Theorem qusgrp2 13830

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 2232	. . . 4 |- ( u e. V |-> [ u ] .~ ) = ( u e. V |-> [ u ] .~ )
4		qusgrp2.r	. . . . 5 |- ( ph -> .~ Er V )
5		basfn 13271	. . . . . . 7 |- Base Fn _V
6		qusgrp2.x	. . . . . . . 8 |- ( ph -> R e. X )
7	6	elexd 2827	. . . . . . 7 |- ( ph -> R e. _V )
8		funfvex 5687	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5458	. . . . . . 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 2309	. . . . 5 |- ( ph -> V e. _V )
12		erex 6791	. . . . 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 13536	. . 3 |- ( ph -> U = ( ( u e. V |-> [ u ] .~ ) "s R ) )
15		qusgrp2.p	. . 3 |- ( ph -> .+ = ( +g ` R ) )
16	1, 2, 3, 13, 6	quslem 13537	. . 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 1231	. . . 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 13544	. . 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 6815	. . . 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 6778	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) e. V )
26	21, 24, 3, 25	divsfvalg 13542	. . . 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 6780	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( x .+ ( y .+ z ) ) e. V )
28	21, 24, 3, 27	divsfvalg 13542	. . . 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 2275	. . 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 6815	. . . 4 |- ( ( ph /\ x e. V ) -> [ ( .0. .+ x ) ] .~ = [ x ] .~ )
34	11	adantr 276	. . . . 5 |- ( ( ph /\ x e. V ) -> V e. _V )
35	31, 32	ercl 6778	. . . . 5 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) e. V )
36	31, 34, 3, 35	divsfvalg 13542	. . . 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 13542	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( u e. V |-> [ u ] .~ ) ` x ) = [ x ] .~ )
39	33, 36, 38	3eqtr4d 2275	. . 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 6779	. . . . 5 |- ( ( ph /\ x e. V ) -> .0. .~ ( N .+ x ) )
43	31, 42	erthi 6815	. . . 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 13542	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( u e. V |-> [ u ] .~ ) ` .0. ) = [ .0. ] .~ )
46	31, 41	ercl 6778	. . . . 5 |- ( ( ph /\ x e. V ) -> ( N .+ x ) e. V )
47	31, 34, 3, 46	divsfvalg 13542	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( u e. V |-> [ u ] .~ ) ` ( N .+ x ) ) = [ ( N .+ x ) ] .~ )
48	43, 45, 47	3eqtr4rd 2276	. . 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 13827	. 2 |- ( ph -> ( U e. Grp /\ ( ( u e. V |-> [ u ] .~ ) ` .0. ) = ( 0g ` U ) ) )
50	4, 11, 3, 30	divsfvalg 13542	. . . . 5 |- ( ph -> ( ( u e. V |-> [ u ] .~ ) ` .0. ) = [ .0. ] .~ )
51	50	eqcomd 2238	. . . 4 |- ( ph -> [ .0. ] .~ = ( ( u e. V |-> [ u ] .~ ) ` .0. ) )
52	51	eqeq1d 2241	. . 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 1005   = wceq 1398   e. wcel 2203   _Vcvv 2813   class class class wbr 4109   |-> cmpt 4171   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   Er wer 6764   [cec 6765   /.cqs 6766   Basecbs 13212   +g cplusg 13290   0gc0g 13469   /._s cqus 13513   Grpcgrp 13713

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-tp 3697  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-er 6767  df-ec 6769  df-qs 6773  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-mulr 13304  df-0g 13471  df-iimas 13515  df-qus 13516  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716

This theorem is referenced by:  qusgrp  13949