Theorem qusrng 13457

Description: The quotient structure of a non-unital ring is a non-unital ring (qusring2 13565 analog). (Contributed by AV, 23-Feb-2025.)

Hypotheses

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

Assertion

Ref	Expression
qusrng	|- ( ph -> U e. Rng )

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

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

Proof of Theorem qusrng

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

Step	Hyp	Ref	Expression
1		qusrng.u	. . 3 |- ( ph -> U = ( R /.s .~ ) )
2		qusrng.v	. . 3 |- ( ph -> V = ( Base ` R ) )
3		eqid 2193	. . 3 |- ( u e. V |-> [ u ] .~ ) = ( u e. V |-> [ u ] .~ )
4		qusrng.r	. . . 4 |- ( ph -> .~ Er V )
5		basfn 12679	. . . . . 6 |- Base Fn _V
6		qusrng.x	. . . . . . 7 |- ( ph -> R e. Rng )
7	6	elexd 2773	. . . . . 6 |- ( ph -> R e. _V )
8		funfvex 5572	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5355	. . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
10	5, 7, 9	sylancr 414	. . . . 5 |- ( ph -> ( Base ` R ) e. _V )
11	2, 10	eqeltrd 2270	. . . 4 |- ( ph -> V e. _V )
12		erex 6613	. . . 4 |- ( .~ Er V -> ( V e. _V -> .~ e. _V ) )
13	4, 11, 12	sylc 62	. . 3 |- ( ph -> .~ e. _V )
14	1, 2, 3, 13, 6	qusval 12909	. 2 |- ( ph -> U = ( ( u e. V |-> [ u ] .~ ) "s R ) )
15		qusrng.p	. 2 |- .+ = ( +g ` R )
16		qusrng.t	. 2 |- .x. = ( .r ` R )
17	1, 2, 3, 13, 6	quslem 12910	. 2 |- ( ph -> ( u e. V |-> [ u ] .~ ) : V -onto-> ( V /. .~ ) )
18	6	adantr 276	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> R e. Rng )
19		simprl 529	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> x e. V )
20	2	eleq2d 2263	. . . . . . 7 |- ( ph -> ( x e. V <-> x e. ( Base ` R ) ) )
21	20	adantr 276	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x e. V <-> x e. ( Base ` R ) ) )
22	19, 21	mpbid 147	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> x e. ( Base ` R ) )
23		simprr 531	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> y e. V )
24	2	eleq2d 2263	. . . . . . 7 |- ( ph -> ( y e. V <-> y e. ( Base ` R ) ) )
25	24	adantr 276	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( y e. V <-> y e. ( Base ` R ) ) )
26	23, 25	mpbid 147	. . . . 5 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> y e. ( Base ` R ) )
27		eqid 2193	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
28	27, 15	rngacl 13441	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) )
29	18, 22, 26, 28	syl3anc 1249	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .+ y ) e. ( Base ` R ) )
30	2	eleq2d 2263	. . . . 5 |- ( ph -> ( ( x .+ y ) e. V <-> ( x .+ y ) e. ( Base ` R ) ) )
31	30	adantr 276	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( ( x .+ y ) e. V <-> ( x .+ y ) e. ( Base ` R ) ) )
32	29, 31	mpbird 167	. . 3 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .+ y ) e. V )
33		qusrng.e1	. . 3 |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .+ b ) .~ ( p .+ q ) ) )
34	4, 11, 3, 32, 33	ercpbl 12917	. 2 |- ( ( 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 ) ) ) )
35	27, 16	rngcl 13443	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .x. y ) e. ( Base ` R ) )
36	18, 22, 26, 35	syl3anc 1249	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .x. y ) e. ( Base ` R ) )
37	2	eleq2d 2263	. . . . 5 |- ( ph -> ( ( x .x. y ) e. V <-> ( x .x. y ) e. ( Base ` R ) ) )
38	37	adantr 276	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( ( x .x. y ) e. V <-> ( x .x. y ) e. ( Base ` R ) ) )
39	36, 38	mpbird 167	. . 3 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .x. y ) e. V )
40		qusrng.e2	. . 3 |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )
41	4, 11, 3, 39, 40	ercpbl 12917	. 2 |- ( ( 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 ) ) ) )
42	14, 2, 15, 16, 17, 34, 41, 6	imasrng 13455	1 |- ( ph -> U e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   _Vcvv 2760   class class class wbr 4030   |-> cmpt 4091   Fn wfn 5250   ` cfv 5255  (class class class)co 5919   Er wer 6586   [cec 6587   /.cqs 6588   Basecbs 12621   +g cplusg 12698   .rcmulr 12699   /._s cqus 12886  Rngcrng 13431

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-tp 3627  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-er 6589  df-ec 6591  df-qs 6595  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-iimas 12888  df-qus 12889  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-cmn 13359  df-abl 13360  df-mgp 13420  df-rng 13432

This theorem is referenced by:  qus2idrng  14024