Theorem qusring2 13622

Description: The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
qusring2.u	|- ( ph -> U = ( R /.s .~ ) )
qusring2.v	|- ( ph -> V = ( Base ` R ) )
qusring2.p	|- .+ = ( +g ` R )
qusring2.t	|- .x. = ( .r ` R )
qusring2.o	|- .1. = ( 1r ` R )
qusring2.r	|- ( ph -> .~ Er V )
qusring2.e1	|- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .+ b ) .~ ( p .+ q ) ) )
qusring2.e2	|- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )
qusring2.x	|- ( ph -> R e. Ring )

Assertion

Ref	Expression
qusring2	|- ( ph -> ( U e. Ring /\ [ .1. ] .~ = ( 1r ` U ) ) )

Distinct variable groups:   q, p, .+   .1. , p, q   a, b, p, q, U   V, a, b, p, q   .~ , a, b, p, q   ph, a, b, p, q   .x. , p, q   R, p, q

Allowed substitution hints:   .+ ( a, b)   R( a, b)   .x. ( a, b)   .1. ( a, b)

Proof of Theorem qusring2

Dummy variables u x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusring2.u	. . . 4 |- ( ph -> U = ( R /.s .~ ) )
2		qusring2.v	. . . 4 |- ( ph -> V = ( Base ` R ) )
3		eqid 2196	. . . 4 |- ( u e. V |-> [ u ] .~ ) = ( u e. V |-> [ u ] .~ )
4		qusring2.r	. . . . 5 |- ( ph -> .~ Er V )
5		basfn 12736	. . . . . . 7 |- Base Fn _V
6		qusring2.x	. . . . . . . 8 |- ( ph -> R e. Ring )
7	6	elexd 2776	. . . . . . 7 |- ( ph -> R e. _V )
8		funfvex 5575	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5358	. . . . . . 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 2273	. . . . 5 |- ( ph -> V e. _V )
12		erex 6616	. . . . 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 12966	. . 3 |- ( ph -> U = ( ( u e. V |-> [ u ] .~ ) "s R ) )
15		qusring2.p	. . 3 |- .+ = ( +g ` R )
16		qusring2.t	. . 3 |- .x. = ( .r ` R )
17		qusring2.o	. . 3 |- .1. = ( 1r ` R )
18	1, 2, 3, 13, 6	quslem 12967	. . 3 |- ( ph -> ( u e. V |-> [ u ] .~ ) : V -onto-> ( V /. .~ ) )
19	6	adantr 276	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> R e. Ring )
20		simprl 529	. . . . . . 7 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> x e. V )
21	2	adantr 276	. . . . . . 7 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> V = ( Base ` R ) )
22	20, 21	eleqtrd 2275	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> x e. ( Base ` R ) )
23		simprr 531	. . . . . . 7 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> y e. V )
24	23, 21	eleqtrd 2275	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> y e. ( Base ` R ) )
25		eqid 2196	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
26	25, 15	ringacl 13586	. . . . . 6 |- ( ( R e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) )
27	19, 22, 24, 26	syl3anc 1249	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .+ y ) e. ( Base ` R ) )
28	27, 21	eleqtrrd 2276	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .+ y ) e. V )
29		qusring2.e1	. . . 4 |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .+ b ) .~ ( p .+ q ) ) )
30	4, 11, 3, 28, 29	ercpbl 12974	. . 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 ) ) ) )
31	25, 16	ringcl 13569	. . . . . 6 |- ( ( R e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .x. y ) e. ( Base ` R ) )
32	19, 22, 24, 31	syl3anc 1249	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .x. y ) e. ( Base ` R ) )
33	32, 21	eleqtrrd 2276	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .x. y ) e. V )
34		qusring2.e2	. . . 4 |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )
35	4, 11, 3, 33, 34	ercpbl 12974	. . 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 .x. b ) ) = ( ( u e. V |-> [ u ] .~ ) ` ( p .x. q ) ) ) )
36	14, 2, 15, 16, 17, 18, 30, 35, 6	imasring 13620	. 2 |- ( ph -> ( U e. Ring /\ ( ( u e. V |-> [ u ] .~ ) ` .1. ) = ( 1r ` U ) ) )
37		ringsrg 13603	. . . . . . . 8 |- ( R e. Ring -> R e. SRing )
38	25, 17	srgidcl 13532	. . . . . . . 8 |- ( R e. SRing -> .1. e. ( Base ` R ) )
39	6, 37, 38	3syl 17	. . . . . . 7 |- ( ph -> .1. e. ( Base ` R ) )
40	39, 2	eleqtrrd 2276	. . . . . 6 |- ( ph -> .1. e. V )
41	4, 11, 3, 40	divsfvalg 12972	. . . . 5 |- ( ph -> ( ( u e. V |-> [ u ] .~ ) ` .1. ) = [ .1. ] .~ )
42	41	eqcomd 2202	. . . 4 |- ( ph -> [ .1. ] .~ = ( ( u e. V |-> [ u ] .~ ) ` .1. ) )
43	42	eqeq1d 2205	. . 3 |- ( ph -> ( [ .1. ] .~ = ( 1r ` U ) <-> ( ( u e. V |-> [ u ] .~ ) ` .1. ) = ( 1r ` U ) ) )
44	43	anbi2d 464	. 2 |- ( ph -> ( ( U e. Ring /\ [ .1. ] .~ = ( 1r ` U ) ) <-> ( U e. Ring /\ ( ( u e. V |-> [ u ] .~ ) ` .1. ) = ( 1r ` U ) ) ) )
45	36, 44	mpbird 167	1 |- ( ph -> ( U e. Ring /\ [ .1. ] .~ = ( 1r ` U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   class class class wbr 4033   |-> cmpt 4094   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Er wer 6589   [cec 6590   /.cqs 6591   Basecbs 12678   +g cplusg 12755   .rcmulr 12756   /._s cqus 12943   1rcur 13515  SRingcsrg 13519   Ringcrg 13552

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-er 6592  df-ec 6594  df-qs 6598  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-iimas 12945  df-qus 12946  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-cmn 13416  df-abl 13417  df-mgp 13477  df-ur 13516  df-srg 13520  df-ring 13554

This theorem is referenced by:  qus1  14082