Theorem qusgrp2 13449

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 2205	. . . 4 |- ( u e. V |-> [ u ] .~ ) = ( u e. V |-> [ u ] .~ )
4		qusgrp2.r	. . . . 5 |- ( ph -> .~ Er V )
5		basfn 12890	. . . . . . 7 |- Base Fn _V
6		qusgrp2.x	. . . . . . . 8 |- ( ph -> R e. X )
7	6	elexd 2785	. . . . . . 7 |- ( ph -> R e. _V )
8		funfvex 5593	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5376	. . . . . . 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 2282	. . . . 5 |- ( ph -> V e. _V )
12		erex 6644	. . . . 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 13155	. . 3 |- ( ph -> U = ( ( u e. V |-> [ u ] .~ ) "s R ) )
15		qusgrp2.p	. . 3 |- ( ph -> .+ = ( +g ` R ) )
16	1, 2, 3, 13, 6	quslem 13156	. . 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 1207	. . . 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 13163	. . 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 6668	. . . 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 6631	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) e. V )
26	21, 24, 3, 25	divsfvalg 13161	. . . 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 6633	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( x .+ ( y .+ z ) ) e. V )
28	21, 24, 3, 27	divsfvalg 13161	. . . 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 2248	. . 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 6668	. . . 4 |- ( ( ph /\ x e. V ) -> [ ( .0. .+ x ) ] .~ = [ x ] .~ )
34	11	adantr 276	. . . . 5 |- ( ( ph /\ x e. V ) -> V e. _V )
35	31, 32	ercl 6631	. . . . 5 |- ( ( ph /\ x e. V ) -> ( .0. .+ x ) e. V )
36	31, 34, 3, 35	divsfvalg 13161	. . . 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 13161	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( u e. V |-> [ u ] .~ ) ` x ) = [ x ] .~ )
39	33, 36, 38	3eqtr4d 2248	. . 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 6632	. . . . 5 |- ( ( ph /\ x e. V ) -> .0. .~ ( N .+ x ) )
43	31, 42	erthi 6668	. . . 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 13161	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( u e. V |-> [ u ] .~ ) ` .0. ) = [ .0. ] .~ )
46	31, 41	ercl 6631	. . . . 5 |- ( ( ph /\ x e. V ) -> ( N .+ x ) e. V )
47	31, 34, 3, 46	divsfvalg 13161	. . . 4 |- ( ( ph /\ x e. V ) -> ( ( u e. V |-> [ u ] .~ ) ` ( N .+ x ) ) = [ ( N .+ x ) ] .~ )
48	43, 45, 47	3eqtr4rd 2249	. . 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 13446	. 2 |- ( ph -> ( U e. Grp /\ ( ( u e. V |-> [ u ] .~ ) ` .0. ) = ( 0g ` U ) ) )
50	4, 11, 3, 30	divsfvalg 13161	. . . . 5 |- ( ph -> ( ( u e. V |-> [ u ] .~ ) ` .0. ) = [ .0. ] .~ )
51	50	eqcomd 2211	. . . 4 |- ( ph -> [ .0. ] .~ = ( ( u e. V |-> [ u ] .~ ) ` .0. ) )
52	51	eqeq1d 2214	. . 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 981   = wceq 1373   e. wcel 2176   _Vcvv 2772   class class class wbr 4044   |-> cmpt 4105   Fn wfn 5266   ` cfv 5271  (class class class)co 5944   Er wer 6617   [cec 6618   /.cqs 6619   Basecbs 12832   +g cplusg 12909   0gc0g 13088   /._s cqus 13132   Grpcgrp 13332

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-tp 3641  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-er 6620  df-ec 6622  df-qs 6626  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-mulr 12923  df-0g 13090  df-iimas 13134  df-qus 13135  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335

This theorem is referenced by:  qusgrp  13568