Theorem qusrng 14257

Description: The quotient structure of a non-unital ring is a non-unital ring (qusring2 14371 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 2238	. . 3 |- ( u e. V |-> [ u ] .~ ) = ( u e. V |-> [ u ] .~ )
4		qusrng.r	. . . 4 |- ( ph -> .~ Er V )
5		basfn 13411	. . . . . 6 |- Base Fn _V
6		qusrng.x	. . . . . . 7 |- ( ph -> R e. Rng )
7	6	elexd 2835	. . . . . 6 |- ( ph -> R e. _V )
8		funfvex 5712	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5483	. . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
10	5, 7, 9	sylancr 418	. . . . 5 |- ( ph -> ( Base ` R ) e. _V )
11	2, 10	eqeltrd 2315	. . . 4 |- ( ph -> V e. _V )
12		erex 6831	. . . 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 13644	. 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 13645	. 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 535	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> x e. V )
20	2	eleq2d 2308	. . . . . . 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 537	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> y e. V )
24	2	eleq2d 2308	. . . . . . 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 2238	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
28	27, 15	rngacl 14241	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) )
29	18, 22, 26, 28	syl3anc 1278	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .+ y ) e. ( Base ` R ) )
30	2	eleq2d 2308	. . . . 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 13652	. 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 14243	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .x. y ) e. ( Base ` R ) )
36	18, 22, 26, 35	syl3anc 1278	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .x. y ) e. ( Base ` R ) )
37	2	eleq2d 2308	. . . . 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 13652	. 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 14255	1 |- ( ph -> U e. Rng )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   |-> cmpt 4192   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   Er wer 6804   [cec 6805   /.cqs 6806   Basecbs 13352   +g cplusg 13431   .rcmulr 13432   /._s cqus 13623  Rngcrng 14231

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-er 6807  df-ec 6809  df-qs 6813  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-0g 13612  df-iimas 13624  df-qus 13625  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-cmn 14089  df-abl 14090  df-mgp 14218  df-rng 14232

This theorem is used by:  qus2idrng  14862