Theorem imasgrp 13050

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 2268	. . . 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 2268	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. ( Base ` R ) )
13		eqid 2189	. . . . 5 |- ( Base ` R ) = ( Base ` R )
14		eqid 2189	. . . . 5 |- ( +g ` R ) = ( +g ` R )
15	13, 14	grpcl 12950	. . . 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 5912	. . 3 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) = ( x ( +g ` R ) y ) )
19	16, 18, 9	3eltr4d 2273	. 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 2268	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z e. ( Base ` R ) )
26	13, 14	grpass 12951	. . . . 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 2190	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z = z )
31	28, 29, 30	oveq123d 5916	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( ( x ( +g ` R ) y ) ( +g ` R ) z ) )
32		eqidd 2190	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> x = x )
33	28	oveqd 5912	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( y .+ z ) = ( y ( +g ` R ) z ) )
34	28, 32, 33	oveq123d 5916	. . . 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 2232	. . 3 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
36	35	fveq2d 5538	. 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 12970	. . . 4 |- ( R e. Grp -> .0. e. ( Base ` R ) )
39	6, 38	syl 14	. . 3 |- ( ph -> .0. e. ( Base ` R ) )
40	39, 2	eleqtrrd 2269	. 2 |- ( ph -> .0. e. V )
41	3	adantr 276	. . . . 5 |- ( ( ph /\ x e. V ) -> .+ = ( +g ` R ) )
42	41	oveqd 5912	. . . 4 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) = ( .0. ( +g ` R ) x ) )
43	2	eleq2d 2259	. . . . . 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 12972	. . . . 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 2222	. . 3 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) = x )
48	47	fveq2d 5538	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( .0. .+ x ) ) = ( F ` x ) )
49		eqid 2189	. . . . 5 |- ( invg ` R ) = ( invg ` R )
50	13, 49	grpinvcl 12989	. . . 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 2269	. 2 |- ( ( ph /\ x e. V ) -> ( ( invg ` R ) ` x ) e. V )
54	41	oveqd 5912	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( ( invg ` R ) ` x ) .+ x ) = ( ( ( invg ` R ) ` x ) ( +g ` R ) x ) )
55	13, 14, 37, 49	grplinv 12991	. . . . 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 2222	. . 3 |- ( ( ph /\ x e. V ) -> ( ( ( invg ` R ) ` x ) .+ x ) = .0. )
58	57	fveq2d 5538	. 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 13049	1 |- ( ph -> ( U e. Grp /\ ( F ` .0. ) = ( 0g ` U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160   -onto->wfo 5233   ` cfv 5235  (class class class)co 5895   Basecbs 12511   +g cplusg 12586   0gc0g 12758   "_s cimas 12773   Grpcgrp 12942   invgcminusg 12943

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-pre-ltirr 7952  ax-pre-lttrn 7954  ax-pre-ltadd 7956

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-tp 3615  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-pnf 8023  df-mnf 8024  df-ltxr 8026  df-inn 8949  df-2 9007  df-3 9008  df-ndx 12514  df-slot 12515  df-base 12517  df-plusg 12599  df-mulr 12600  df-0g 12760  df-iimas 12776  df-mgm 12829  df-sgrp 12862  df-mnd 12875  df-grp 12945  df-minusg 12946

This theorem is referenced by:  imasgrpf1  13051  imasabl  13270  imasring  13411