Theorem qusrng 13970

Description: The quotient structure of a non-unital ring is a non-unital ring (qusring2 14078 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 2231	. . 3 |- ( u e. V |-> [ u ] .~ ) = ( u e. V |-> [ u ] .~ )
4		qusrng.r	. . . 4 |- ( ph -> .~ Er V )
5		basfn 13140	. . . . . 6 |- Base Fn _V
6		qusrng.x	. . . . . . 7 |- ( ph -> R e. Rng )
7	6	elexd 2816	. . . . . 6 |- ( ph -> R e. _V )
8		funfvex 5656	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5432	. . . . . 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 2308	. . . 4 |- ( ph -> V e. _V )
12		erex 6725	. . . 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 13405	. 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 13406	. 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 531	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> x e. V )
20	2	eleq2d 2301	. . . . . . 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 533	. . . . . 6 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> y e. V )
24	2	eleq2d 2301	. . . . . . 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 2231	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
28	27, 15	rngacl 13954	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) )
29	18, 22, 26, 28	syl3anc 1273	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .+ y ) e. ( Base ` R ) )
30	2	eleq2d 2301	. . . . 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 13413	. 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 13956	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .x. y ) e. ( Base ` R ) )
36	18, 22, 26, 35	syl3anc 1273	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .x. y ) e. ( Base ` R ) )
37	2	eleq2d 2301	. . . . 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 13413	. 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 13968	1 |- ( ph -> U e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   |-> cmpt 4150   Fn wfn 5321   ` cfv 5326  (class class class)co 6017   Er wer 6698   [cec 6699   /.cqs 6700   Basecbs 13081   +g cplusg 13159   .rcmulr 13160   /._s cqus 13382  Rngcrng 13944

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-er 6701  df-ec 6703  df-qs 6707  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-iimas 13384  df-qus 13385  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-cmn 13872  df-abl 13873  df-mgp 13933  df-rng 13945

This theorem is referenced by:  qus2idrng  14538