Theorem qusrng 13795

Description: The quotient structure of a non-unital ring is a non-unital ring (qusring2 13903 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 2206	. . 3 |- ( u e. V |-> [ u ] .~ ) = ( u e. V |-> [ u ] .~ )
4		qusrng.r	. . . 4 |- ( ph -> .~ Er V )
5		basfn 12965	. . . . . 6 |- Base Fn _V
6		qusrng.x	. . . . . . 7 |- ( ph -> R e. Rng )
7	6	elexd 2787	. . . . . 6 |- ( ph -> R e. _V )
8		funfvex 5606	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
9	8	funfni 5385	. . . . . 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 2283	. . . 4 |- ( ph -> V e. _V )
12		erex 6657	. . . 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 13230	. 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 13231	. 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 2276	. . . . . . 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 2276	. . . . . . 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 2206	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
28	27, 15	rngacl 13779	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) )
29	18, 22, 26, 28	syl3anc 1250	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .+ y ) e. ( Base ` R ) )
30	2	eleq2d 2276	. . . . 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 13238	. 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 13781	. . . . 5 |- ( ( R e. Rng /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .x. y ) e. ( Base ` R ) )
36	18, 22, 26, 35	syl3anc 1250	. . . 4 |- ( ( ph /\ ( x e. V /\ y e. V ) ) -> ( x .x. y ) e. ( Base ` R ) )
37	2	eleq2d 2276	. . . . 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 13238	. 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 13793	1 |- ( ph -> U e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2177   _Vcvv 2773   class class class wbr 4051   |-> cmpt 4113   Fn wfn 5275   ` cfv 5280  (class class class)co 5957   Er wer 6630   [cec 6631   /.cqs 6632   Basecbs 12907   +g cplusg 12984   .rcmulr 12985   /._s cqus 13207  Rngcrng 13769

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-pre-ltirr 8057  ax-pre-lttrn 8059  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-tp 3646  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-er 6633  df-ec 6635  df-qs 6639  df-pnf 8129  df-mnf 8130  df-ltxr 8132  df-inn 9057  df-2 9115  df-3 9116  df-ndx 12910  df-slot 12911  df-base 12913  df-sets 12914  df-plusg 12997  df-mulr 12998  df-0g 13165  df-iimas 13209  df-qus 13210  df-mgm 13263  df-sgrp 13309  df-mnd 13324  df-grp 13410  df-minusg 13411  df-cmn 13697  df-abl 13698  df-mgp 13758  df-rng 13770

This theorem is referenced by:  qus2idrng  14362