Theorem qqhghm 30032

Description: The QQHom homomorphism is a group homomorphism if the target structure is a division ring. (Contributed by Thierry Arnoux, 9-Nov-2017.)

Hypotheses

Ref	Expression
qqhval2.0	|- B = ( Base ` R )
qqhval2.1	|- ./ = (/r ` R )
qqhval2.2	|- L = ( ZRHom ` R )
qqhrhm.1	|- Q = (ℂfld ↾s QQ )

Assertion

Ref	Expression
qqhghm	|- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q GrpHom R ) )

Proof of Theorem qqhghm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qqhrhm.1	. . 3 |- Q = (ℂfld ↾s QQ )
2	1	qrngbas 25308	. 2 |- QQ = ( Base ` Q )
3		qqhval2.0	. 2 |- B = ( Base ` R )
4		qex 11800	. . 3 |- QQ e. _V
5		cnfldadd 19751	. . . 4 |- + = ( +g ` ℂfld )
6	1, 5	ressplusg 15993	. . 3 |- ( QQ e. _V -> + = ( +g ` Q ) )
7	4, 6	ax-mp 5	. 2 |- + = ( +g ` Q )
8		eqid 2622	. 2 |- ( +g ` R ) = ( +g ` R )
9	1	qdrng 25309	. . 3 |- Q e. DivRing
10		drnggrp 18755	. . 3 |- ( Q e. DivRing -> Q e. Grp )
11	9, 10	mp1i 13	. 2 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> Q e. Grp )
12		drnggrp 18755	. . 3 |- ( R e. DivRing -> R e. Grp )
13	12	adantr 481	. 2 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> R e. Grp )
14		qqhval2.1	. . 3 |- ./ = (/r ` R )
15		qqhval2.2	. . 3 |- L = ( ZRHom ` R )
16	3, 14, 15	qqhf 30030	. 2 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> B )
17		drngring 18754	. . . . 5 |- ( R e. DivRing -> R e. Ring )
18	17	ad2antrr 762	. . . 4 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> R e. Ring )
19	17	adantr 481	. . . . . . 7 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> R e. Ring )
20	15	zrhrhm 19860	. . . . . . 7 |- ( R e. Ring -> L e. (ℤring RingHom R ) )
21		zringbas 19824	. . . . . . . 8 |- ZZ = ( Base ` ℤring )
22	21, 3	rhmf 18726	. . . . . . 7 |- ( L e. (ℤring RingHom R ) -> L : ZZ --> B )
23	19, 20, 22	3syl 18	. . . . . 6 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> L : ZZ --> B )
24	23	adantr 481	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> L : ZZ --> B )
25		qnumcl 15448	. . . . . . 7 |- ( x e. QQ -> (numer ` x ) e. ZZ )
26	25	ad2antrl 764	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` x ) e. ZZ )
27		qdencl 15449	. . . . . . . 8 |- ( y e. QQ -> (denom ` y ) e. NN )
28	27	ad2antll 765	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) e. NN )
29	28	nnzd 11481	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) e. ZZ )
30	26, 29	zmulcld 11488	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` x ) x. (denom ` y ) ) e. ZZ )
31	24, 30	ffvelrnd 6360	. . . 4 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` ( (numer ` x ) x. (denom ` y ) ) ) e. B )
32		qnumcl 15448	. . . . . . 7 |- ( y e. QQ -> (numer ` y ) e. ZZ )
33	32	ad2antll 765	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` y ) e. ZZ )
34		qdencl 15449	. . . . . . . 8 |- ( x e. QQ -> (denom ` x ) e. NN )
35	34	ad2antrl 764	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. NN )
36	35	nnzd 11481	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. ZZ )
37	33, 36	zmulcld 11488	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` y ) x. (denom ` x ) ) e. ZZ )
38	24, 37	ffvelrnd 6360	. . . 4 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` ( (numer ` y ) x. (denom ` x ) ) ) e. B )
39	18, 20	syl 17	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> L e. (ℤring RingHom R ) )
40		zringmulr 19827	. . . . . . 7 |- x. = ( .r ` ℤring )
41		eqid 2622	. . . . . . 7 |- ( .r ` R ) = ( .r ` R )
42	21, 40, 41	rhmmul 18727	. . . . . 6 |- ( ( L e. (ℤring RingHom R ) /\ (denom ` x ) e. ZZ /\ (denom ` y ) e. ZZ ) -> ( L ` ( (denom ` x ) x. (denom ` y ) ) ) = ( ( L ` (denom ` x ) ) ( .r ` R ) ( L ` (denom ` y ) ) ) )
43	39, 36, 29, 42	syl3anc 1326	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` ( (denom ` x ) x. (denom ` y ) ) ) = ( ( L ` (denom ` x ) ) ( .r ` R ) ( L ` (denom ` y ) ) ) )
44		simpl 473	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( R e. DivRing /\ (chr ` R ) = 0 ) )
45	35	nnne0d 11065	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) =/= 0 )
46		eqid 2622	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
47	3, 15, 46	elzrhunit 30023	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( (denom ` x ) e. ZZ /\ (denom ` x ) =/= 0 ) ) -> ( L ` (denom ` x ) ) e. (Unit ` R ) )
48	44, 36, 45, 47	syl12anc 1324	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (denom ` x ) ) e. (Unit ` R ) )
49	28	nnne0d 11065	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) =/= 0 )
50	3, 15, 46	elzrhunit 30023	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( (denom ` y ) e. ZZ /\ (denom ` y ) =/= 0 ) ) -> ( L ` (denom ` y ) ) e. (Unit ` R ) )
51	44, 29, 49, 50	syl12anc 1324	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (denom ` y ) ) e. (Unit ` R ) )
52		eqid 2622	. . . . . . 7 |- (Unit ` R ) = (Unit ` R )
53	52, 41	unitmulcl 18664	. . . . . 6 |- ( ( R e. Ring /\ ( L ` (denom ` x ) ) e. (Unit ` R ) /\ ( L ` (denom ` y ) ) e. (Unit ` R ) ) -> ( ( L ` (denom ` x ) ) ( .r ` R ) ( L ` (denom ` y ) ) ) e. (Unit ` R ) )
54	18, 48, 51, 53	syl3anc 1326	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( L ` (denom ` x ) ) ( .r ` R ) ( L ` (denom ` y ) ) ) e. (Unit ` R ) )
55	43, 54	eqeltrd 2701	. . . 4 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` ( (denom ` x ) x. (denom ` y ) ) ) e. (Unit ` R ) )
56	3, 52, 8, 14	dvrdir 29790	. . . 4 |- ( ( R e. Ring /\ ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) e. B /\ ( L ` ( (numer ` y ) x. (denom ` x ) ) ) e. B /\ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) e. (Unit ` R ) ) ) -> ( ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) ( +g ` R ) ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) = ( ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) ( +g ` R ) ( ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) ) )
57	18, 31, 38, 55, 56	syl13anc 1328	. . 3 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) ( +g ` R ) ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) = ( ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) ( +g ` R ) ( ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) ) )
58		qeqnumdivden 15454	. . . . . . . 8 |- ( x e. QQ -> x = ( (numer ` x ) / (denom ` x ) ) )
59	58	ad2antrl 764	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> x = ( (numer ` x ) / (denom ` x ) ) )
60		qeqnumdivden 15454	. . . . . . . 8 |- ( y e. QQ -> y = ( (numer ` y ) / (denom ` y ) ) )
61	60	ad2antll 765	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> y = ( (numer ` y ) / (denom ` y ) ) )
62	59, 61	oveq12d 6668	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( x + y ) = ( ( (numer ` x ) / (denom ` x ) ) + ( (numer ` y ) / (denom ` y ) ) ) )
63	26	zcnd 11483	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` x ) e. CC )
64	36	zcnd 11483	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. CC )
65	33	zcnd 11483	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` y ) e. CC )
66	29	zcnd 11483	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) e. CC )
67	63, 64, 65, 66, 45, 49	divadddivd 10845	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( (numer ` x ) / (denom ` x ) ) + ( (numer ` y ) / (denom ` y ) ) ) = ( ( ( (numer ` x ) x. (denom ` y ) ) + ( (numer ` y ) x. (denom ` x ) ) ) / ( (denom ` x ) x. (denom ` y ) ) ) )
68	62, 67	eqtrd 2656	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( x + y ) = ( ( ( (numer ` x ) x. (denom ` y ) ) + ( (numer ` y ) x. (denom ` x ) ) ) / ( (denom ` x ) x. (denom ` y ) ) ) )
69	68	fveq2d 6195	. . . 4 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( x + y ) ) = ( (QQHom ` R ) ` ( ( ( (numer ` x ) x. (denom ` y ) ) + ( (numer ` y ) x. (denom ` x ) ) ) / ( (denom ` x ) x. (denom ` y ) ) ) ) )
70	30, 37	zaddcld 11486	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( (numer ` x ) x. (denom ` y ) ) + ( (numer ` y ) x. (denom ` x ) ) ) e. ZZ )
71	36, 29	zmulcld 11488	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (denom ` x ) x. (denom ` y ) ) e. ZZ )
72	64, 66, 45, 49	mulne0d 10679	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (denom ` x ) x. (denom ` y ) ) =/= 0 )
73	3, 14, 15	qqhvq 30031	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( ( ( (numer ` x ) x. (denom ` y ) ) + ( (numer ` y ) x. (denom ` x ) ) ) e. ZZ /\ ( (denom ` x ) x. (denom ` y ) ) e. ZZ /\ ( (denom ` x ) x. (denom ` y ) ) =/= 0 ) ) -> ( (QQHom ` R ) ` ( ( ( (numer ` x ) x. (denom ` y ) ) + ( (numer ` y ) x. (denom ` x ) ) ) / ( (denom ` x ) x. (denom ` y ) ) ) ) = ( ( L ` ( ( (numer ` x ) x. (denom ` y ) ) + ( (numer ` y ) x. (denom ` x ) ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
74	44, 70, 71, 72, 73	syl13anc 1328	. . . 4 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( ( ( (numer ` x ) x. (denom ` y ) ) + ( (numer ` y ) x. (denom ` x ) ) ) / ( (denom ` x ) x. (denom ` y ) ) ) ) = ( ( L ` ( ( (numer ` x ) x. (denom ` y ) ) + ( (numer ` y ) x. (denom ` x ) ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
75		rhmghm 18725	. . . . . 6 |- ( L e. (ℤring RingHom R ) -> L e. (ℤring GrpHom R ) )
76	39, 75	syl 17	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> L e. (ℤring GrpHom R ) )
77		zringplusg 19825	. . . . . . 7 |- + = ( +g ` ℤring )
78	21, 77, 8	ghmlin 17665	. . . . . 6 |- ( ( L e. (ℤring GrpHom R ) /\ ( (numer ` x ) x. (denom ` y ) ) e. ZZ /\ ( (numer ` y ) x. (denom ` x ) ) e. ZZ ) -> ( L ` ( ( (numer ` x ) x. (denom ` y ) ) + ( (numer ` y ) x. (denom ` x ) ) ) ) = ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) ( +g ` R ) ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ) )
79	78	oveq1d 6665	. . . . 5 |- ( ( L e. (ℤring GrpHom R ) /\ ( (numer ` x ) x. (denom ` y ) ) e. ZZ /\ ( (numer ` y ) x. (denom ` x ) ) e. ZZ ) -> ( ( L ` ( ( (numer ` x ) x. (denom ` y ) ) + ( (numer ` y ) x. (denom ` x ) ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) = ( ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) ( +g ` R ) ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
80	76, 30, 37, 79	syl3anc 1326	. . . 4 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( L ` ( ( (numer ` x ) x. (denom ` y ) ) + ( (numer ` y ) x. (denom ` x ) ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) = ( ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) ( +g ` R ) ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
81	69, 74, 80	3eqtrd 2660	. . 3 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( x + y ) ) = ( ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) ( +g ` R ) ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
82	58	fveq2d 6195	. . . . . 6 |- ( x e. QQ -> ( (QQHom ` R ) ` x ) = ( (QQHom ` R ) ` ( (numer ` x ) / (denom ` x ) ) ) )
83	82	ad2antrl 764	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` x ) = ( (QQHom ` R ) ` ( (numer ` x ) / (denom ` x ) ) ) )
84	3, 14, 15	qqhvq 30031	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( (numer ` x ) e. ZZ /\ (denom ` x ) e. ZZ /\ (denom ` x ) =/= 0 ) ) -> ( (QQHom ` R ) ` ( (numer ` x ) / (denom ` x ) ) ) = ( ( L ` (numer ` x ) ) ./ ( L ` (denom ` x ) ) ) )
85	44, 26, 36, 45, 84	syl13anc 1328	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( (numer ` x ) / (denom ` x ) ) ) = ( ( L ` (numer ` x ) ) ./ ( L ` (denom ` x ) ) ) )
86	52, 21, 14, 40	rhmdvd 29821	. . . . . 6 |- ( ( L e. (ℤring RingHom R ) /\ ( (numer ` x ) e. ZZ /\ (denom ` x ) e. ZZ /\ (denom ` y ) e. ZZ ) /\ ( ( L ` (denom ` x ) ) e. (Unit ` R ) /\ ( L ` (denom ` y ) ) e. (Unit ` R ) ) ) -> ( ( L ` (numer ` x ) ) ./ ( L ` (denom ` x ) ) ) = ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
87	39, 26, 36, 29, 48, 51, 86	syl132anc 1344	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( L ` (numer ` x ) ) ./ ( L ` (denom ` x ) ) ) = ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
88	83, 85, 87	3eqtrd 2660	. . . 4 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` x ) = ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
89	60	fveq2d 6195	. . . . . 6 |- ( y e. QQ -> ( (QQHom ` R ) ` y ) = ( (QQHom ` R ) ` ( (numer ` y ) / (denom ` y ) ) ) )
90	89	ad2antll 765	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` y ) = ( (QQHom ` R ) ` ( (numer ` y ) / (denom ` y ) ) ) )
91	52, 21, 14, 40	rhmdvd 29821	. . . . . . 7 |- ( ( L e. (ℤring RingHom R ) /\ ( (numer ` y ) e. ZZ /\ (denom ` y ) e. ZZ /\ (denom ` x ) e. ZZ ) /\ ( ( L ` (denom ` y ) ) e. (Unit ` R ) /\ ( L ` (denom ` x ) ) e. (Unit ` R ) ) ) -> ( ( L ` (numer ` y ) ) ./ ( L ` (denom ` y ) ) ) = ( ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ./ ( L ` ( (denom ` y ) x. (denom ` x ) ) ) ) )
92	39, 33, 29, 36, 51, 48, 91	syl132anc 1344	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( L ` (numer ` y ) ) ./ ( L ` (denom ` y ) ) ) = ( ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ./ ( L ` ( (denom ` y ) x. (denom ` x ) ) ) ) )
93	3, 14, 15	qqhvq 30031	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( (numer ` y ) e. ZZ /\ (denom ` y ) e. ZZ /\ (denom ` y ) =/= 0 ) ) -> ( (QQHom ` R ) ` ( (numer ` y ) / (denom ` y ) ) ) = ( ( L ` (numer ` y ) ) ./ ( L ` (denom ` y ) ) ) )
94	44, 33, 29, 49, 93	syl13anc 1328	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( (numer ` y ) / (denom ` y ) ) ) = ( ( L ` (numer ` y ) ) ./ ( L ` (denom ` y ) ) ) )
95	64, 66	mulcomd 10061	. . . . . . . 8 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (denom ` x ) x. (denom ` y ) ) = ( (denom ` y ) x. (denom ` x ) ) )
96	95	fveq2d 6195	. . . . . . 7 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` ( (denom ` x ) x. (denom ` y ) ) ) = ( L ` ( (denom ` y ) x. (denom ` x ) ) ) )
97	96	oveq2d 6666	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) = ( ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ./ ( L ` ( (denom ` y ) x. (denom ` x ) ) ) ) )
98	92, 94, 97	3eqtr4d 2666	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( (numer ` y ) / (denom ` y ) ) ) = ( ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
99	90, 98	eqtrd 2656	. . . 4 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` y ) = ( ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
100	88, 99	oveq12d 6668	. . 3 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( (QQHom ` R ) ` x ) ( +g ` R ) ( (QQHom ` R ) ` y ) ) = ( ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) ( +g ` R ) ( ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) ) )
101	57, 81, 100	3eqtr4d 2666	. 2 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( x + y ) ) = ( ( (QQHom ` R ) ` x ) ( +g ` R ) ( (QQHom ` R ) ` y ) ) )
102	2, 3, 7, 8, 11, 13, 16, 101	isghmd 17669	1 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q GrpHom R ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   + caddc 9939   x. cmul 9941   / cdiv 10684   NNcn 11020   ZZcz 11377   QQcq 11788  numercnumer 15441  denomcdenom 15442   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942   0gc0g 16100   Grpcgrp 17422   GrpHom cghm 17657   Ringcrg 18547  Unitcui 18639  /_rcdvr 18682   RingHom crh 18712   DivRingcdr 18747  ℂ_fldccnfld 19746  ℤ_ringzring 19818   ZRHomczrh 19848  chrcchr 19850  QQHomcqqh 30016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-numer 15443  df-denom 15444  df-gz 15634  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-od 17948  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-subrg 18778  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-chr 19854  df-qqh 30017

This theorem is referenced by:  qqhcn  30035  qqhucn  30036