Theorem imasgrp 13648

Description: The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)

Hypotheses

Ref	Expression
imasgrp.u	|- ( ph -> U = ( F "s R ) )
imasgrp.v	|- ( ph -> V = ( Base ` R ) )
imasgrp.p	|- ( ph -> .+ = ( +g ` R ) )
imasgrp.f	|- ( ph -> F : V -onto-> B )
imasgrp.e	|- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .+ b ) ) = ( F ` ( p .+ q ) ) ) )
imasgrp.r	|- ( ph -> R e. Grp )
imasgrp.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
imasgrp	|- ( ph -> ( U e. Grp /\ ( F ` .0. ) = ( 0g ` U ) ) )

Distinct variable groups:   q, p, B   a, b, p, q, ph   R, p, q   F, a, b, p, q   .+ , p, q   U, a, b, p, q   V, a, b, p, q   .0. , p, q

Allowed substitution hints:   B( a, b)   .+ ( a, b)   R( a, b)   .0. ( a, b)

Proof of Theorem imasgrp

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasgrp.u	. 2 |- ( ph -> U = ( F "s R ) )
2		imasgrp.v	. 2 |- ( ph -> V = ( Base ` R ) )
3		imasgrp.p	. 2 |- ( ph -> .+ = ( +g ` R ) )
4		imasgrp.f	. 2 |- ( ph -> F : V -onto-> B )
5		imasgrp.e	. 2 |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .+ b ) ) = ( F ` ( p .+ q ) ) ) )
6		imasgrp.r	. 2 |- ( ph -> R e. Grp )
7	6	3ad2ant1 1042	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> R e. Grp )
8		simp2 1022	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. V )
9	2	3ad2ant1 1042	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> V = ( Base ` R ) )
10	8, 9	eleqtrd 2308	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. ( Base ` R ) )
11		simp3 1023	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. V )
12	11, 9	eleqtrd 2308	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. ( Base ` R ) )
13		eqid 2229	. . . . 5 |- ( Base ` R ) = ( Base ` R )
14		eqid 2229	. . . . 5 |- ( +g ` R ) = ( +g ` R )
15	13, 14	grpcl 13541	. . . 4 |- ( ( R e. Grp /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( +g ` R ) y ) e. ( Base ` R ) )
16	7, 10, 12, 15	syl3anc 1271	. . 3 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x ( +g ` R ) y ) e. ( Base ` R ) )
17	3	3ad2ant1 1042	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> .+ = ( +g ` R ) )
18	17	oveqd 6018	. . 3 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) = ( x ( +g ` R ) y ) )
19	16, 18, 9	3eltr4d 2313	. 2 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. V )
20	6	adantr 276	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> R e. Grp )
21	10	3adant3r3 1238	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> x e. ( Base ` R ) )
22	12	3adant3r3 1238	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> y e. ( Base ` R ) )
23		simpr3 1029	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z e. V )
24	2	adantr 276	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> V = ( Base ` R ) )
25	23, 24	eleqtrd 2308	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z e. ( Base ` R ) )
26	13, 14	grpass 13542	. . . . 5 |- ( ( R e. Grp /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( +g ` R ) y ) ( +g ` R ) z ) = ( x ( +g ` R ) ( y ( +g ` R ) z ) ) )
27	20, 21, 22, 25, 26	syl13anc 1273	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x ( +g ` R ) y ) ( +g ` R ) z ) = ( x ( +g ` R ) ( y ( +g ` R ) z ) ) )
28	3	adantr 276	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> .+ = ( +g ` R ) )
29	18	3adant3r3 1238	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( x .+ y ) = ( x ( +g ` R ) y ) )
30		eqidd 2230	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z = z )
31	28, 29, 30	oveq123d 6022	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( ( x ( +g ` R ) y ) ( +g ` R ) z ) )
32		eqidd 2230	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> x = x )
33	28	oveqd 6018	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( y .+ z ) = ( y ( +g ` R ) z ) )
34	28, 32, 33	oveq123d 6022	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( x .+ ( y .+ z ) ) = ( x ( +g ` R ) ( y ( +g ` R ) z ) ) )
35	27, 31, 34	3eqtr4d 2272	. . 3 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
36	35	fveq2d 5631	. 2 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( F ` ( ( x .+ y ) .+ z ) ) = ( F ` ( x .+ ( y .+ z ) ) ) )
37		imasgrp.z	. . . . 5 |- .0. = ( 0g ` R )
38	13, 37	grpidcl 13562	. . . 4 |- ( R e. Grp -> .0. e. ( Base ` R ) )
39	6, 38	syl 14	. . 3 |- ( ph -> .0. e. ( Base ` R ) )
40	39, 2	eleqtrrd 2309	. 2 |- ( ph -> .0. e. V )
41	3	adantr 276	. . . . 5 |- ( ( ph /\ x e. V ) -> .+ = ( +g ` R ) )
42	41	oveqd 6018	. . . 4 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) = ( .0. ( +g ` R ) x ) )
43	2	eleq2d 2299	. . . . . 6 |- ( ph -> ( x e. V <-> x e. ( Base ` R ) ) )
44	43	biimpa 296	. . . . 5 |- ( ( ph /\ x e. V ) -> x e. ( Base ` R ) )
45	13, 14, 37	grplid 13564	. . . . 5 |- ( ( R e. Grp /\ x e. ( Base ` R ) ) -> ( .0. ( +g ` R ) x ) = x )
46	6, 44, 45	syl2an2r 597	. . . 4 |- ( ( ph /\ x e. V ) -> ( .0. ( +g ` R ) x ) = x )
47	42, 46	eqtrd 2262	. . 3 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) = x )
48	47	fveq2d 5631	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( .0. .+ x ) ) = ( F ` x ) )
49		eqid 2229	. . . . 5 |- ( invg ` R ) = ( invg ` R )
50	13, 49	grpinvcl 13581	. . . 4 |- ( ( R e. Grp /\ x e. ( Base ` R ) ) -> ( ( invg ` R ) ` x ) e. ( Base ` R ) )
51	6, 44, 50	syl2an2r 597	. . 3 |- ( ( ph /\ x e. V ) -> ( ( invg ` R ) ` x ) e. ( Base ` R ) )
52	2	adantr 276	. . 3 |- ( ( ph /\ x e. V ) -> V = ( Base ` R ) )
53	51, 52	eleqtrrd 2309	. 2 |- ( ( ph /\ x e. V ) -> ( ( invg ` R ) ` x ) e. V )
54	41	oveqd 6018	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( ( invg ` R ) ` x ) .+ x ) = ( ( ( invg ` R ) ` x ) ( +g ` R ) x ) )
55	13, 14, 37, 49	grplinv 13583	. . . . 5 |- ( ( R e. Grp /\ x e. ( Base ` R ) ) -> ( ( ( invg ` R ) ` x ) ( +g ` R ) x ) = .0. )
56	6, 44, 55	syl2an2r 597	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( ( invg ` R ) ` x ) ( +g ` R ) x ) = .0. )
57	54, 56	eqtrd 2262	. . 3 |- ( ( ph /\ x e. V ) -> ( ( ( invg ` R ) ` x ) .+ x ) = .0. )
58	57	fveq2d 5631	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( ( ( invg ` R ) ` x ) .+ x ) ) = ( F ` .0. ) )
59	1, 2, 3, 4, 5, 6, 19, 36, 40, 48, 53, 58	imasgrp2 13647	1 |- ( ph -> ( U e. Grp /\ ( F ` .0. ) = ( 0g ` U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   -onto->wfo 5316   ` cfv 5318  (class class class)co 6001   Basecbs 13032   +g cplusg 13110   0gc0g 13289   "_s cimas 13332   Grpcgrp 13533   invgcminusg 13534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-mulr 13124  df-0g 13291  df-iimas 13335  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537

This theorem is referenced by:  imasgrpf1  13649  imasabl  13873  imasring  14027