Theorem efgtlen 18139

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
efgtlen	|- ( ( X e. W /\ A e. ran ( T ` X ) ) -> ( # ` A ) = ( ( # ` X ) + 2 ) )

Distinct variable groups:   y, z   v, n, w, y, z   n, M, v, w   n, W, v, w, y, z   y, .~ , z   n, I, v, w, y, z

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

Proof of Theorem efgtlen

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . . . . 8 |- W = ( _I ` Word ( I X. 2o ) )
2		efgval.r	. . . . . . . 8 |- .~ = ( ~FG ` I )
3		efgval2.m	. . . . . . . 8 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
4		efgval2.t	. . . . . . . 8 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
5	1, 2, 3, 4	efgtf 18135	. . . . . . 7 |- ( 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 ) )
6	5	simpld 475	. . . . . 6 |- ( X e. W -> ( T ` X ) = ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) )
7	6	rneqd 5353	. . . . 5 |- ( X e. W -> ran ( T ` X ) = ran ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) )
8	7	eleq2d 2687	. . . 4 |- ( X e. W -> ( A e. ran ( T ` X ) <-> A e. ran ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) ) )
9		eqid 2622	. . . . 5 |- ( 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 ) "> >. ) )
10		ovex 6678	. . . . 5 |- ( X splice <. a , a , <" b ( M ` b ) "> >. ) e. _V
11	9, 10	elrnmpt2 6773	. . . 4 |- ( A e. ran ( a e. ( 0 ... ( # ` X ) ) , b e. ( I X. 2o ) |-> ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) <-> E. a e. ( 0 ... ( # ` X ) ) E. b e. ( I X. 2o ) A = ( X splice <. a , a , <" b ( M ` b ) "> >. ) )
12	8, 11	syl6bb 276	. . 3 |- ( X e. W -> ( A e. ran ( T ` X ) <-> E. a e. ( 0 ... ( # ` X ) ) E. b e. ( I X. 2o ) A = ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) )
13		fviss 6256	. . . . . . . . 9 |- ( _I ` Word ( I X. 2o ) ) C_ Word ( I X. 2o )
14	1, 13	eqsstri 3635	. . . . . . . 8 |- W C_ Word ( I X. 2o )
15		simpl 473	. . . . . . . 8 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> X e. W )
16	14, 15	sseldi 3601	. . . . . . 7 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> X e. Word ( I X. 2o ) )
17		elfzuz 12338	. . . . . . . . 9 |- ( a e. ( 0 ... ( # ` X ) ) -> a e. ( ZZ>= ` 0 ) )
18	17	ad2antrl 764	. . . . . . . 8 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> a e. ( ZZ>= ` 0 ) )
19		eluzfz2b 12350	. . . . . . . 8 |- ( a e. ( ZZ>= ` 0 ) <-> a e. ( 0 ... a ) )
20	18, 19	sylib 208	. . . . . . 7 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> a e. ( 0 ... a ) )
21		simprl 794	. . . . . . 7 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> a e. ( 0 ... ( # ` X ) ) )
22		simprr 796	. . . . . . . 8 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> b e. ( I X. 2o ) )
23	3	efgmf 18126	. . . . . . . . . 10 |- M : ( I X. 2o ) --> ( I X. 2o )
24	23	ffvelrni 6358	. . . . . . . . 9 |- ( b e. ( I X. 2o ) -> ( M ` b ) e. ( I X. 2o ) )
25	22, 24	syl 17	. . . . . . . 8 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> ( M ` b ) e. ( I X. 2o ) )
26	22, 25	s2cld 13616	. . . . . . 7 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> <" b ( M ` b ) "> e. Word ( I X. 2o ) )
27	16, 20, 21, 26	spllen 13505	. . . . . 6 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> ( # ` ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) = ( ( # ` X ) + ( ( # ` <" b ( M ` b ) "> ) - ( a - a ) ) ) )
28		s2len 13634	. . . . . . . . . 10 |- ( # ` <" b ( M ` b ) "> ) = 2
29	28	a1i 11	. . . . . . . . 9 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> ( # ` <" b ( M ` b ) "> ) = 2 )
30		eluzelcn 11699	. . . . . . . . . . 11 |- ( a e. ( ZZ>= ` 0 ) -> a e. CC )
31	18, 30	syl 17	. . . . . . . . . 10 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> a e. CC )
32	31	subidd 10380	. . . . . . . . 9 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> ( a - a ) = 0 )
33	29, 32	oveq12d 6668	. . . . . . . 8 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> ( ( # ` <" b ( M ` b ) "> ) - ( a - a ) ) = ( 2 - 0 ) )
34		2cn 11091	. . . . . . . . 9 |- 2 e. CC
35	34	subid1i 10353	. . . . . . . 8 |- ( 2 - 0 ) = 2
36	33, 35	syl6eq 2672	. . . . . . 7 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> ( ( # ` <" b ( M ` b ) "> ) - ( a - a ) ) = 2 )
37	36	oveq2d 6666	. . . . . 6 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> ( ( # ` X ) + ( ( # ` <" b ( M ` b ) "> ) - ( a - a ) ) ) = ( ( # ` X ) + 2 ) )
38	27, 37	eqtrd 2656	. . . . 5 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> ( # ` ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) = ( ( # ` X ) + 2 ) )
39		fveq2 6191	. . . . . 6 |- ( A = ( X splice <. a , a , <" b ( M ` b ) "> >. ) -> ( # ` A ) = ( # ` ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) )
40	39	eqeq1d 2624	. . . . 5 |- ( A = ( X splice <. a , a , <" b ( M ` b ) "> >. ) -> ( ( # ` A ) = ( ( # ` X ) + 2 ) <-> ( # ` ( X splice <. a , a , <" b ( M ` b ) "> >. ) ) = ( ( # ` X ) + 2 ) ) )
41	38, 40	syl5ibrcom 237	. . . 4 |- ( ( X e. W /\ ( a e. ( 0 ... ( # ` X ) ) /\ b e. ( I X. 2o ) ) ) -> ( A = ( X splice <. a , a , <" b ( M ` b ) "> >. ) -> ( # ` A ) = ( ( # ` X ) + 2 ) ) )
42	41	rexlimdvva 3038	. . 3 |- ( X e. W -> ( E. a e. ( 0 ... ( # ` X ) ) E. b e. ( I X. 2o ) A = ( X splice <. a , a , <" b ( M ` b ) "> >. ) -> ( # ` A ) = ( ( # ` X ) + 2 ) ) )
43	12, 42	sylbid 230	. 2 |- ( X e. W -> ( A e. ran ( T ` X ) -> ( # ` A ) = ( ( # ` X ) + 2 ) ) )
44	43	imp 445	1 |- ( ( X e. W /\ A e. ran ( T ` X ) ) -> ( # ` A ) = ( ( # ` X ) + 2 ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   <.cop 4183   <.cotp 4185   |-> cmpt 4729   _I cid 5023   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   CCcc 9934   0cc0 9936   + caddc 9939   - cmin 10266   2c2 11070   ZZ>=cuz 11687   ...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-2 11079  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:  efgsfo  18152  efgredlemg  18155  efgredlemd  18157  efgredlem  18160  frgpnabllem1  18276