Theorem efgi 16329

Description: Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
efgval.w	|- W = ( _I ` Word ( I X. 2o ) )
efgval.r	|- .~ = ( ~FG ` I )

Assertion

Ref	Expression
efgi	|- ( ( ( A e. W /\ N e. ( 0 ... ( # ` A ) ) ) /\ ( J e. I /\ K e. 2o ) ) -> A .~ ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) )

Proof of Theorem efgi

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

Step	Hyp	Ref	Expression
1		fveq2 5792	. . . . . . . . . . 11 |- ( u = A -> ( # ` u ) = ( # ` A ) )
2	1	oveq2d 6209	. . . . . . . . . 10 |- ( u = A -> ( 0 ... ( # ` u ) ) = ( 0 ... ( # ` A ) ) )
3		id 22	. . . . . . . . . . . 12 |- ( u = A -> u = A )
4		oveq1 6200	. . . . . . . . . . . 12 |- ( u = A -> ( u splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) = ( A splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) )
5	3, 4	breq12d 4406	. . . . . . . . . . 11 |- ( u = A -> ( u r ( u splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) <-> A r ( A splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) )
6	5	2ralbidv 2871	. . . . . . . . . 10 |- ( u = A -> ( A. a e. I A. b e. 2o u r ( u splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) <-> A. a e. I A. b e. 2o A r ( A splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) )
7	2, 6	raleqbidv 3030	. . . . . . . . 9 |- ( u = A -> ( A. i e. ( 0 ... ( # ` u ) ) A. a e. I A. b e. 2o u r ( u splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) <-> A. i e. ( 0 ... ( # ` A ) ) A. a e. I A. b e. 2o A r ( A splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) )
8	7	rspcv 3168	. . . . . . . 8 |- ( A e. W -> ( A. u e. W A. i e. ( 0 ... ( # ` u ) ) A. a e. I A. b e. 2o u r ( u splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) -> A. i e. ( 0 ... ( # ` A ) ) A. a e. I A. b e. 2o A r ( A splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) )
9		oteq1 4169	. . . . . . . . . . . . 13 |- ( i = N -> <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. = <. N , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. )
10		oteq2 4170	. . . . . . . . . . . . 13 |- ( i = N -> <. N , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. = <. N , N , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. )
11	9, 10	eqtrd 2492	. . . . . . . . . . . 12 |- ( i = N -> <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. = <. N , N , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. )
12	11	oveq2d 6209	. . . . . . . . . . 11 |- ( i = N -> ( A splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) = ( A splice <. N , N , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) )
13	12	breq2d 4405	. . . . . . . . . 10 |- ( i = N -> ( A r ( A splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) <-> A r ( A splice <. N , N , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) )
14	13	2ralbidv 2871	. . . . . . . . 9 |- ( i = N -> ( A. a e. I A. b e. 2o A r ( A splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) <-> A. a e. I A. b e. 2o A r ( A splice <. N , N , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) )
15	14	rspcv 3168	. . . . . . . 8 |- ( N e. ( 0 ... ( # ` A ) ) -> ( A. i e. ( 0 ... ( # ` A ) ) A. a e. I A. b e. 2o A r ( A splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) -> A. a e. I A. b e. 2o A r ( A splice <. N , N , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) )
16	8, 15	sylan9 657	. . . . . . 7 |- ( ( A e. W /\ N e. ( 0 ... ( # ` A ) ) ) -> ( A. u e. W A. i e. ( 0 ... ( # ` u ) ) A. a e. I A. b e. 2o u r ( u splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) -> A. a e. I A. b e. 2o A r ( A splice <. N , N , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) )
17		opeq1 4160	. . . . . . . . . . . 12 |- ( a = J -> <. a , b >. = <. J , b >. )
18		opeq1 4160	. . . . . . . . . . . 12 |- ( a = J -> <. a , ( 1o \ b ) >. = <. J , ( 1o \ b ) >. )
19	17, 18	s2eqd 12600	. . . . . . . . . . 11 |- ( a = J -> <" <. a , b >. <. a , ( 1o \ b ) >. "> = <" <. J , b >. <. J , ( 1o \ b ) >. "> )
20	19	oteq3d 4174	. . . . . . . . . 10 |- ( a = J -> <. N , N , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. = <. N , N , <" <. J , b >. <. J , ( 1o \ b ) >. "> >. )
21	20	oveq2d 6209	. . . . . . . . 9 |- ( a = J -> ( A splice <. N , N , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) = ( A splice <. N , N , <" <. J , b >. <. J , ( 1o \ b ) >. "> >. ) )
22	21	breq2d 4405	. . . . . . . 8 |- ( a = J -> ( A r ( A splice <. N , N , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) <-> A r ( A splice <. N , N , <" <. J , b >. <. J , ( 1o \ b ) >. "> >. ) ) )
23		opeq2 4161	. . . . . . . . . . . . 13 |- ( b = K -> <. J , b >. = <. J , K >. )
24		difeq2 3569	. . . . . . . . . . . . . 14 |- ( b = K -> ( 1o \ b ) = ( 1o \ K ) )
25	24	opeq2d 4167	. . . . . . . . . . . . 13 |- ( b = K -> <. J , ( 1o \ b ) >. = <. J , ( 1o \ K ) >. )
26	23, 25	s2eqd 12600	. . . . . . . . . . . 12 |- ( b = K -> <" <. J , b >. <. J , ( 1o \ b ) >. "> = <" <. J , K >. <. J , ( 1o \ K ) >. "> )
27	26	oteq3d 4174	. . . . . . . . . . 11 |- ( b = K -> <. N , N , <" <. J , b >. <. J , ( 1o \ b ) >. "> >. = <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. )
28	27	oveq2d 6209	. . . . . . . . . 10 |- ( b = K -> ( A splice <. N , N , <" <. J , b >. <. J , ( 1o \ b ) >. "> >. ) = ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) )
29	28	breq2d 4405	. . . . . . . . 9 |- ( b = K -> ( A r ( A splice <. N , N , <" <. J , b >. <. J , ( 1o \ b ) >. "> >. ) <-> A r ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) ) )
30		df-br 4394	. . . . . . . . 9 |- ( A r ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) <-> <. A , ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) >. e. r )
31	29, 30	syl6bb 261	. . . . . . . 8 |- ( b = K -> ( A r ( A splice <. N , N , <" <. J , b >. <. J , ( 1o \ b ) >. "> >. ) <-> <. A , ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) >. e. r ) )
32	22, 31	rspc2v 3179	. . . . . . 7 |- ( ( J e. I /\ K e. 2o ) -> ( A. a e. I A. b e. 2o A r ( A splice <. N , N , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) -> <. A , ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) >. e. r ) )
33	16, 32	sylan9 657	. . . . . 6 |- ( ( ( A e. W /\ N e. ( 0 ... ( # ` A ) ) ) /\ ( J e. I /\ K e. 2o ) ) -> ( A. u e. W A. i e. ( 0 ... ( # ` u ) ) A. a e. I A. b e. 2o u r ( u splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) -> <. A , ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) >. e. r ) )
34	33	adantld 467	. . . . 5 |- ( ( ( A e. W /\ N e. ( 0 ... ( # ` A ) ) ) /\ ( J e. I /\ K e. 2o ) ) -> ( ( r Er W /\ A. u e. W A. i e. ( 0 ... ( # ` u ) ) A. a e. I A. b e. 2o u r ( u splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) -> <. A , ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) >. e. r ) )
35	34	alrimiv 1686	. . . 4 |- ( ( ( A e. W /\ N e. ( 0 ... ( # ` A ) ) ) /\ ( J e. I /\ K e. 2o ) ) -> A. r ( ( r Er W /\ A. u e. W A. i e. ( 0 ... ( # ` u ) ) A. a e. I A. b e. 2o u r ( u splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) -> <. A , ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) >. e. r ) )
36		opex 4657	. . . . 5 |- <. A , ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) >. e. _V
37	36	elintab 4240	. . . 4 |- ( <. A , ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) >. e. |^| { r | ( r Er W /\ A. u e. W A. i e. ( 0 ... ( # ` u ) ) A. a e. I A. b e. 2o u r ( u splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) } <-> A. r ( ( r Er W /\ A. u e. W A. i e. ( 0 ... ( # ` u ) ) A. a e. I A. b e. 2o u r ( u splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) -> <. A , ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) >. e. r ) )
38	35, 37	sylibr 212	. . 3 |- ( ( ( A e. W /\ N e. ( 0 ... ( # ` A ) ) ) /\ ( J e. I /\ K e. 2o ) ) -> <. A , ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) >. e. |^| { r | ( r Er W /\ A. u e. W A. i e. ( 0 ... ( # ` u ) ) A. a e. I A. b e. 2o u r ( u splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) } )
39		efgval.w	. . . 4 |- W = ( _I ` Word ( I X. 2o ) )
40		efgval.r	. . . 4 |- .~ = ( ~FG ` I )
41	39, 40	efgval 16327	. . 3 |- .~ = |^| { r | ( r Er W /\ A. u e. W A. i e. ( 0 ... ( # ` u ) ) A. a e. I A. b e. 2o u r ( u splice <. i , i , <" <. a , b >. <. a , ( 1o \ b ) >. "> >. ) ) }
42	38, 41	syl6eleqr 2550	. 2 |- ( ( ( A e. W /\ N e. ( 0 ... ( # ` A ) ) ) /\ ( J e. I /\ K e. 2o ) ) -> <. A , ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) >. e. .~ )
43		df-br 4394	. 2 |- ( A .~ ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) <-> <. A , ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) >. e. .~ )
44	42, 43	sylibr 212	1 |- ( ( ( A e. W /\ N e. ( 0 ... ( # ` A ) ) ) /\ ( J e. I /\ K e. 2o ) ) -> A .~ ( A splice <. N , N , <" <. J , K >. <. J , ( 1o \ K ) >. "> >. ) )

Syntax hints:   -> wi 4   /\ wa 369   A.wal 1368   = wceq 1370   e. wcel 1758   {cab 2436   A.wral 2795   \ cdif 3426   <.cop 3984   <.cotp 3986   |^|cint 4229   class class class wbr 4393   _I cid 4732   X. cxp 4939   ` cfv 5519  (class class class)co 6193   1oc1o 7016   2oc2o 7017   Er wer 7201   0cc0 9386   ...cfz 11547   #chash 12213  Word cword 12332   splice csplice 12337   <"cs2 12579   ~_FG cefg 16316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-ot 3987  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-hash 12214  df-word 12340  df-concat 12342  df-s1 12343  df-substr 12344  df-splice 12345  df-s2 12586  df-efg 16319

This theorem is referenced by:  efgi0  16330  efgi1  16331