Theorem imasgrp 13562

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 1021	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> R e. Grp )
8		simp2 1001	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. V )
9	2	3ad2ant1 1021	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> V = ( Base ` R ) )
10	8, 9	eleqtrd 2286	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> x e. ( Base ` R ) )
11		simp3 1002	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. V )
12	11, 9	eleqtrd 2286	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> y e. ( Base ` R ) )
13		eqid 2207	. . . . 5 |- ( Base ` R ) = ( Base ` R )
14		eqid 2207	. . . . 5 |- ( +g ` R ) = ( +g ` R )
15	13, 14	grpcl 13455	. . . 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 1250	. . 3 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x ( +g ` R ) y ) e. ( Base ` R ) )
17	3	3ad2ant1 1021	. . . 4 |- ( ( ph /\ x e. V /\ y e. V ) -> .+ = ( +g ` R ) )
18	17	oveqd 5984	. . 3 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) = ( x ( +g ` R ) y ) )
19	16, 18, 9	3eltr4d 2291	. 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 1217	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> x e. ( Base ` R ) )
22	12	3adant3r3 1217	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> y e. ( Base ` R ) )
23		simpr3 1008	. . . . . 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 2286	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z e. ( Base ` R ) )
26	13, 14	grpass 13456	. . . . 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 1252	. . . 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 1217	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( x .+ y ) = ( x ( +g ` R ) y ) )
30		eqidd 2208	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z = z )
31	28, 29, 30	oveq123d 5988	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( ( x ( +g ` R ) y ) ( +g ` R ) z ) )
32		eqidd 2208	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> x = x )
33	28	oveqd 5984	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( y .+ z ) = ( y ( +g ` R ) z ) )
34	28, 32, 33	oveq123d 5988	. . . 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 2250	. . 3 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
36	35	fveq2d 5603	. 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 13476	. . . 4 |- ( R e. Grp -> .0. e. ( Base ` R ) )
39	6, 38	syl 14	. . 3 |- ( ph -> .0. e. ( Base ` R ) )
40	39, 2	eleqtrrd 2287	. 2 |- ( ph -> .0. e. V )
41	3	adantr 276	. . . . 5 |- ( ( ph /\ x e. V ) -> .+ = ( +g ` R ) )
42	41	oveqd 5984	. . . 4 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) = ( .0. ( +g ` R ) x ) )
43	2	eleq2d 2277	. . . . . 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 13478	. . . . 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 2240	. . 3 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) = x )
48	47	fveq2d 5603	. 2 |- ( ( ph /\ x e. V ) -> ( F ` ( .0. .+ x ) ) = ( F ` x ) )
49		eqid 2207	. . . . 5 |- ( invg ` R ) = ( invg ` R )
50	13, 49	grpinvcl 13495	. . . 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 2287	. 2 |- ( ( ph /\ x e. V ) -> ( ( invg ` R ) ` x ) e. V )
54	41	oveqd 5984	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( ( invg ` R ) ` x ) .+ x ) = ( ( ( invg ` R ) ` x ) ( +g ` R ) x ) )
55	13, 14, 37, 49	grplinv 13497	. . . . 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 2240	. . 3 |- ( ( ph /\ x e. V ) -> ( ( ( invg ` R ) ` x ) .+ x ) = .0. )
58	57	fveq2d 5603	. 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 13561	1 |- ( ph -> ( U e. Grp /\ ( F ` .0. ) = ( 0g ` U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2178   -onto->wfo 5288   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   0gc0g 13203   "_s cimas 13246   Grpcgrp 13447   invgcminusg 13448

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-mulr 13038  df-0g 13205  df-iimas 13249  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451

This theorem is referenced by:  imasgrpf1  13563  imasabl  13787  imasring  13941