Theorem frgpuptinv 17470

Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
frgpup.b	|- B = ( Base ` H )
frgpup.n	|- N = ( invg ` H )
frgpup.t	|- T = ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) )
frgpup.h	|- ( ph -> H e. Grp )
frgpup.i	|- ( ph -> I e. V )
frgpup.a	|- ( ph -> F : I --> B )
frgpuptinv.m	|- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )

Assertion

Ref	Expression
frgpuptinv	|- ( ( ph /\ A e. ( I X. 2o ) ) -> ( T ` ( M ` A ) ) = ( N ` ( T ` A ) ) )

Distinct variable groups:   y, z, A   y, F, z   y, N, z   y, B, z   ph, y, z   y, I, z

Allowed substitution hints:   T( y, z)   H( y, z)   M( y, z)   V( y, z)

Proof of Theorem frgpuptinv

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

Step	Hyp	Ref	Expression
1		elxp2 4871	. . 3 |- ( A e. ( I X. 2o ) <-> E. a e. I E. b e. 2o A = <. a , b >. )
2		frgpuptinv.m	. . . . . . . . . 10 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
3	2	efgmval 17411	. . . . . . . . 9 |- ( ( a e. I /\ b e. 2o ) -> ( a M b ) = <. a , ( 1o \ b ) >. )
4	3	adantl 472	. . . . . . . 8 |- ( ( ph /\ ( a e. I /\ b e. 2o ) ) -> ( a M b ) = <. a , ( 1o \ b ) >. )
5	4	fveq2d 5892	. . . . . . 7 |- ( ( ph /\ ( a e. I /\ b e. 2o ) ) -> ( T ` ( a M b ) ) = ( T ` <. a , ( 1o \ b ) >. ) )
6		df-ov 6318	. . . . . . 7 |- ( a T ( 1o \ b ) ) = ( T ` <. a , ( 1o \ b ) >. )
7	5, 6	syl6eqr 2514	. . . . . 6 |- ( ( ph /\ ( a e. I /\ b e. 2o ) ) -> ( T ` ( a M b ) ) = ( a T ( 1o \ b ) ) )
8		elpri 3997	. . . . . . . . 9 |- ( b e. { (/) , 1o } -> ( b = (/) \/ b = 1o ) )
9		df2o3 7221	. . . . . . . . 9 |- 2o = { (/) , 1o }
10	8, 9	eleq2s 2558	. . . . . . . 8 |- ( b e. 2o -> ( b = (/) \/ b = 1o ) )
11		simpr 467	. . . . . . . . . . . 12 |- ( ( ph /\ a e. I ) -> a e. I )
12		1on 7215	. . . . . . . . . . . . . . 15 |- 1o e. On
13	12	elexi 3067	. . . . . . . . . . . . . 14 |- 1o e. _V
14	13	prid2 4094	. . . . . . . . . . . . 13 |- 1o e. { (/) , 1o }
15	14, 9	eleqtrri 2539	. . . . . . . . . . . 12 |- 1o e. 2o
16		1n0 7223	. . . . . . . . . . . . . . . 16 |- 1o =/= (/)
17		neeq1 2698	. . . . . . . . . . . . . . . 16 |- ( z = 1o -> ( z =/= (/) <-> 1o =/= (/) ) )
18	16, 17	mpbiri 241	. . . . . . . . . . . . . . 15 |- ( z = 1o -> z =/= (/) )
19		ifnefalse 3905	. . . . . . . . . . . . . . 15 |- ( z =/= (/) -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) = ( N ` ( F ` y ) ) )
20	18, 19	syl 17	. . . . . . . . . . . . . 14 |- ( z = 1o -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) = ( N ` ( F ` y ) ) )
21		fveq2 5888	. . . . . . . . . . . . . . 15 |- ( y = a -> ( F ` y ) = ( F ` a ) )
22	21	fveq2d 5892	. . . . . . . . . . . . . 14 |- ( y = a -> ( N ` ( F ` y ) ) = ( N ` ( F ` a ) ) )
23	20, 22	sylan9eqr 2518	. . . . . . . . . . . . 13 |- ( ( y = a /\ z = 1o ) -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) = ( N ` ( F ` a ) ) )
24		frgpup.t	. . . . . . . . . . . . 13 |- T = ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) )
25		fvex 5898	. . . . . . . . . . . . 13 |- ( N ` ( F ` a ) ) e. _V
26	23, 24, 25	ovmpt2a 6454	. . . . . . . . . . . 12 |- ( ( a e. I /\ 1o e. 2o ) -> ( a T 1o ) = ( N ` ( F ` a ) ) )
27	11, 15, 26	sylancl 673	. . . . . . . . . . 11 |- ( ( ph /\ a e. I ) -> ( a T 1o ) = ( N ` ( F ` a ) ) )
28		0ex 4549	. . . . . . . . . . . . . . 15 |- (/) e. _V
29	28	prid1 4093	. . . . . . . . . . . . . 14 |- (/) e. { (/) , 1o }
30	29, 9	eleqtrri 2539	. . . . . . . . . . . . 13 |- (/) e. 2o
31		iftrue 3899	. . . . . . . . . . . . . . 15 |- ( z = (/) -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) = ( F ` y ) )
32	31, 21	sylan9eqr 2518	. . . . . . . . . . . . . 14 |- ( ( y = a /\ z = (/) ) -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) = ( F ` a ) )
33		fvex 5898	. . . . . . . . . . . . . 14 |- ( F ` a ) e. _V
34	32, 24, 33	ovmpt2a 6454	. . . . . . . . . . . . 13 |- ( ( a e. I /\ (/) e. 2o ) -> ( a T (/) ) = ( F ` a ) )
35	11, 30, 34	sylancl 673	. . . . . . . . . . . 12 |- ( ( ph /\ a e. I ) -> ( a T (/) ) = ( F ` a ) )
36	35	fveq2d 5892	. . . . . . . . . . 11 |- ( ( ph /\ a e. I ) -> ( N ` ( a T (/) ) ) = ( N ` ( F ` a ) ) )
37	27, 36	eqtr4d 2499	. . . . . . . . . 10 |- ( ( ph /\ a e. I ) -> ( a T 1o ) = ( N ` ( a T (/) ) ) )
38		difeq2 3557	. . . . . . . . . . . . 13 |- ( b = (/) -> ( 1o \ b ) = ( 1o \ (/) ) )
39		dif0 3849	. . . . . . . . . . . . 13 |- ( 1o \ (/) ) = 1o
40	38, 39	syl6eq 2512	. . . . . . . . . . . 12 |- ( b = (/) -> ( 1o \ b ) = 1o )
41	40	oveq2d 6331	. . . . . . . . . . 11 |- ( b = (/) -> ( a T ( 1o \ b ) ) = ( a T 1o ) )
42		oveq2 6323	. . . . . . . . . . . 12 |- ( b = (/) -> ( a T b ) = ( a T (/) ) )
43	42	fveq2d 5892	. . . . . . . . . . 11 |- ( b = (/) -> ( N ` ( a T b ) ) = ( N ` ( a T (/) ) ) )
44	41, 43	eqeq12d 2477	. . . . . . . . . 10 |- ( b = (/) -> ( ( a T ( 1o \ b ) ) = ( N ` ( a T b ) ) <-> ( a T 1o ) = ( N ` ( a T (/) ) ) ) )
45	37, 44	syl5ibrcom 230	. . . . . . . . 9 |- ( ( ph /\ a e. I ) -> ( b = (/) -> ( a T ( 1o \ b ) ) = ( N ` ( a T b ) ) ) )
46	37	fveq2d 5892	. . . . . . . . . . 11 |- ( ( ph /\ a e. I ) -> ( N ` ( a T 1o ) ) = ( N ` ( N ` ( a T (/) ) ) ) )
47		frgpup.h	. . . . . . . . . . . . 13 |- ( ph -> H e. Grp )
48	47	adantr 471	. . . . . . . . . . . 12 |- ( ( ph /\ a e. I ) -> H e. Grp )
49		frgpup.a	. . . . . . . . . . . . . 14 |- ( ph -> F : I --> B )
50	49	ffvelrnda 6045	. . . . . . . . . . . . 13 |- ( ( ph /\ a e. I ) -> ( F ` a ) e. B )
51	35, 50	eqeltrd 2540	. . . . . . . . . . . 12 |- ( ( ph /\ a e. I ) -> ( a T (/) ) e. B )
52		frgpup.b	. . . . . . . . . . . . 13 |- B = ( Base ` H )
53		frgpup.n	. . . . . . . . . . . . 13 |- N = ( invg ` H )
54	52, 53	grpinvinv 16770	. . . . . . . . . . . 12 |- ( ( H e. Grp /\ ( a T (/) ) e. B ) -> ( N ` ( N ` ( a T (/) ) ) ) = ( a T (/) ) )
55	48, 51, 54	syl2anc 671	. . . . . . . . . . 11 |- ( ( ph /\ a e. I ) -> ( N ` ( N ` ( a T (/) ) ) ) = ( a T (/) ) )
56	46, 55	eqtr2d 2497	. . . . . . . . . 10 |- ( ( ph /\ a e. I ) -> ( a T (/) ) = ( N ` ( a T 1o ) ) )
57		difeq2 3557	. . . . . . . . . . . . 13 |- ( b = 1o -> ( 1o \ b ) = ( 1o \ 1o ) )
58		difid 3847	. . . . . . . . . . . . 13 |- ( 1o \ 1o ) = (/)
59	57, 58	syl6eq 2512	. . . . . . . . . . . 12 |- ( b = 1o -> ( 1o \ b ) = (/) )
60	59	oveq2d 6331	. . . . . . . . . . 11 |- ( b = 1o -> ( a T ( 1o \ b ) ) = ( a T (/) ) )
61		oveq2 6323	. . . . . . . . . . . 12 |- ( b = 1o -> ( a T b ) = ( a T 1o ) )
62	61	fveq2d 5892	. . . . . . . . . . 11 |- ( b = 1o -> ( N ` ( a T b ) ) = ( N ` ( a T 1o ) ) )
63	60, 62	eqeq12d 2477	. . . . . . . . . 10 |- ( b = 1o -> ( ( a T ( 1o \ b ) ) = ( N ` ( a T b ) ) <-> ( a T (/) ) = ( N ` ( a T 1o ) ) ) )
64	56, 63	syl5ibrcom 230	. . . . . . . . 9 |- ( ( ph /\ a e. I ) -> ( b = 1o -> ( a T ( 1o \ b ) ) = ( N ` ( a T b ) ) ) )
65	45, 64	jaod 386	. . . . . . . 8 |- ( ( ph /\ a e. I ) -> ( ( b = (/) \/ b = 1o ) -> ( a T ( 1o \ b ) ) = ( N ` ( a T b ) ) ) )
66	10, 65	syl5 33	. . . . . . 7 |- ( ( ph /\ a e. I ) -> ( b e. 2o -> ( a T ( 1o \ b ) ) = ( N ` ( a T b ) ) ) )
67	66	impr 629	. . . . . 6 |- ( ( ph /\ ( a e. I /\ b e. 2o ) ) -> ( a T ( 1o \ b ) ) = ( N ` ( a T b ) ) )
68	7, 67	eqtrd 2496	. . . . 5 |- ( ( ph /\ ( a e. I /\ b e. 2o ) ) -> ( T ` ( a M b ) ) = ( N ` ( a T b ) ) )
69		fveq2 5888	. . . . . . . 8 |- ( A = <. a , b >. -> ( M ` A ) = ( M ` <. a , b >. ) )
70		df-ov 6318	. . . . . . . 8 |- ( a M b ) = ( M ` <. a , b >. )
71	69, 70	syl6eqr 2514	. . . . . . 7 |- ( A = <. a , b >. -> ( M ` A ) = ( a M b ) )
72	71	fveq2d 5892	. . . . . 6 |- ( A = <. a , b >. -> ( T ` ( M ` A ) ) = ( T ` ( a M b ) ) )
73		fveq2 5888	. . . . . . . 8 |- ( A = <. a , b >. -> ( T ` A ) = ( T ` <. a , b >. ) )
74		df-ov 6318	. . . . . . . 8 |- ( a T b ) = ( T ` <. a , b >. )
75	73, 74	syl6eqr 2514	. . . . . . 7 |- ( A = <. a , b >. -> ( T ` A ) = ( a T b ) )
76	75	fveq2d 5892	. . . . . 6 |- ( A = <. a , b >. -> ( N ` ( T ` A ) ) = ( N ` ( a T b ) ) )
77	72, 76	eqeq12d 2477	. . . . 5 |- ( A = <. a , b >. -> ( ( T ` ( M ` A ) ) = ( N ` ( T ` A ) ) <-> ( T ` ( a M b ) ) = ( N ` ( a T b ) ) ) )
78	68, 77	syl5ibrcom 230	. . . 4 |- ( ( ph /\ ( a e. I /\ b e. 2o ) ) -> ( A = <. a , b >. -> ( T ` ( M ` A ) ) = ( N ` ( T ` A ) ) ) )
79	78	rexlimdvva 2898	. . 3 |- ( ph -> ( E. a e. I E. b e. 2o A = <. a , b >. -> ( T ` ( M ` A ) ) = ( N ` ( T ` A ) ) ) )
80	1, 79	syl5bi 225	. 2 |- ( ph -> ( A e. ( I X. 2o ) -> ( T ` ( M ` A ) ) = ( N ` ( T ` A ) ) ) )
81	80	imp 435	1 |- ( ( ph /\ A e. ( I X. 2o ) ) -> ( T ` ( M ` A ) ) = ( N ` ( T ` A ) ) )

Syntax hints:   -> wi 4   \/ wo 374   /\ wa 375   = wceq 1455   e. wcel 1898   =/= wne 2633   E.wrex 2750   \ cdif 3413   (/)c0 3743   ifcif 3893   {cpr 3982   <.cop 3986   X. cxp 4851   Oncon0 5442   -->wf 5597   ` cfv 5601  (class class class)co 6315   |-> cmpt2 6317   1oc1o 7201   2oc2o 7202   Basecbs 15170   Grpcgrp 16718   invgcminusg 16719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-1o 7208  df-2o 7209  df-0g 15389  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-grp 16722  df-minusg 16723

This theorem is referenced by:  frgpuplem  17471