Theorem efgval2 18137

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
efgval2	|- .~ = |^| { r | ( r Er W /\ A. x e. W ran ( T ` x ) C_ [ x ] r ) }

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

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

Proof of Theorem efgval2

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

Step	Hyp	Ref	Expression
1		efgval.w	. . 3 |- W = ( _I ` Word ( I X. 2o ) )
2		efgval.r	. . 3 |- .~ = ( ~FG ` I )
3	1, 2	efgval 18130	. 2 |- .~ = |^| { r | ( r Er W /\ A. x e. W A. m e. ( 0 ... ( # ` x ) ) A. a e. I A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) }
4		efgval2.m	. . . . . . . . . . 11 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
5		efgval2.t	. . . . . . . . . . 11 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
6	1, 2, 4, 5	efgtf 18135	. . . . . . . . . 10 |- ( x e. W -> ( ( T ` x ) = ( m e. ( 0 ... ( # ` x ) ) , u e. ( I X. 2o ) |-> ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) /\ ( T ` x ) : ( ( 0 ... ( # ` x ) ) X. ( I X. 2o ) ) --> W ) )
7	6	simpld 475	. . . . . . . . 9 |- ( x e. W -> ( T ` x ) = ( m e. ( 0 ... ( # ` x ) ) , u e. ( I X. 2o ) |-> ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) )
8	7	rneqd 5353	. . . . . . . 8 |- ( x e. W -> ran ( T ` x ) = ran ( m e. ( 0 ... ( # ` x ) ) , u e. ( I X. 2o ) |-> ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) )
9	8	sseq1d 3632	. . . . . . 7 |- ( x e. W -> ( ran ( T ` x ) C_ [ x ] r <-> ran ( m e. ( 0 ... ( # ` x ) ) , u e. ( I X. 2o ) |-> ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) C_ [ x ] r ) )
10		dfss3 3592	. . . . . . . 8 |- ( ran ( m e. ( 0 ... ( # ` x ) ) , u e. ( I X. 2o ) |-> ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) C_ [ x ] r <-> A. a e. ran ( m e. ( 0 ... ( # ` x ) ) , u e. ( I X. 2o ) |-> ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) a e. [ x ] r )
11		ovex 6678	. . . . . . . . . . 11 |- ( x splice <. m , m , <" u ( M ` u ) "> >. ) e. _V
12	11	rgen2w 2925	. . . . . . . . . 10 |- A. m e. ( 0 ... ( # ` x ) ) A. u e. ( I X. 2o ) ( x splice <. m , m , <" u ( M ` u ) "> >. ) e. _V
13		eqid 2622	. . . . . . . . . . 11 |- ( m e. ( 0 ... ( # ` x ) ) , u e. ( I X. 2o ) |-> ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) = ( m e. ( 0 ... ( # ` x ) ) , u e. ( I X. 2o ) |-> ( x splice <. m , m , <" u ( M ` u ) "> >. ) )
14		vex 3203	. . . . . . . . . . . . 13 |- a e. _V
15		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
16	14, 15	elec 7786	. . . . . . . . . . . 12 |- ( a e. [ x ] r <-> x r a )
17		breq2 4657	. . . . . . . . . . . 12 |- ( a = ( x splice <. m , m , <" u ( M ` u ) "> >. ) -> ( x r a <-> x r ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) )
18	16, 17	syl5bb 272	. . . . . . . . . . 11 |- ( a = ( x splice <. m , m , <" u ( M ` u ) "> >. ) -> ( a e. [ x ] r <-> x r ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) )
19	13, 18	ralrnmpt2 6775	. . . . . . . . . 10 |- ( A. m e. ( 0 ... ( # ` x ) ) A. u e. ( I X. 2o ) ( x splice <. m , m , <" u ( M ` u ) "> >. ) e. _V -> ( A. a e. ran ( m e. ( 0 ... ( # ` x ) ) , u e. ( I X. 2o ) |-> ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) a e. [ x ] r <-> A. m e. ( 0 ... ( # ` x ) ) A. u e. ( I X. 2o ) x r ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) )
20	12, 19	ax-mp 5	. . . . . . . . 9 |- ( A. a e. ran ( m e. ( 0 ... ( # ` x ) ) , u e. ( I X. 2o ) |-> ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) a e. [ x ] r <-> A. m e. ( 0 ... ( # ` x ) ) A. u e. ( I X. 2o ) x r ( x splice <. m , m , <" u ( M ` u ) "> >. ) )
21		id 22	. . . . . . . . . . . . . . . 16 |- ( u = <. a , b >. -> u = <. a , b >. )
22		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( u = <. a , b >. -> ( M ` u ) = ( M ` <. a , b >. ) )
23		df-ov 6653	. . . . . . . . . . . . . . . . 17 |- ( a M b ) = ( M ` <. a , b >. )
24	22, 23	syl6eqr 2674	. . . . . . . . . . . . . . . 16 |- ( u = <. a , b >. -> ( M ` u ) = ( a M b ) )
25	21, 24	s2eqd 13608	. . . . . . . . . . . . . . 15 |- ( u = <. a , b >. -> <" u ( M ` u ) "> = <" <. a , b >. ( a M b ) "> )
26	25	oteq3d 4416	. . . . . . . . . . . . . 14 |- ( u = <. a , b >. -> <. m , m , <" u ( M ` u ) "> >. = <. m , m , <" <. a , b >. ( a M b ) "> >. )
27	26	oveq2d 6666	. . . . . . . . . . . . 13 |- ( u = <. a , b >. -> ( x splice <. m , m , <" u ( M ` u ) "> >. ) = ( x splice <. m , m , <" <. a , b >. ( a M b ) "> >. ) )
28	27	breq2d 4665	. . . . . . . . . . . 12 |- ( u = <. a , b >. -> ( x r ( x splice <. m , m , <" u ( M ` u ) "> >. ) <-> x r ( x splice <. m , m , <" <. a , b >. ( a M b ) "> >. ) ) )
29	28	ralxp 5263	. . . . . . . . . . 11 |- ( A. u e. ( I X. 2o ) x r ( x splice <. m , m , <" u ( M ` u ) "> >. ) <-> A. a e. I A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. ( a M b ) "> >. ) )
30		eqidd 2623	. . . . . . . . . . . . . . . . 17 |- ( ( a e. I /\ b e. 2o ) -> <. a , b >. = <. a , b >. )
31	4	efgmval 18125	. . . . . . . . . . . . . . . . 17 |- ( ( a e. I /\ b e. 2o ) -> ( a M b ) = <. a , ( 1o \ b ) >. )
32	30, 31	s2eqd 13608	. . . . . . . . . . . . . . . 16 |- ( ( a e. I /\ b e. 2o ) -> <" <. a , b >. ( a M b ) "> = <" <. a , b >. <. a , ( 1o \ b ) >. "> )
33	32	oteq3d 4416	. . . . . . . . . . . . . . 15 |- ( ( a e. I /\ b e. 2o ) -> <. m , m , <" <. a , b >. ( a M b ) "> >. = <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. )
34	33	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( ( a e. I /\ b e. 2o ) -> ( x splice <. m , m , <" <. a , b >. ( a M b ) "> >. ) = ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) )
35	34	breq2d 4665	. . . . . . . . . . . . 13 |- ( ( a e. I /\ b e. 2o ) -> ( x r ( x splice <. m , m , <" <. a , b >. ( a M b ) "> >. ) <-> x r ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) )
36	35	ralbidva 2985	. . . . . . . . . . . 12 |- ( a e. I -> ( A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. ( a M b ) "> >. ) <-> A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) )
37	36	ralbiia 2979	. . . . . . . . . . 11 |- ( A. a e. I A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. ( a M b ) "> >. ) <-> A. a e. I A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) )
38	29, 37	bitri 264	. . . . . . . . . 10 |- ( A. u e. ( I X. 2o ) x r ( x splice <. m , m , <" u ( M ` u ) "> >. ) <-> A. a e. I A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) )
39	38	ralbii 2980	. . . . . . . . 9 |- ( A. m e. ( 0 ... ( # ` x ) ) A. u e. ( I X. 2o ) x r ( x splice <. m , m , <" u ( M ` u ) "> >. ) <-> A. m e. ( 0 ... ( # ` x ) ) A. a e. I A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) )
40	20, 39	bitri 264	. . . . . . . 8 |- ( A. a e. ran ( m e. ( 0 ... ( # ` x ) ) , u e. ( I X. 2o ) |-> ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) a e. [ x ] r <-> A. m e. ( 0 ... ( # ` x ) ) A. a e. I A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) )
41	10, 40	bitri 264	. . . . . . 7 |- ( ran ( m e. ( 0 ... ( # ` x ) ) , u e. ( I X. 2o ) |-> ( x splice <. m , m , <" u ( M ` u ) "> >. ) ) C_ [ x ] r <-> A. m e. ( 0 ... ( # ` x ) ) A. a e. I A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) )
42	9, 41	syl6bb 276	. . . . . 6 |- ( x e. W -> ( ran ( T ` x ) C_ [ x ] r <-> A. m e. ( 0 ... ( # ` x ) ) A. a e. I A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) )
43	42	ralbiia 2979	. . . . 5 |- ( A. x e. W ran ( T ` x ) C_ [ x ] r <-> A. x e. W A. m e. ( 0 ... ( # ` x ) ) A. a e. I A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) )
44	43	anbi2i 730	. . . 4 |- ( ( r Er W /\ A. x e. W ran ( T ` x ) C_ [ x ] r ) <-> ( r Er W /\ A. x e. W A. m e. ( 0 ... ( # ` x ) ) A. a e. I A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) )
45	44	abbii 2739	. . 3 |- { r | ( r Er W /\ A. x e. W ran ( T ` x ) C_ [ x ] r ) } = { r | ( r Er W /\ A. x e. W A. m e. ( 0 ... ( # ` x ) ) A. a e. I A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) }
46	45	inteqi 4479	. 2 |- |^| { r | ( r Er W /\ A. x e. W ran ( T ` x ) C_ [ x ] r ) } = |^| { r | ( r Er W /\ A. x e. W A. m e. ( 0 ... ( # ` x ) ) A. a e. I A. b e. 2o x r ( x splice <. m , m , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) }
47	3, 46	eqtr4i 2647	1 |- .~ = |^| { r | ( r Er W /\ A. x e. W ran ( T ` x ) C_ [ x ] r ) }

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   <.cop 4183   <.cotp 4185   |^|cint 4475   class class class wbr 4653   |-> 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   Er wer 7739   [cec 7740   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-ec 7744  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  df-efg 18122

This theorem is referenced by:  efgi2  18138  efgrelexlemb  18163  efgcpbllemb  18168