Theorem frgpnabllem2 18277

Description: Lemma for frgpnabl 18278. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
frgpnabl.g	|- G = (freeGrp ` I )
frgpnabl.w	|- W = ( _I ` Word ( I X. 2o ) )
frgpnabl.r	|- .~ = ( ~FG ` I )
frgpnabl.p	|- .+ = ( +g ` G )
frgpnabl.m	|- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
frgpnabl.t	|- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
frgpnabl.d	|- D = ( W \ U_ x e. W ran ( T ` x ) )
frgpnabl.u	|- U = (varFGrp ` I )
frgpnabl.i	|- ( ph -> I e. _V )
frgpnabl.a	|- ( ph -> A e. I )
frgpnabl.b	|- ( ph -> B e. I )
frgpnabl.n	|- ( ph -> ( ( U ` A ) .+ ( U ` B ) ) = ( ( U ` B ) .+ ( U ` A ) ) )

Assertion

Ref	Expression
frgpnabllem2	|- ( ph -> A = B )

Distinct variable groups:   x, A   v, n, w, x, y, z, I   ph, x   x, .~ , y, z   x, B   n, W, v, w, x, y, z   x, G   n, M, v, w, x   x, T

Allowed substitution hints:   ph( y, z, w, v, n)   A( y, z, w, v, n)   B( y, z, w, v, n)   D( x, y, z, w, v, n)   .+ ( x, y, z, w, v, n)   .~ ( w, v, n)   T( y, z, w, v, n)   U( x, y, z, w, v, n)   G( y, z, w, v, n)   M( y, z)

Proof of Theorem frgpnabllem2

Dummy variables d m t k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgpnabl.a	. 2 |- ( ph -> A e. I )
2		0ex 4790	. . 3 |- (/) e. _V
3	2	a1i 11	. 2 |- ( ph -> (/) e. _V )
4		frgpnabl.d	. . . . . . . 8 |- D = ( W \ U_ x e. W ran ( T ` x ) )
5		difss 3737	. . . . . . . 8 |- ( W \ U_ x e. W ran ( T ` x ) ) C_ W
6	4, 5	eqsstri 3635	. . . . . . 7 |- D C_ W
7		inss1 3833	. . . . . . . 8 |- ( D i^i ( ( U ` B ) .+ ( U ` A ) ) ) C_ D
8		frgpnabl.g	. . . . . . . . 9 |- G = (freeGrp ` I )
9		frgpnabl.w	. . . . . . . . 9 |- W = ( _I ` Word ( I X. 2o ) )
10		frgpnabl.r	. . . . . . . . 9 |- .~ = ( ~FG ` I )
11		frgpnabl.p	. . . . . . . . 9 |- .+ = ( +g ` G )
12		frgpnabl.m	. . . . . . . . 9 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
13		frgpnabl.t	. . . . . . . . 9 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
14		frgpnabl.u	. . . . . . . . 9 |- U = (varFGrp ` I )
15		frgpnabl.i	. . . . . . . . 9 |- ( ph -> I e. _V )
16		frgpnabl.b	. . . . . . . . 9 |- ( ph -> B e. I )
17	8, 9, 10, 11, 12, 13, 4, 14, 15, 16, 1	frgpnabllem1 18276	. . . . . . . 8 |- ( ph -> <" <. B , (/) >. <. A , (/) >. "> e. ( D i^i ( ( U ` B ) .+ ( U ` A ) ) ) )
18	7, 17	sseldi 3601	. . . . . . 7 |- ( ph -> <" <. B , (/) >. <. A , (/) >. "> e. D )
19	6, 18	sseldi 3601	. . . . . 6 |- ( ph -> <" <. B , (/) >. <. A , (/) >. "> e. W )
20		eqid 2622	. . . . . . 7 |- ( m e. { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) ) = ( m e. { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) )
21	9, 10, 12, 13, 4, 20	efgredeu 18165	. . . . . 6 |- ( <" <. B , (/) >. <. A , (/) >. "> e. W -> E! d e. D d .~ <" <. B , (/) >. <. A , (/) >. "> )
22		reurmo 3161	. . . . . 6 |- ( E! d e. D d .~ <" <. B , (/) >. <. A , (/) >. "> -> E* d e. D d .~ <" <. B , (/) >. <. A , (/) >. "> )
23	19, 21, 22	3syl 18	. . . . 5 |- ( ph -> E* d e. D d .~ <" <. B , (/) >. <. A , (/) >. "> )
24		inss1 3833	. . . . . 6 |- ( D i^i ( ( U ` A ) .+ ( U ` B ) ) ) C_ D
25	8, 9, 10, 11, 12, 13, 4, 14, 15, 1, 16	frgpnabllem1 18276	. . . . . 6 |- ( ph -> <" <. A , (/) >. <. B , (/) >. "> e. ( D i^i ( ( U ` A ) .+ ( U ` B ) ) ) )
26	24, 25	sseldi 3601	. . . . 5 |- ( ph -> <" <. A , (/) >. <. B , (/) >. "> e. D )
27	9, 10	efger 18131	. . . . . . . . 9 |- .~ Er W
28	27	a1i 11	. . . . . . . 8 |- ( ph -> .~ Er W )
29	8	frgpgrp 18175	. . . . . . . . . . 11 |- ( I e. _V -> G e. Grp )
30	15, 29	syl 17	. . . . . . . . . 10 |- ( ph -> G e. Grp )
31		eqid 2622	. . . . . . . . . . . . 13 |- ( Base ` G ) = ( Base ` G )
32	10, 14, 8, 31	vrgpf 18181	. . . . . . . . . . . 12 |- ( I e. _V -> U : I --> ( Base ` G ) )
33	15, 32	syl 17	. . . . . . . . . . 11 |- ( ph -> U : I --> ( Base ` G ) )
34	33, 1	ffvelrnd 6360	. . . . . . . . . 10 |- ( ph -> ( U ` A ) e. ( Base ` G ) )
35	33, 16	ffvelrnd 6360	. . . . . . . . . 10 |- ( ph -> ( U ` B ) e. ( Base ` G ) )
36	31, 11	grpcl 17430	. . . . . . . . . 10 |- ( ( G e. Grp /\ ( U ` A ) e. ( Base ` G ) /\ ( U ` B ) e. ( Base ` G ) ) -> ( ( U ` A ) .+ ( U ` B ) ) e. ( Base ` G ) )
37	30, 34, 35, 36	syl3anc 1326	. . . . . . . . 9 |- ( ph -> ( ( U ` A ) .+ ( U ` B ) ) e. ( Base ` G ) )
38		eqid 2622	. . . . . . . . . . . 12 |- (freeMnd ` ( I X. 2o ) ) = (freeMnd ` ( I X. 2o ) )
39	8, 38, 10	frgpval 18171	. . . . . . . . . . 11 |- ( I e. _V -> G = ( (freeMnd ` ( I X. 2o ) ) /.s .~ ) )
40	15, 39	syl 17	. . . . . . . . . 10 |- ( ph -> G = ( (freeMnd ` ( I X. 2o ) ) /.s .~ ) )
41		2on 7568	. . . . . . . . . . . . . 14 |- 2o e. On
42		xpexg 6960	. . . . . . . . . . . . . 14 |- ( ( I e. _V /\ 2o e. On ) -> ( I X. 2o ) e. _V )
43	15, 41, 42	sylancl 694	. . . . . . . . . . . . 13 |- ( ph -> ( I X. 2o ) e. _V )
44		wrdexg 13315	. . . . . . . . . . . . 13 |- ( ( I X. 2o ) e. _V -> Word ( I X. 2o ) e. _V )
45		fvi 6255	. . . . . . . . . . . . 13 |- (Word ( I X. 2o ) e. _V -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
46	43, 44, 45	3syl 18	. . . . . . . . . . . 12 |- ( ph -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
47	9, 46	syl5eq 2668	. . . . . . . . . . 11 |- ( ph -> W = Word ( I X. 2o ) )
48		eqid 2622	. . . . . . . . . . . . 13 |- ( Base ` (freeMnd ` ( I X. 2o ) ) ) = ( Base ` (freeMnd ` ( I X. 2o ) ) )
49	38, 48	frmdbas 17389	. . . . . . . . . . . 12 |- ( ( I X. 2o ) e. _V -> ( Base ` (freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) )
50	43, 49	syl 17	. . . . . . . . . . 11 |- ( ph -> ( Base ` (freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) )
51	47, 50	eqtr4d 2659	. . . . . . . . . 10 |- ( ph -> W = ( Base ` (freeMnd ` ( I X. 2o ) ) ) )
52		fvex 6201	. . . . . . . . . . . 12 |- ( ~FG ` I ) e. _V
53	10, 52	eqeltri 2697	. . . . . . . . . . 11 |- .~ e. _V
54	53	a1i 11	. . . . . . . . . 10 |- ( ph -> .~ e. _V )
55		fvexd 6203	. . . . . . . . . 10 |- ( ph -> (freeMnd ` ( I X. 2o ) ) e. _V )
56	40, 51, 54, 55	qusbas 16205	. . . . . . . . 9 |- ( ph -> ( W /. .~ ) = ( Base ` G ) )
57	37, 56	eleqtrrd 2704	. . . . . . . 8 |- ( ph -> ( ( U ` A ) .+ ( U ` B ) ) e. ( W /. .~ ) )
58		inss2 3834	. . . . . . . . 9 |- ( D i^i ( ( U ` A ) .+ ( U ` B ) ) ) C_ ( ( U ` A ) .+ ( U ` B ) )
59	58, 25	sseldi 3601	. . . . . . . 8 |- ( ph -> <" <. A , (/) >. <. B , (/) >. "> e. ( ( U ` A ) .+ ( U ` B ) ) )
60		qsel 7826	. . . . . . . 8 |- ( ( .~ Er W /\ ( ( U ` A ) .+ ( U ` B ) ) e. ( W /. .~ ) /\ <" <. A , (/) >. <. B , (/) >. "> e. ( ( U ` A ) .+ ( U ` B ) ) ) -> ( ( U ` A ) .+ ( U ` B ) ) = [ <" <. A , (/) >. <. B , (/) >. "> ] .~ )
61	28, 57, 59, 60	syl3anc 1326	. . . . . . 7 |- ( ph -> ( ( U ` A ) .+ ( U ` B ) ) = [ <" <. A , (/) >. <. B , (/) >. "> ] .~ )
62		inss2 3834	. . . . . . . . . 10 |- ( D i^i ( ( U ` B ) .+ ( U ` A ) ) ) C_ ( ( U ` B ) .+ ( U ` A ) )
63	62, 17	sseldi 3601	. . . . . . . . 9 |- ( ph -> <" <. B , (/) >. <. A , (/) >. "> e. ( ( U ` B ) .+ ( U ` A ) ) )
64		frgpnabl.n	. . . . . . . . 9 |- ( ph -> ( ( U ` A ) .+ ( U ` B ) ) = ( ( U ` B ) .+ ( U ` A ) ) )
65	63, 64	eleqtrrd 2704	. . . . . . . 8 |- ( ph -> <" <. B , (/) >. <. A , (/) >. "> e. ( ( U ` A ) .+ ( U ` B ) ) )
66		qsel 7826	. . . . . . . 8 |- ( ( .~ Er W /\ ( ( U ` A ) .+ ( U ` B ) ) e. ( W /. .~ ) /\ <" <. B , (/) >. <. A , (/) >. "> e. ( ( U ` A ) .+ ( U ` B ) ) ) -> ( ( U ` A ) .+ ( U ` B ) ) = [ <" <. B , (/) >. <. A , (/) >. "> ] .~ )
67	28, 57, 65, 66	syl3anc 1326	. . . . . . 7 |- ( ph -> ( ( U ` A ) .+ ( U ` B ) ) = [ <" <. B , (/) >. <. A , (/) >. "> ] .~ )
68	61, 67	eqtr3d 2658	. . . . . 6 |- ( ph -> [ <" <. A , (/) >. <. B , (/) >. "> ] .~ = [ <" <. B , (/) >. <. A , (/) >. "> ] .~ )
69	6, 26	sseldi 3601	. . . . . . 7 |- ( ph -> <" <. A , (/) >. <. B , (/) >. "> e. W )
70	28, 69	erth 7791	. . . . . 6 |- ( ph -> ( <" <. A , (/) >. <. B , (/) >. "> .~ <" <. B , (/) >. <. A , (/) >. "> <-> [ <" <. A , (/) >. <. B , (/) >. "> ] .~ = [ <" <. B , (/) >. <. A , (/) >. "> ] .~ ) )
71	68, 70	mpbird 247	. . . . 5 |- ( ph -> <" <. A , (/) >. <. B , (/) >. "> .~ <" <. B , (/) >. <. A , (/) >. "> )
72	28, 19	erref 7762	. . . . 5 |- ( ph -> <" <. B , (/) >. <. A , (/) >. "> .~ <" <. B , (/) >. <. A , (/) >. "> )
73		breq1 4656	. . . . . 6 |- ( d = <" <. A , (/) >. <. B , (/) >. "> -> ( d .~ <" <. B , (/) >. <. A , (/) >. "> <-> <" <. A , (/) >. <. B , (/) >. "> .~ <" <. B , (/) >. <. A , (/) >. "> ) )
74		breq1 4656	. . . . . 6 |- ( d = <" <. B , (/) >. <. A , (/) >. "> -> ( d .~ <" <. B , (/) >. <. A , (/) >. "> <-> <" <. B , (/) >. <. A , (/) >. "> .~ <" <. B , (/) >. <. A , (/) >. "> ) )
75	73, 74	rmoi 3530	. . . . 5 |- ( ( E* d e. D d .~ <" <. B , (/) >. <. A , (/) >. "> /\ ( <" <. A , (/) >. <. B , (/) >. "> e. D /\ <" <. A , (/) >. <. B , (/) >. "> .~ <" <. B , (/) >. <. A , (/) >. "> ) /\ ( <" <. B , (/) >. <. A , (/) >. "> e. D /\ <" <. B , (/) >. <. A , (/) >. "> .~ <" <. B , (/) >. <. A , (/) >. "> ) ) -> <" <. A , (/) >. <. B , (/) >. "> = <" <. B , (/) >. <. A , (/) >. "> )
76	23, 26, 71, 18, 72, 75	syl122anc 1335	. . . 4 |- ( ph -> <" <. A , (/) >. <. B , (/) >. "> = <" <. B , (/) >. <. A , (/) >. "> )
77	76	fveq1d 6193	. . 3 |- ( ph -> ( <" <. A , (/) >. <. B , (/) >. "> ` 0 ) = ( <" <. B , (/) >. <. A , (/) >. "> ` 0 ) )
78		opex 4932	. . . 4 |- <. A , (/) >. e. _V
79		s2fv0 13632	. . . 4 |- ( <. A , (/) >. e. _V -> ( <" <. A , (/) >. <. B , (/) >. "> ` 0 ) = <. A , (/) >. )
80	78, 79	ax-mp 5	. . 3 |- ( <" <. A , (/) >. <. B , (/) >. "> ` 0 ) = <. A , (/) >.
81		opex 4932	. . . 4 |- <. B , (/) >. e. _V
82		s2fv0 13632	. . . 4 |- ( <. B , (/) >. e. _V -> ( <" <. B , (/) >. <. A , (/) >. "> ` 0 ) = <. B , (/) >. )
83	81, 82	ax-mp 5	. . 3 |- ( <" <. B , (/) >. <. A , (/) >. "> ` 0 ) = <. B , (/) >.
84	77, 80, 83	3eqtr3g 2679	. 2 |- ( ph -> <. A , (/) >. = <. B , (/) >. )
85		opthg 4946	. . 3 |- ( ( A e. I /\ (/) e. _V ) -> ( <. A , (/) >. = <. B , (/) >. <-> ( A = B /\ (/) = (/) ) ) )
86	85	simprbda 653	. 2 |- ( ( ( A e. I /\ (/) e. _V ) /\ <. A , (/) >. = <. B , (/) >. ) -> A = B )
87	1, 3, 84, 86	syl21anc 1325	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E!wreu 2914   E*wrmo 2915   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   (/)c0 3915   {csn 4177   <.cop 4183   <.cotp 4185   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   ran crn 5115   Oncon0 5723   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   Er wer 7739   [cec 7740   /.cqs 7741   0cc0 9936   1c1 9937   - cmin 10266   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   splice csplice 13296   <"cs2 13586   Basecbs 15857   +g cplusg 15941   /._s cqus 16165  freeMndcfrmd 17384   Grpcgrp 17422   ~_FG cefg 18119  freeGrpcfrgp 18120  var_FGrpcvrgp 18121

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-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  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-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  df-s2 13593  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-0g 16102  df-imas 16168  df-qus 16169  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-frmd 17386  df-grp 17425  df-efg 18122  df-frgp 18123  df-vrgp 18124

This theorem is referenced by:  frgpnabl  18278