Theorem crnghomfo 16154

Description: The image of a homomorphism from a commutative ring is commutative.

Hypotheses

Ref	Expression
crnghomfo.1	|- G = (1st` R)
crnghomfo.2	|- X = ran G
crnghomfo.3	|- J = (1st` S)
crnghomfo.4	|- Y = ran J

Assertion

Ref	Expression
crnghomfo	|- (((R e. CRing /\ S e. Ring) /\ (F e. (R RngHom S) /\ F:X-onto->Y)) -> S e. CRing)

Proof of Theorem crnghomfo

Step	Hyp	Ref	Expression
1		crnghomfo.3	. . 3 |- J = (1st` S)
2		eqid 1884	. . 3 |- (2nd` S) = (2nd` S)
3		crnghomfo.4	. . 3 |- Y = ran J
4	1, 2, 3	iscring2 16146	. 2 |- (S e. CRing <-> (S e. Ring /\ A.y e. Y A.z e. Y (y(2nd` S)z) = (z(2nd` S)y)))
5		simplr 449	. 2 |- (((R e. CRing /\ S e. Ring) /\ (F e. (R RngHom S) /\ F:X-onto->Y)) -> S e. Ring)
6		foelrn 15685	. . . . . . . 8 |- ((F:X-onto->Y /\ y e. Y) -> E.w e. X y = (F` w))
7	6	ex 402	. . . . . . 7 |- (F:X-onto->Y -> (y e. Y -> E.w e. X y = (F` w)))
8		foelrn 15685	. . . . . . . 8 |- ((F:X-onto->Y /\ z e. Y) -> E.x e. X z = (F` x))
9	8	ex 402	. . . . . . 7 |- (F:X-onto->Y -> (z e. Y -> E.x e. X z = (F` x)))
10	7, 9	anim12d 617	. . . . . 6 |- (F:X-onto->Y -> ((y e. Y /\ z e. Y) -> (E.w e. X y = (F` w) /\ E.x e. X z = (F` x))))
11		reeanv 2249	. . . . . 6 |- (E.w e. X E.x e. X (y = (F` w) /\ z = (F` x)) <-> (E.w e. X y = (F` w) /\ E.x e. X z = (F` x)))
12	10, 11	syl6ibr 230	. . . . 5 |- (F:X-onto->Y -> ((y e. Y /\ z e. Y) -> E.w e. X E.x e. X (y = (F` w) /\ z = (F` x))))
13	12	ad2antll 443	. . . 4 |- (((R e. CRing /\ S e. Ring) /\ (F e. (R RngHom S) /\ F:X-onto->Y)) -> ((y e. Y /\ z e. Y) -> E.w e. X E.x e. X (y = (F` w) /\ z = (F` x))))
14		opreq12 4891	. . . . . . . . . 10 |- ((y = (F` w) /\ z = (F` x)) -> (y(2nd` S)z) = ((F` w)(2nd` S)(F` x)))
15		opreq12 4891	. . . . . . . . . . 11 |- ((z = (F` x) /\ y = (F` w)) -> (z(2nd` S)y) = ((F` x)(2nd` S)(F` w)))
16	15	ancoms 484	. . . . . . . . . 10 |- ((y = (F` w) /\ z = (F` x)) -> (z(2nd` S)y) = ((F` x)(2nd` S)(F` w)))
17	14, 16	eqeq12d 1899	. . . . . . . . 9 |- ((y = (F` w) /\ z = (F` x)) -> ((y(2nd` S)z) = (z(2nd` S)y) <-> ((F` w)(2nd` S)(F` x)) = ((F` x)(2nd` S)(F` w))))
18		crnghomfo.1	. . . . . . . . . . . . . 14 |- G = (1st` R)
19		eqid 1884	. . . . . . . . . . . . . 14 |- (2nd` R) = (2nd` R)
20		crnghomfo.2	. . . . . . . . . . . . . 14 |- X = ran G
21	18, 19, 20	crngcom 16149	. . . . . . . . . . . . 13 |- ((R e. CRing /\ w e. X /\ x e. X) -> (w(2nd` R)x) = (x(2nd` R)w))
22	21	3expb 1068	. . . . . . . . . . . 12 |- ((R e. CRing /\ (w e. X /\ x e. X)) -> (w(2nd` R)x) = (x(2nd` R)w))
23	22	3ad2antl1 1038	. . . . . . . . . . 11 |- (((R e. CRing /\ S e. Ring /\ F e. (R RngHom S)) /\ (w e. X /\ x e. X)) -> (w(2nd` R)x) = (x(2nd` R)w))
24	23	fveq2d 4685	. . . . . . . . . 10 |- (((R e. CRing /\ S e. Ring /\ F e. (R RngHom S)) /\ (w e. X /\ x e. X)) -> (F` (w(2nd` R)x)) = (F` (x(2nd` R)w)))
25	18, 20, 19, 2	rnghommul 16124	. . . . . . . . . . 11 |- (((R e. Ring /\ S e. Ring /\ F e. (R RngHom S)) /\ (w e. X /\ x e. X)) -> (F` (w(2nd` R)x)) = ((F` w)(2nd` S)(F` x)))
26		crngrng 16148	. . . . . . . . . . 11 |- (R e. CRing -> R e. Ring)
27	25, 26	syl3anl1 1145	. . . . . . . . . 10 |- (((R e. CRing /\ S e. Ring /\ F e. (R RngHom S)) /\ (w e. X /\ x e. X)) -> (F` (w(2nd` R)x)) = ((F` w)(2nd` S)(F` x)))
28	18, 20, 19, 2	rnghommul 16124	. . . . . . . . . . . 12 |- (((R e. Ring /\ S e. Ring /\ F e. (R RngHom S)) /\ (x e. X /\ w e. X)) -> (F` (x(2nd` R)w)) = ((F` x)(2nd` S)(F` w)))
29	28	ancom2s 545	. . . . . . . . . . 11 |- (((R e. Ring /\ S e. Ring /\ F e. (R RngHom S)) /\ (w e. X /\ x e. X)) -> (F` (x(2nd` R)w)) = ((F` x)(2nd` S)(F` w)))
30	29, 26	syl3anl1 1145	. . . . . . . . . 10 |- (((R e. CRing /\ S e. Ring /\ F e. (R RngHom S)) /\ (w e. X /\ x e. X)) -> (F` (x(2nd` R)w)) = ((F` x)(2nd` S)(F` w)))
31	24, 27, 30	3eqtr3d 1934	. . . . . . . . 9 |- (((R e. CRing /\ S e. Ring /\ F e. (R RngHom S)) /\ (w e. X /\ x e. X)) -> ((F` w)(2nd` S)(F` x)) = ((F` x)(2nd` S)(F` w)))
32	17, 31	syl5cbir 228	. . . . . . . 8 |- (((R e. CRing /\ S e. Ring /\ F e. (R RngHom S)) /\ (w e. X /\ x e. X)) -> ((y = (F` w) /\ z = (F` x)) -> (y(2nd` S)z) = (z(2nd` S)y)))
33	32	ex 402	. . . . . . 7 |- ((R e. CRing /\ S e. Ring /\ F e. (R RngHom S)) -> ((w e. X /\ x e. X) -> ((y = (F` w) /\ z = (F` x)) -> (y(2nd` S)z) = (z(2nd` S)y))))
34	33	3expa 1067	. . . . . 6 |- (((R e. CRing /\ S e. Ring) /\ F e. (R RngHom S)) -> ((w e. X /\ x e. X) -> ((y = (F` w) /\ z = (F` x)) -> (y(2nd` S)z) = (z(2nd` S)y))))
35	34	adantrr 431	. . . . 5 |- (((R e. CRing /\ S e. Ring) /\ (F e. (R RngHom S) /\ F:X-onto->Y)) -> ((w e. X /\ x e. X) -> ((y = (F` w) /\ z = (F` x)) -> (y(2nd` S)z) = (z(2nd` S)y))))
36	35	r19.23advv 2218	. . . 4 |- (((R e. CRing /\ S e. Ring) /\ (F e. (R RngHom S) /\ F:X-onto->Y)) -> (E.w e. X E.x e. X (y = (F` w) /\ z = (F` x)) -> (y(2nd` S)z) = (z(2nd` S)y)))
37	13, 36	syld 30	. . 3 |- (((R e. CRing /\ S e. Ring) /\ (F e. (R RngHom S) /\ F:X-onto->Y)) -> ((y e. Y /\ z e. Y) -> (y(2nd` S)z) = (z(2nd` S)y)))
38	37	r19.21aivv 2183	. 2 |- (((R e. CRing /\ S e. Ring) /\ (F e. (R RngHom S) /\ F:X-onto->Y)) -> A.y e. Y A.z e. Y (y(2nd` S)z) = (z(2nd` S)y))
39	4, 5, 38	sylanbrc 527	1 |- (((R e. CRing /\ S e. Ring) /\ (F e. (R RngHom S) /\ F:X-onto->Y)) -> S e. CRing)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  ran crn 3987  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Ringcring 9463   RngHom crnghom 16114  CRingccring 16143

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-ring 9464  df-com2 10395  df-rnghom 16117  df-cring 16144

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