This theorem is referenced by:  efgtval  18136  efgval2  18137  efgtlen  18139  efginvrel2  18140  efgsp1  18150  efgredleme  18156  efgredlem  18160  efgrelexlemb  18163  efgcpbllemb  18168  frgpnabllem1  18276