Theorem qqhrhm 30033

Description: The QQHom homomorphism is a ring homomorphism if the target structure is a field. If the target structure is a division ring, it is a group homomorphism, but not a ring homomorphism, because it does not preserve the ring multiplication operation. (Contributed by Thierry Arnoux, 29-Oct-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
qqhrhm	|- ( ( R e. Field /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q RingHom R ) )

Proof of Theorem qqhrhm

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	1	qrng1 25311	. 2 |- 1 = ( 1r ` Q )
4		eqid 2622	. 2 |- ( 1r ` R ) = ( 1r ` R )
5		qex 11800	. . 3 |- QQ e. _V
6		cnfldmul 19752	. . . 4 |- x. = ( .r ` ℂfld )
7	1, 6	ressmulr 16006	. . 3 |- ( QQ e. _V -> x. = ( .r ` Q ) )
8	5, 7	ax-mp 5	. 2 |- x. = ( .r ` Q )
9		eqid 2622	. 2 |- ( .r ` R ) = ( .r ` R )
10	1	qdrng 25309	. . 3 |- Q e. DivRing
11		drngring 18754	. . 3 |- ( Q e. DivRing -> Q e. Ring )
12	10, 11	mp1i 13	. 2 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> Q e. Ring )
13		isfld 18756	. . . . 5 |- ( R e. Field <-> ( R e. DivRing /\ R e. CRing ) )
14	13	simplbi 476	. . . 4 |- ( R e. Field -> R e. DivRing )
15	14	adantr 481	. . 3 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> R e. DivRing )
16		drngring 18754	. . 3 |- ( R e. DivRing -> R e. Ring )
17	15, 16	syl 17	. 2 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> R e. Ring )
18		qqhval2.0	. . . 4 |- B = ( Base ` R )
19		qqhval2.1	. . . 4 |- ./ = (/r ` R )
20		qqhval2.2	. . . 4 |- L = ( ZRHom ` R )
21	18, 19, 20	qqh1 30029	. . 3 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( (QQHom ` R ) ` 1 ) = ( 1r ` R ) )
22	14, 21	sylan 488	. 2 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> ( (QQHom ` R ) ` 1 ) = ( 1r ` R ) )
23		eqid 2622	. . . 4 |- (Unit ` R ) = (Unit ` R )
24		eqid 2622	. . . 4 |- ( +g ` R ) = ( +g ` R )
25	13	simprbi 480	. . . . 5 |- ( R e. Field -> R e. CRing )
26	25	ad2antrr 762	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> R e. CRing )
27	20	zrhrhm 19860	. . . . . . 7 |- ( R e. Ring -> L e. (ℤring RingHom R ) )
28		zringbas 19824	. . . . . . . 8 |- ZZ = ( Base ` ℤring )
29	28, 18	rhmf 18726	. . . . . . 7 |- ( L e. (ℤring RingHom R ) -> L : ZZ --> B )
30	17, 27, 29	3syl 18	. . . . . 6 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> L : ZZ --> B )
31	30	adantr 481	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> L : ZZ --> B )
32		qnumcl 15448	. . . . . 6 |- ( x e. QQ -> (numer ` x ) e. ZZ )
33	32	ad2antrl 764	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` x ) e. ZZ )
34	31, 33	ffvelrnd 6360	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (numer ` x ) ) e. B )
35	14	ad2antrr 762	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> R e. DivRing )
36		simplr 792	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (chr ` R ) = 0 )
37	35, 36	jca 554	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( R e. DivRing /\ (chr ` R ) = 0 ) )
38		qdencl 15449	. . . . . . 7 |- ( x e. QQ -> (denom ` x ) e. NN )
39	38	ad2antrl 764	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. NN )
40	39	nnzd 11481	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. ZZ )
41	39	nnne0d 11065	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) =/= 0 )
42		eqid 2622	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
43	18, 20, 42	elzrhunit 30023	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( (denom ` x ) e. ZZ /\ (denom ` x ) =/= 0 ) ) -> ( L ` (denom ` x ) ) e. (Unit ` R ) )
44	37, 40, 41, 43	syl12anc 1324	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (denom ` x ) ) e. (Unit ` R ) )
45		qnumcl 15448	. . . . . 6 |- ( y e. QQ -> (numer ` y ) e. ZZ )
46	45	ad2antll 765	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` y ) e. ZZ )
47	31, 46	ffvelrnd 6360	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (numer ` y ) ) e. B )
48		qdencl 15449	. . . . . . 7 |- ( y e. QQ -> (denom ` y ) e. NN )
49	48	ad2antll 765	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) e. NN )
50	49	nnzd 11481	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) e. ZZ )
51	49	nnne0d 11065	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) =/= 0 )
52	18, 20, 42	elzrhunit 30023	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( (denom ` y ) e. ZZ /\ (denom ` y ) =/= 0 ) ) -> ( L ` (denom ` y ) ) e. (Unit ` R ) )
53	37, 50, 51, 52	syl12anc 1324	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (denom ` y ) ) e. (Unit ` R ) )
54	18, 23, 24, 19, 9, 26, 34, 44, 47, 53	rdivmuldivd 29791	. . 3 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( ( L ` (numer ` x ) ) ./ ( L ` (denom ` x ) ) ) ( .r ` R ) ( ( L ` (numer ` y ) ) ./ ( L ` (denom ` y ) ) ) ) = ( ( ( L ` (numer ` x ) ) ( .r ` R ) ( L ` (numer ` y ) ) ) ./ ( ( L ` (denom ` x ) ) ( .r ` R ) ( L ` (denom ` y ) ) ) ) )
55		qeqnumdivden 15454	. . . . . . 7 |- ( x e. QQ -> x = ( (numer ` x ) / (denom ` x ) ) )
56	55	fveq2d 6195	. . . . . 6 |- ( x e. QQ -> ( (QQHom ` R ) ` x ) = ( (QQHom ` R ) ` ( (numer ` x ) / (denom ` x ) ) ) )
57	56	ad2antrl 764	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` x ) = ( (QQHom ` R ) ` ( (numer ` x ) / (denom ` x ) ) ) )
58	18, 19, 20	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 ) ) ) )
59	37, 33, 40, 41, 58	syl13anc 1328	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( (numer ` x ) / (denom ` x ) ) ) = ( ( L ` (numer ` x ) ) ./ ( L ` (denom ` x ) ) ) )
60	57, 59	eqtrd 2656	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` x ) = ( ( L ` (numer ` x ) ) ./ ( L ` (denom ` x ) ) ) )
61		qeqnumdivden 15454	. . . . . . 7 |- ( y e. QQ -> y = ( (numer ` y ) / (denom ` y ) ) )
62	61	fveq2d 6195	. . . . . 6 |- ( y e. QQ -> ( (QQHom ` R ) ` y ) = ( (QQHom ` R ) ` ( (numer ` y ) / (denom ` y ) ) ) )
63	62	ad2antll 765	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` y ) = ( (QQHom ` R ) ` ( (numer ` y ) / (denom ` y ) ) ) )
64	18, 19, 20	qqhvq 30031	. . . . . 6 |- ( ( ( 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 ) ) ) )
65	37, 46, 50, 51, 64	syl13anc 1328	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( (numer ` y ) / (denom ` y ) ) ) = ( ( L ` (numer ` y ) ) ./ ( L ` (denom ` y ) ) ) )
66	63, 65	eqtrd 2656	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` y ) = ( ( L ` (numer ` y ) ) ./ ( L ` (denom ` y ) ) ) )
67	60, 66	oveq12d 6668	. . 3 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( (QQHom ` R ) ` x ) ( .r ` R ) ( (QQHom ` R ) ` y ) ) = ( ( ( L ` (numer ` x ) ) ./ ( L ` (denom ` x ) ) ) ( .r ` R ) ( ( L ` (numer ` y ) ) ./ ( L ` (denom ` y ) ) ) ) )
68	55	ad2antrl 764	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> x = ( (numer ` x ) / (denom ` x ) ) )
69	61	ad2antll 765	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> y = ( (numer ` y ) / (denom ` y ) ) )
70	68, 69	oveq12d 6668	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( x x. y ) = ( ( (numer ` x ) / (denom ` x ) ) x. ( (numer ` y ) / (denom ` y ) ) ) )
71	33	zcnd 11483	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` x ) e. CC )
72	40	zcnd 11483	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. CC )
73	46	zcnd 11483	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` y ) e. CC )
74	50	zcnd 11483	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) e. CC )
75	71, 72, 73, 74, 41, 51	divmuldivd 10842	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( (numer ` x ) / (denom ` x ) ) x. ( (numer ` y ) / (denom ` y ) ) ) = ( ( (numer ` x ) x. (numer ` y ) ) / ( (denom ` x ) x. (denom ` y ) ) ) )
76	70, 75	eqtrd 2656	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( x x. y ) = ( ( (numer ` x ) x. (numer ` y ) ) / ( (denom ` x ) x. (denom ` y ) ) ) )
77	76	fveq2d 6195	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( x x. y ) ) = ( (QQHom ` R ) ` ( ( (numer ` x ) x. (numer ` y ) ) / ( (denom ` x ) x. (denom ` y ) ) ) ) )
78	33, 46	zmulcld 11488	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` x ) x. (numer ` y ) ) e. ZZ )
79	40, 50	zmulcld 11488	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (denom ` x ) x. (denom ` y ) ) e. ZZ )
80	72, 74, 41, 51	mulne0d 10679	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (denom ` x ) x. (denom ` y ) ) =/= 0 )
81	18, 19, 20	qqhvq 30031	. . . . 5 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( ( (numer ` x ) x. (numer ` y ) ) e. ZZ /\ ( (denom ` x ) x. (denom ` y ) ) e. ZZ /\ ( (denom ` x ) x. (denom ` y ) ) =/= 0 ) ) -> ( (QQHom ` R ) ` ( ( (numer ` x ) x. (numer ` y ) ) / ( (denom ` x ) x. (denom ` y ) ) ) ) = ( ( L ` ( (numer ` x ) x. (numer ` y ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
82	37, 78, 79, 80, 81	syl13anc 1328	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( ( (numer ` x ) x. (numer ` y ) ) / ( (denom ` x ) x. (denom ` y ) ) ) ) = ( ( L ` ( (numer ` x ) x. (numer ` y ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
83	35, 16	syl 17	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> R e. Ring )
84	83, 27	syl 17	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> L e. (ℤring RingHom R ) )
85		zringmulr 19827	. . . . . . 7 |- x. = ( .r ` ℤring )
86	28, 85, 9	rhmmul 18727	. . . . . 6 |- ( ( L e. (ℤring RingHom R ) /\ (numer ` x ) e. ZZ /\ (numer ` y ) e. ZZ ) -> ( L ` ( (numer ` x ) x. (numer ` y ) ) ) = ( ( L ` (numer ` x ) ) ( .r ` R ) ( L ` (numer ` y ) ) ) )
87	84, 33, 46, 86	syl3anc 1326	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` ( (numer ` x ) x. (numer ` y ) ) ) = ( ( L ` (numer ` x ) ) ( .r ` R ) ( L ` (numer ` y ) ) ) )
88	28, 85, 9	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 ) ) ) )
89	84, 40, 50, 88	syl3anc 1326	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` ( (denom ` x ) x. (denom ` y ) ) ) = ( ( L ` (denom ` x ) ) ( .r ` R ) ( L ` (denom ` y ) ) ) )
90	87, 89	oveq12d 6668	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( L ` ( (numer ` x ) x. (numer ` y ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) = ( ( ( L ` (numer ` x ) ) ( .r ` R ) ( L ` (numer ` y ) ) ) ./ ( ( L ` (denom ` x ) ) ( .r ` R ) ( L ` (denom ` y ) ) ) ) )
91	77, 82, 90	3eqtrd 2660	. . 3 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( x x. y ) ) = ( ( ( L ` (numer ` x ) ) ( .r ` R ) ( L ` (numer ` y ) ) ) ./ ( ( L ` (denom ` x ) ) ( .r ` R ) ( L ` (denom ` y ) ) ) ) )
92	54, 67, 91	3eqtr4rd 2667	. 2 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( x x. y ) ) = ( ( (QQHom ` R ) ` x ) ( .r ` R ) ( (QQHom ` R ) ` y ) ) )
93		cnfldadd 19751	. . . 4 |- + = ( +g ` ℂfld )
94	1, 93	ressplusg 15993	. . 3 |- ( QQ e. _V -> + = ( +g ` Q ) )
95	5, 94	ax-mp 5	. 2 |- + = ( +g ` Q )
96	18, 19, 20	qqhf 30030	. . 3 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> B )
97	14, 96	sylan 488	. 2 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> B )
98	33, 50	zmulcld 11488	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` x ) x. (denom ` y ) ) e. ZZ )
99	31, 98	ffvelrnd 6360	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` ( (numer ` x ) x. (denom ` y ) ) ) e. B )
100	46, 40	zmulcld 11488	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` y ) x. (denom ` x ) ) e. ZZ )
101	31, 100	ffvelrnd 6360	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` ( (numer ` y ) x. (denom ` x ) ) ) e. B )
102	23, 9	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 ) )
103	83, 44, 53, 102	syl3anc 1326	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( L ` (denom ` x ) ) ( .r ` R ) ( L ` (denom ` y ) ) ) e. (Unit ` R ) )
104	89, 103	eqeltrd 2701	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` ( (denom ` x ) x. (denom ` y ) ) ) e. (Unit ` R ) )
105	18, 23, 24, 19	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 ) ) ) ) ) )
106	83, 99, 101, 104, 105	syl13anc 1328	. . 3 |- ( ( ( R e. Field /\ (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 ) ) ) ) ) )
107	68, 69	oveq12d 6668	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( x + y ) = ( ( (numer ` x ) / (denom ` x ) ) + ( (numer ` y ) / (denom ` y ) ) ) )
108	71, 72, 73, 74, 41, 51	divadddivd 10845	. . . . . 6 |- ( ( ( R e. Field /\ (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 ) ) ) )
109	107, 108	eqtrd 2656	. . . . 5 |- ( ( ( R e. Field /\ (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 ) ) ) )
110	109	fveq2d 6195	. . . 4 |- ( ( ( R e. Field /\ (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 ) ) ) ) )
111	98, 100	zaddcld 11486	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( ( (numer ` x ) x. (denom ` y ) ) + ( (numer ` y ) x. (denom ` x ) ) ) e. ZZ )
112	18, 19, 20	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 ) ) ) ) )
113	37, 111, 79, 80, 112	syl13anc 1328	. . . 4 |- ( ( ( R e. Field /\ (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 ) ) ) ) )
114		rhmghm 18725	. . . . . 6 |- ( L e. (ℤring RingHom R ) -> L e. (ℤring GrpHom R ) )
115	84, 114	syl 17	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> L e. (ℤring GrpHom R ) )
116		zringplusg 19825	. . . . . . 7 |- + = ( +g ` ℤring )
117	28, 116, 24	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 ) ) ) ) )
118	117	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 ) ) ) ) )
119	115, 98, 100, 118	syl3anc 1326	. . . 4 |- ( ( ( R e. Field /\ (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 ) ) ) ) )
120	110, 113, 119	3eqtrd 2660	. . 3 |- ( ( ( R e. Field /\ (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 ) ) ) ) )
121	23, 28, 19, 85	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 ) ) ) ) )
122	84, 33, 40, 50, 44, 53, 121	syl132anc 1344	. . . . 5 |- ( ( ( R e. Field /\ (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 ) ) ) ) )
123	57, 59, 122	3eqtrd 2660	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` x ) = ( ( L ` ( (numer ` x ) x. (denom ` y ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
124	23, 28, 19, 85	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 ) ) ) ) )
125	84, 46, 50, 40, 53, 44, 124	syl132anc 1344	. . . . . 6 |- ( ( ( R e. Field /\ (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 ) ) ) ) )
126	72, 74	mulcomd 10061	. . . . . . . 8 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (denom ` x ) x. (denom ` y ) ) = ( (denom ` y ) x. (denom ` x ) ) )
127	126	fveq2d 6195	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` ( (denom ` x ) x. (denom ` y ) ) ) = ( L ` ( (denom ` y ) x. (denom ` x ) ) ) )
128	127	oveq2d 6666	. . . . . 6 |- ( ( ( R e. Field /\ (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 ) ) ) ) )
129	125, 65, 128	3eqtr4d 2666	. . . . 5 |- ( ( ( R e. Field /\ (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 ) ) ) ) )
130	63, 129	eqtrd 2656	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` y ) = ( ( L ` ( (numer ` y ) x. (denom ` x ) ) ) ./ ( L ` ( (denom ` x ) x. (denom ` y ) ) ) ) )
131	123, 130	oveq12d 6668	. . 3 |- ( ( ( R e. Field /\ (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 ) ) ) ) ) )
132	106, 120, 131	3eqtr4d 2666	. 2 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` ( x + y ) ) = ( ( (QQHom ` R ) ` x ) ( +g ` R ) ( (QQHom ` R ) ` y ) ) )
133	2, 3, 4, 8, 9, 12, 17, 22, 92, 18, 95, 24, 97, 132	isrhmd 18729	1 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q RingHom 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   1c1 9937   + 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   GrpHom cghm 17657   1rcur 18501   Ringcrg 18547   CRingccrg 18548  Unitcui 18639  /_rcdvr 18682   RingHom crh 18712   DivRingcdr 18747  Fieldcfield 18748  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-field 18750  df-subrg 18778  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-chr 19854  df-qqh 30017

This theorem is referenced by: (None)