Theorem qusring2 14210

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 2232	. . . 4 |- ( u e. V |-> [ u ] .~ ) = ( u e. V |-> [ u ] .~ )
4		qusring2.r	. . . . 5 |- ( ph -> .~ Er V )
5		basfn 13271	. . . . . . 7 |- Base Fn _V
6		qusring2.x	. . . . . . . 8 |- ( ph -> R e. Ring )
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		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 13537	. . 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 531	. . . . . . 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 2311	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> x e. ( Base ` R ) )
23		simprr 533	. . . . . . 7 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> y e. V )
24	23, 21	eleqtrd 2311	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> y e. ( Base ` R ) )
25		eqid 2232	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
26	25, 15	ringacl 14174	. . . . . 6 |- ( ( R e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) )
27	19, 22, 24, 26	syl3anc 1274	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .+ y ) e. ( Base ` R ) )
28	27, 21	eleqtrrd 2312	. . . 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 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 ) ) ) )
31	25, 16	ringcl 14157	. . . . . 6 |- ( ( R e. Ring /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .x. y ) e. ( Base ` R ) )
32	19, 22, 24, 31	syl3anc 1274	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .x. y ) e. ( Base ` R ) )
33	32, 21	eleqtrrd 2312	. . . 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 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 .x. b ) ) = ( ( u e. V |-> [ u ] .~ ) ` ( p .x. q ) ) ) )
36	14, 2, 15, 16, 17, 18, 30, 35, 6	imasring 14208	. 2 |- ( ph -> ( U e. Ring /\ ( ( u e. V |-> [ u ] .~ ) ` .1. ) = ( 1r ` U ) ) )
37		ringsrg 14191	. . . . . . . 8 |- ( R e. Ring -> R e. SRing )
38	25, 17	srgidcl 14120	. . . . . . . 8 |- ( R e. SRing -> .1. e. ( Base ` R ) )
39	6, 37, 38	3syl 17	. . . . . . 7 |- ( ph -> .1. e. ( Base ` R ) )
40	39, 2	eleqtrrd 2312	. . . . . 6 |- ( ph -> .1. e. V )
41	4, 11, 3, 40	divsfvalg 13542	. . . . 5 |- ( ph -> ( ( u e. V |-> [ u ] .~ ) ` .1. ) = [ .1. ] .~ )
42	41	eqcomd 2238	. . . 4 |- ( ph -> [ .1. ] .~ = ( ( u e. V |-> [ u ] .~ ) ` .1. ) )
43	42	eqeq1d 2241	. . 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 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   .rcmulr 13291   /._s cqus 13513   1rcur 14103  SRingcsrg 14107   Ringcrg 14140

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-sets 13219  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  df-minusg 13717  df-cmn 14003  df-abl 14004  df-mgp 14065  df-ur 14104  df-srg 14108  df-ring 14142

This theorem is referenced by:  qus1  14674