Theorem rnghomadd 16123

Description: Ring homomorphisms preserve addition.

Hypotheses

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

Assertion

Ref	Expression
rnghomadd	|- (((R e. Ring /\ S e. Ring /\ F e. (R RngHom S)) /\ (A e. X /\ B e. X)) -> (F` (AGB)) = ((F` A)J(F` B)))

Proof of Theorem rnghomadd

Step	Hyp	Ref	Expression
1		opreq1 4889	. . . . . 6 |- (x = A -> (xGy) = (AGy))
2	1	fveq2d 4685	. . . . 5 |- (x = A -> (F` (xGy)) = (F` (AGy)))
3		fveq2 4681	. . . . . 6 |- (x = A -> (F` x) = (F` A))
4	3	opreq1d 4897	. . . . 5 |- (x = A -> ((F` x)J(F` y)) = ((F` A)J(F` y)))
5	2, 4	eqeq12d 1899	. . . 4 |- (x = A -> ((F` (xGy)) = ((F` x)J(F` y)) <-> (F` (AGy)) = ((F` A)J(F` y))))
6		opreq2 4890	. . . . . 6 |- (y = B -> (AGy) = (AGB))
7	6	fveq2d 4685	. . . . 5 |- (y = B -> (F` (AGy)) = (F` (AGB)))
8		fveq2 4681	. . . . . 6 |- (y = B -> (F` y) = (F` B))
9	8	opreq2d 4898	. . . . 5 |- (y = B -> ((F` A)J(F` y)) = ((F` A)J(F` B)))
10	7, 9	eqeq12d 1899	. . . 4 |- (y = B -> ((F` (AGy)) = ((F` A)J(F` y)) <-> (F` (AGB)) = ((F` A)J(F` B))))
11	5, 10	rcla42v 2384	. . 3 |- ((A e. X /\ B e. X) -> (A.x e. X A.y e. X (F` (xGy)) = ((F` x)J(F` y)) -> (F` (AGB)) = ((F` A)J(F` B))))
12	11	impcom 378	. 2 |- ((A.x e. X A.y e. X (F` (xGy)) = ((F` x)J(F` y)) /\ (A e. X /\ B e. X)) -> (F` (AGB)) = ((F` A)J(F` B)))
13		rnghomadd.1	. . . . . . 7 |- G = (1st` R)
14		eqid 1884	. . . . . . 7 |- (2nd` R) = (2nd` R)
15		rnghomadd.2	. . . . . . 7 |- X = ran G
16		eqid 1884	. . . . . . 7 |- (Id` (2nd` R)) = (Id` (2nd` R))
17		rnghomadd.3	. . . . . . 7 |- J = (1st` S)
18		eqid 1884	. . . . . . 7 |- (2nd` S) = (2nd` S)
19		eqid 1884	. . . . . . 7 |- ran J = ran J
20		eqid 1884	. . . . . . 7 |- (Id` (2nd` S)) = (Id` (2nd` S))
21	13, 14, 15, 16, 17, 18, 19, 20	isrnghom 16119	. . . . . 6 |- ((R e. Ring /\ S e. Ring) -> (F e. (R RngHom S) <-> (F:X-->ran J /\ (F` (Id` (2nd` R))) = (Id` (2nd` S)) /\ A.x e. X A.y e. X ((F` (xGy)) = ((F` x)J(F` y)) /\ (F` (x(2nd` R)y)) = ((F` x)(2nd` S)(F` y))))))
22	21	biimpa 460	. . . . 5 |- (((R e. Ring /\ S e. Ring) /\ F e. (R RngHom S)) -> (F:X-->ran J /\ (F` (Id` (2nd` R))) = (Id` (2nd` S)) /\ A.x e. X A.y e. X ((F` (xGy)) = ((F` x)J(F` y)) /\ (F` (x(2nd` R)y)) = ((F` x)(2nd` S)(F` y)))))
23	22	simp3d 890	. . . 4 |- (((R e. Ring /\ S e. Ring) /\ F e. (R RngHom S)) -> A.x e. X A.y e. X ((F` (xGy)) = ((F` x)J(F` y)) /\ (F` (x(2nd` R)y)) = ((F` x)(2nd` S)(F` y))))
24	23	3impa 1062	. . 3 |- ((R e. Ring /\ S e. Ring /\ F e. (R RngHom S)) -> A.x e. X A.y e. X ((F` (xGy)) = ((F` x)J(F` y)) /\ (F` (x(2nd` R)y)) = ((F` x)(2nd` S)(F` y))))
25		simpl 346	. . . . 5 |- (((F` (xGy)) = ((F` x)J(F` y)) /\ (F` (x(2nd` R)y)) = ((F` x)(2nd` S)(F` y))) -> (F` (xGy)) = ((F` x)J(F` y)))
26	25	ralimi 2168	. . . 4 |- (A.y e. X ((F` (xGy)) = ((F` x)J(F` y)) /\ (F` (x(2nd` R)y)) = ((F` x)(2nd` S)(F` y))) -> A.y e. X (F` (xGy)) = ((F` x)J(F` y)))
27	26	ralimi 2168	. . 3 |- (A.x e. X A.y e. X ((F` (xGy)) = ((F` x)J(F` y)) /\ (F` (x(2nd` R)y)) = ((F` x)(2nd` S)(F` y))) -> A.x e. X A.y e. X (F` (xGy)) = ((F` x)J(F` y)))
28	24, 27	syl 12	. 2 |- ((R e. Ring /\ S e. Ring /\ F e. (R RngHom S)) -> A.x e. X A.y e. X (F` (xGy)) = ((F` x)J(F` y)))
29	12, 28	sylan 497	1 |- (((R e. Ring /\ S e. Ring /\ F e. (R RngHom S)) /\ (A e. X /\ B e. X)) -> (F` (AGB)) = ((F` A)J(F` B)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463   RngHom crnghom 16114

This theorem is referenced by:  rnggrphom 16125  rnghomco 16128  rngisocnv 16135  keridl 16180

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-fv 4014  df-opr 4886  df-oprab 4887  df-rnghom 16117

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