Theorem qusring2 13943

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 2207	. . . 4 |- ( u e. V |-> [ u ] .~ ) = ( u e. V |-> [ u ] .~ )
4		qusring2.r	. . . . 5 |- ( ph -> .~ Er V )
5		basfn 13005	. . . . . . 7 |- Base Fn _V
6		qusring2.x	. . . . . . . 8 |- ( ph -> R e. Ring )
7	6	elexd 2790	. . . . . . 7 |- ( ph -> R e. _V )
8		funfvex 5616	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5395	. . . . . . 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 2284	. . . . 5 |- ( ph -> V e. _V )
12		erex 6667	. . . . 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 13270	. . 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 13271	. . 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 2286	. . . . . 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 2286	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> y e. ( Base ` R ) )
25		eqid 2207	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
26	25, 15	ringacl 13907	. . . . . 6 |- ( ( R e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) )
27	19, 22, 24, 26	syl3anc 1250	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .+ y ) e. ( Base ` R ) )
28	27, 21	eleqtrrd 2287	. . . 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 13278	. . 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 13890	. . . . . 6 |- ( ( R e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .x. y ) e. ( Base ` R ) )
32	19, 22, 24, 31	syl3anc 1250	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .x. y ) e. ( Base ` R ) )
33	32, 21	eleqtrrd 2287	. . . 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 13278	. . 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 13941	. 2 |- ( ph -> ( U e. Ring /\ ( ( u e. V |-> [ u ] .~ ) ` .1. ) = ( 1r ` U ) ) )
37		ringsrg 13924	. . . . . . . 8 |- ( R e. Ring -> R e. SRing )
38	25, 17	srgidcl 13853	. . . . . . . 8 |- ( R e. SRing -> .1. e. ( Base ` R ) )
39	6, 37, 38	3syl 17	. . . . . . 7 |- ( ph -> .1. e. ( Base ` R ) )
40	39, 2	eleqtrrd 2287	. . . . . 6 |- ( ph -> .1. e. V )
41	4, 11, 3, 40	divsfvalg 13276	. . . . 5 |- ( ph -> ( ( u e. V |-> [ u ] .~ ) ` .1. ) = [ .1. ] .~ )
42	41	eqcomd 2213	. . . 4 |- ( ph -> [ .1. ] .~ = ( ( u e. V |-> [ u ] .~ ) ` .1. ) )
43	42	eqeq1d 2216	. . 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 1373   e. wcel 2178   _Vcvv 2776   class class class wbr 4059   |-> cmpt 4121   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   Er wer 6640   [cec 6641   /.cqs 6642   Basecbs 12947   +g cplusg 13024   .rcmulr 13025   /._s cqus 13247   1rcur 13836  SRingcsrg 13840   Ringcrg 13873

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-er 6643  df-ec 6645  df-qs 6649  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-sets 12954  df-plusg 13037  df-mulr 13038  df-0g 13205  df-iimas 13249  df-qus 13250  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-cmn 13737  df-abl 13738  df-mgp 13798  df-ur 13837  df-srg 13841  df-ring 13875

This theorem is referenced by:  qus1  14403