Theorem imasgrp 13241

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 1020	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> R e. Grp )
8		simp2 1000	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. V )
9	2	3ad2ant1 1020	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> V = ( Base ` R ) )
10	8, 9	eleqtrd 2275	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. ( Base ` R ) )
11		simp3 1001	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. V )
12	11, 9	eleqtrd 2275	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. ( Base ` R ) )
13		eqid 2196	. . . . 5 |- ( Base ` R ) = ( Base ` R )
14		eqid 2196	. . . . 5 |- ( +g ` R ) = ( +g ` R )
15	13, 14	grpcl 13140	. . . 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 1249	. . 3 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x ( +g ` R ) y ) e. ( Base ` R ) )
17	3	3ad2ant1 1020	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> .+ = ( +g ` R ) )
18	17	oveqd 5939	. . 3 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) = ( x ( +g ` R ) y ) )
19	16, 18, 9	3eltr4d 2280	. 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 1216	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> x e. ( Base ` R ) )
22	12	3adant3r3 1216	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> y e. ( Base ` R ) )
23		simpr3 1007	. . . . . 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 2275	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z e. ( Base ` R ) )
26	13, 14	grpass 13141	. . . . 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 1251	. . . 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 1216	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( x .+ y ) = ( x ( +g ` R ) y ) )
30		eqidd 2197	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z = z )
31	28, 29, 30	oveq123d 5943	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( ( x ( +g ` R ) y ) ( +g ` R ) z ) )
32		eqidd 2197	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> x = x )
33	28	oveqd 5939	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( y .+ z ) = ( y ( +g ` R ) z ) )
34	28, 32, 33	oveq123d 5943	. . . 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 2239	. . 3 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
36	35	fveq2d 5562	. 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 13161	. . . 4 |- ( R e. Grp -> .0. e. ( Base ` R ) )
39	6, 38	syl 14	. . 3 |- ( ph -> .0. e. ( Base ` R ) )
40	39, 2	eleqtrrd 2276	. 2 |- ( ph -> .0. e. V )
41	3	adantr 276	. . . . 5 |- ( ( ph /\ x e. V ) -> .+ = ( +g ` R ) )
42	41	oveqd 5939	. . . 4 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) = ( .0. ( +g ` R ) x ) )
43	2	eleq2d 2266	. . . . . 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 13163	. . . . 5 |- ( ( R e. Grp /\ x e. ( Base ` R ) ) -> ( .0. ( +g ` R ) x ) = x )
46	6, 44, 45	syl2an2r 595	. . . 4 |- ( ( ph /\ x e. V ) -> ( .0. ( +g ` R ) x ) = x )
47	42, 46	eqtrd 2229	. . 3 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) = x )
48	47	fveq2d 5562	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( .0. .+ x ) ) = ( F ` x ) )
49		eqid 2196	. . . . 5 |- ( invg ` R ) = ( invg ` R )
50	13, 49	grpinvcl 13180	. . . 4 |- ( ( R e. Grp /\ x e. ( Base ` R ) ) -> ( ( invg ` R ) ` x ) e. ( Base ` R ) )
51	6, 44, 50	syl2an2r 595	. . 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 2276	. 2 |- ( ( ph /\ x e. V ) -> ( ( invg ` R ) ` x ) e. V )
54	41	oveqd 5939	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( ( invg ` R ) ` x ) .+ x ) = ( ( ( invg ` R ) ` x ) ( +g ` R ) x ) )
55	13, 14, 37, 49	grplinv 13182	. . . . 5 |- ( ( R e. Grp /\ x e. ( Base ` R ) ) -> ( ( ( invg ` R ) ` x ) ( +g ` R ) x ) = .0. )
56	6, 44, 55	syl2an2r 595	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( ( invg ` R ) ` x ) ( +g ` R ) x ) = .0. )
57	54, 56	eqtrd 2229	. . 3 |- ( ( ph /\ x e. V ) -> ( ( ( invg ` R ) ` x ) .+ x ) = .0. )
58	57	fveq2d 5562	. 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 13240	1 |- ( ph -> ( U e. Grp /\ ( F ` .0. ) = ( 0g ` U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   -onto->wfo 5256   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   0gc0g 12927   "_s cimas 12942   Grpcgrp 13132   invgcminusg 13133

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-mulr 12769  df-0g 12929  df-iimas 12945  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136

This theorem is referenced by:  imasgrpf1  13242  imasabl  13466  imasring  13620