Theorem qusrng 14035

Description: The quotient structure of a non-unital ring is a non-unital ring (qusring2 14143 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 13204	. . . . . 6 |- Base Fn _V
6		qusrng.x	. . . . . . 7 |- ( ph -> R e. Rng )
7	6	elexd 2817	. . . . . 6 |- ( ph -> R e. _V )
8		funfvex 5665	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5439	. . . . . 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 6769	. . . 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 13469	. 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 13470	. 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 14019	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) )
29	18, 22, 26, 28	syl3anc 1274	. . . 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 13477	. 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 14021	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .x. y ) e. ( Base ` R ) )
36	18, 22, 26, 35	syl3anc 1274	. . . 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 13477	. 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 14033	1 |- ( ph -> U e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = 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 13145   +g cplusg 13223   .rcmulr 13224   /._s cqus 13446  Rngcrng 14009

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 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

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 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-3 9245  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-plusg 13236  df-mulr 13237  df-0g 13404  df-iimas 13448  df-qus 13449  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650  df-cmn 13936  df-abl 13937  df-mgp 13998  df-rng 14010

This theorem is referenced by:  qus2idrng  14604