Theorem qusring2 14160

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 2231	. . . 4 |- ( u e. V |-> [ u ] .~ ) = ( u e. V |-> [ u ] .~ )
4		qusring2.r	. . . . 5 |- ( ph -> .~ Er V )
5		basfn 13221	. . . . . . 7 |- Base Fn _V
6		qusring2.x	. . . . . . . 8 |- ( ph -> R e. Ring )
7	6	elexd 2817	. . . . . . 7 |- ( ph -> R e. _V )
8		funfvex 5665	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5439	. . . . . . 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 2308	. . . . 5 |- ( ph -> V e. _V )
12		erex 6769	. . . . 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 13486	. . 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 13487	. . 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 2310	. . . . . 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 2310	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> y e. ( Base ` R ) )
25		eqid 2231	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
26	25, 15	ringacl 14124	. . . . . 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 2311	. . . 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 13494	. . 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 14107	. . . . . 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 2311	. . . 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 13494	. . 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 14158	. 2 |- ( ph -> ( U e. Ring /\ ( ( u e. V |-> [ u ] .~ ) ` .1. ) = ( 1r ` U ) ) )
37		ringsrg 14141	. . . . . . . 8 |- ( R e. Ring -> R e. SRing )
38	25, 17	srgidcl 14070	. . . . . . . 8 |- ( R e. SRing -> .1. e. ( Base ` R ) )
39	6, 37, 38	3syl 17	. . . . . . 7 |- ( ph -> .1. e. ( Base ` R ) )
40	39, 2	eleqtrrd 2311	. . . . . 6 |- ( ph -> .1. e. V )
41	4, 11, 3, 40	divsfvalg 13492	. . . . 5 |- ( ph -> ( ( u e. V |-> [ u ] .~ ) ` .1. ) = [ .1. ] .~ )
42	41	eqcomd 2237	. . . 4 |- ( ph -> [ .1. ] .~ = ( ( u e. V |-> [ u ] .~ ) ` .1. ) )
43	42	eqeq1d 2240	. . 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 2202   _Vcvv 2803   class class class wbr 4093   |-> cmpt 4155   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   Er wer 6742   [cec 6743   /.cqs 6744   Basecbs 13162   +g cplusg 13240   .rcmulr 13241   /._s cqus 13463   1rcur 14053  SRingcsrg 14057   Ringcrg 14090

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-er 6745  df-ec 6747  df-qs 6751  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-plusg 13253  df-mulr 13254  df-0g 13421  df-iimas 13465  df-qus 13466  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-cmn 13953  df-abl 13954  df-mgp 14015  df-ur 14054  df-srg 14058  df-ring 14092

This theorem is referenced by:  qus1  14622