Theorem qusgrp2 13183

Description: Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
qusgrp2.u	|- ( ph -> U = ( R /.s .~ ) )
qusgrp2.v	|- ( ph -> V = ( Base ` R ) )
qusgrp2.p	|- ( ph -> .+ = ( +g ` R ) )
qusgrp2.r	|- ( ph -> .~ Er V )
qusgrp2.x	|- ( ph -> R e. X )
qusgrp2.e	|- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .+ b ) .~ ( p .+ q ) ) )
qusgrp2.1	|- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. V )
qusgrp2.2	|- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) .~ ( x .+ ( y .+ z ) ) )
qusgrp2.3	|- ( ph -> .0. e. V )
qusgrp2.4	|- ( ( ph /\ x e. V ) -> ( .0. .+ x ) .~ x )
qusgrp2.5	|- ( ( ph /\ x e. V ) -> N e. V )
qusgrp2.6	|- ( ( ph /\ x e. V ) -> ( N .+ x ) .~ .0. )

Assertion

Ref	Expression
qusgrp2	|- ( ph -> ( U e. Grp /\ [ .0. ] .~ = ( 0g ` U ) ) )

Distinct variable groups:   a, b, p, q, x, y, z, .~   .0. , a, b, p, q, x   N, p   R, p, q   .+ , a, b, p, q, x, y   ph, a, b, p, q, x, y, z   V, a, b, p, q, x, y, z   U, a, b, p, q, x, y, z

Allowed substitution hints:   .+ ( z)   R( x, y, z, a, b)   N( x, y, z, q, a, b)   X( x, y, z, q, p, a, b)   .0. ( y, z)

Proof of Theorem qusgrp2

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusgrp2.u	. . . 4 |- ( ph -> U = ( R /.s .~ ) )
2		qusgrp2.v	. . . 4 |- ( ph -> V = ( Base ` R ) )
3		eqid 2193	. . . 4 |- ( u e. V |-> [ u ] .~ ) = ( u e. V |-> [ u ] .~ )
4		qusgrp2.r	. . . . 5 |- ( ph -> .~ Er V )
5		basfn 12676	. . . . . . 7 |- Base Fn _V
6		qusgrp2.x	. . . . . . . 8 |- ( ph -> R e. X )
7	6	elexd 2773	. . . . . . 7 |- ( ph -> R e. _V )
8		funfvex 5571	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5354	. . . . . . 7 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
10	5, 7, 9	sylancr 414	. . . . . 6 |- ( ph -> ( Base ` R ) e. _V )
11	2, 10	eqeltrd 2270	. . . . 5 |- ( ph -> V e. _V )
12		erex 6611	. . . . 5 |- ( .~ Er V -> ( V e. _V -> .~ e. _V ) )
13	4, 11, 12	sylc 62	. . . 4 |- ( ph -> .~ e. _V )
14	1, 2, 3, 13, 6	qusval 12906	. . 3 |- ( ph -> U = ( ( u e. V |-> [ u ] .~ ) "s R ) )
15		qusgrp2.p	. . 3 |- ( ph -> .+ = ( +g ` R ) )
16	1, 2, 3, 13, 6	quslem 12907	. . 3 |- ( ph -> ( u e. V |-> [ u ] .~ ) : V -onto-> ( V /. .~ ) )
17		qusgrp2.1	. . . . 5 |- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. V )
18	17	3expb 1206	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .+ y ) e. V )
19		qusgrp2.e	. . . 4 |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .+ b ) .~ ( p .+ q ) ) )
20	4, 11, 3, 18, 19	ercpbl 12914	. . 3 |- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( ( u e. V |-> [ u ] .~ ) ` a ) = ( ( u e. V |-> [ u ] .~ ) ` p ) /\ ( ( u e. V |-> [ u ] .~ ) ` b ) = ( ( u e. V |-> [ u ] .~ ) ` q ) ) -> ( ( u e. V |-> [ u ] .~ ) ` ( a .+ b ) ) = ( ( u e. V |-> [ u ] .~ ) ` ( p .+ q ) ) ) )
21	4	adantr 276	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> .~ Er V )
22		qusgrp2.2	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) .~ ( x .+ ( y .+ z ) ) )
23	21, 22	erthi 6635	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> [ ( ( x .+ y ) .+ z ) ] .~ = [ ( x .+ ( y .+ z ) ) ] .~ )
24	11	adantr 276	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> V e. _V )
25	21, 22	ercl 6598	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) e. V )
26	21, 24, 3, 25	divsfvalg 12912	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( u e. V |-> [ u ] .~ ) ` ( ( x .+ y ) .+ z ) ) = [ ( ( x .+ y ) .+ z ) ] .~ )
27	21, 22	ercl2 6600	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( x .+ ( y .+ z ) ) e. V )
28	21, 24, 3, 27	divsfvalg 12912	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( u e. V |-> [ u ] .~ ) ` ( x .+ ( y .+ z ) ) ) = [ ( x .+ ( y .+ z ) ) ] .~ )
29	23, 26, 28	3eqtr4d 2236	. . 3 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( u e. V |-> [ u ] .~ ) ` ( ( x .+ y ) .+ z ) ) = ( ( u e. V |-> [ u ] .~ ) ` ( x .+ ( y .+ z ) ) ) )
30		qusgrp2.3	. . 3 |- ( ph -> .0. e. V )
31	4	adantr 276	. . . . 5 |- ( ( ph /\ x e. V ) -> .~ Er V )
32		qusgrp2.4	. . . . 5 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) .~ x )
33	31, 32	erthi 6635	. . . 4 |- ( ( ph /\ x e. V ) -> [ ( .0. .+ x ) ] .~ = [ x ] .~ )
34	11	adantr 276	. . . . 5 |- ( ( ph /\ x e. V ) -> V e. _V )
35	31, 32	ercl 6598	. . . . 5 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) e. V )
36	31, 34, 3, 35	divsfvalg 12912	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( u e. V |-> [ u ] .~ ) ` ( .0. .+ x ) ) = [ ( .0. .+ x ) ] .~ )
37		simpr 110	. . . . 5 |- ( ( ph /\ x e. V ) -> x e. V )
38	31, 34, 3, 37	divsfvalg 12912	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( u e. V |-> [ u ] .~ ) ` x ) = [ x ] .~ )
39	33, 36, 38	3eqtr4d 2236	. . 3 |- ( ( ph /\ x e. V ) -> ( ( u e. V |-> [ u ] .~ ) ` ( .0. .+ x ) ) = ( ( u e. V |-> [ u ] .~ ) ` x ) )
40		qusgrp2.5	. . 3 |- ( ( ph /\ x e. V ) -> N e. V )
41		qusgrp2.6	. . . . . 6 |- ( ( ph /\ x e. V ) -> ( N .+ x ) .~ .0. )
42	31, 41	ersym 6599	. . . . 5 |- ( ( ph /\ x e. V ) -> .0. .~ ( N .+ x ) )
43	31, 42	erthi 6635	. . . 4 |- ( ( ph /\ x e. V ) -> [ .0. ] .~ = [ ( N .+ x ) ] .~ )
44	30	adantr 276	. . . . 5 |- ( ( ph /\ x e. V ) -> .0. e. V )
45	31, 34, 3, 44	divsfvalg 12912	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( u e. V |-> [ u ] .~ ) ` .0. ) = [ .0. ] .~ )
46	31, 41	ercl 6598	. . . . 5 |- ( ( ph /\ x e. V ) -> ( N .+ x ) e. V )
47	31, 34, 3, 46	divsfvalg 12912	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( u e. V |-> [ u ] .~ ) ` ( N .+ x ) ) = [ ( N .+ x ) ] .~ )
48	43, 45, 47	3eqtr4rd 2237	. . 3 |- ( ( ph /\ x e. V ) -> ( ( u e. V |-> [ u ] .~ ) ` ( N .+ x ) ) = ( ( u e. V |-> [ u ] .~ ) ` .0. ) )
49	14, 2, 15, 16, 20, 6, 17, 29, 30, 39, 40, 48	imasgrp2 13180	. 2 |- ( ph -> ( U e. Grp /\ ( ( u e. V |-> [ u ] .~ ) ` .0. ) = ( 0g ` U ) ) )
50	4, 11, 3, 30	divsfvalg 12912	. . . . 5 |- ( ph -> ( ( u e. V |-> [ u ] .~ ) ` .0. ) = [ .0. ] .~ )
51	50	eqcomd 2199	. . . 4 |- ( ph -> [ .0. ] .~ = ( ( u e. V |-> [ u ] .~ ) ` .0. ) )
52	51	eqeq1d 2202	. . 3 |- ( ph -> ( [ .0. ] .~ = ( 0g ` U ) <-> ( ( u e. V |-> [ u ] .~ ) ` .0. ) = ( 0g ` U ) ) )
53	52	anbi2d 464	. 2 |- ( ph -> ( ( U e. Grp /\ [ .0. ] .~ = ( 0g ` U ) ) <-> ( U e. Grp /\ ( ( u e. V |-> [ u ] .~ ) ` .0. ) = ( 0g ` U ) ) ) )
54	49, 53	mpbird 167	1 |- ( ph -> ( U e. Grp /\ [ .0. ] .~ = ( 0g ` U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   _Vcvv 2760   class class class wbr 4029   |-> cmpt 4090   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   Er wer 6584   [cec 6585   /.cqs 6586   Basecbs 12618   +g cplusg 12695   0gc0g 12867   /._s cqus 12883   Grpcgrp 13072

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-er 6587  df-ec 6589  df-qs 6593  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-mulr 12709  df-0g 12869  df-iimas 12885  df-qus 12886  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075

This theorem is referenced by:  qusgrp  13302