Theorem qqhrhm 27795

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 23670	. 2 |- QQ = ( Base ` Q )
3	1	qrng1 23673	. 2 |- 1 = ( 1r ` Q )
4		eqid 2467	. 2 |- ( 1r ` R ) = ( 1r ` R )
5		qex 11206	. . 3 |- QQ e. _V
6		cnfldmul 18296	. . . 4 |- x. = ( .r ` ℂfld )
7	1, 6	ressmulr 14625	. . 3 |- ( QQ e. _V -> x. = ( .r ` Q ) )
8	5, 7	ax-mp 5	. 2 |- x. = ( .r ` Q )
9		eqid 2467	. 2 |- ( .r ` R ) = ( .r ` R )
10	1	qdrng 23671	. . 3 |- Q e. DivRing
11		drngrng 17274	. . 3 |- ( Q e. DivRing -> Q e. Ring )
12	10, 11	mp1i 12	. 2 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> Q e. Ring )
13		isfld 17276	. . . . 5 |- ( R e. Field <-> ( R e. DivRing /\ R e. CRing ) )
14	13	simplbi 460	. . . 4 |- ( R e. Field -> R e. DivRing )
15	14	adantr 465	. . 3 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> R e. DivRing )
16		drngrng 17274	. . 3 |- ( R e. DivRing -> R e. Ring )
17	15, 16	syl 16	. 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 27791	. . 3 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( (QQHom ` R ) ` 1 ) = ( 1r ` R ) )
22	14, 21	sylan 471	. 2 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> ( (QQHom ` R ) ` 1 ) = ( 1r ` R ) )
23		eqid 2467	. . . 4 |- (Unit ` R ) = (Unit ` R )
24		eqid 2467	. . . 4 |- ( +g ` R ) = ( +g ` R )
25	13	simprbi 464	. . . . 5 |- ( R e. Field -> R e. CRing )
26	25	ad2antrr 725	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> R e. CRing )
27	20	zrhrhm 18418	. . . . . . 7 |- ( R e. Ring -> L e. (ℤring RingHom R ) )
28		zringbas 18364	. . . . . . . 8 |- ZZ = ( Base ` ℤring )
29	28, 18	rhmf 17247	. . . . . . 7 |- ( L e. (ℤring RingHom R ) -> L : ZZ --> B )
30	17, 27, 29	3syl 20	. . . . . 6 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> L : ZZ --> B )
31	30	adantr 465	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> L : ZZ --> B )
32		qnumcl 14149	. . . . . 6 |- ( x e. QQ -> (numer ` x ) e. ZZ )
33	32	ad2antrl 727	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` x ) e. ZZ )
34	31, 33	ffvelrnd 6033	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (numer ` x ) ) e. B )
35	14	ad2antrr 725	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> R e. DivRing )
36		simplr 754	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (chr ` R ) = 0 )
37	35, 36	jca 532	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( R e. DivRing /\ (chr ` R ) = 0 ) )
38		qdencl 14150	. . . . . . 7 |- ( x e. QQ -> (denom ` x ) e. NN )
39	38	ad2antrl 727	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. NN )
40	39	nnzd 10977	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. ZZ )
41	39	nnne0d 10592	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) =/= 0 )
42		eqid 2467	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
43	18, 20, 42	elzrhunit 27785	. . . . 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 1226	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (denom ` x ) ) e. (Unit ` R ) )
45		qnumcl 14149	. . . . . 6 |- ( y e. QQ -> (numer ` y ) e. ZZ )
46	45	ad2antll 728	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` y ) e. ZZ )
47	31, 46	ffvelrnd 6033	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (numer ` y ) ) e. B )
48		qdencl 14150	. . . . . . 7 |- ( y e. QQ -> (denom ` y ) e. NN )
49	48	ad2antll 728	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) e. NN )
50	49	nnzd 10977	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) e. ZZ )
51	49	nnne0d 10592	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) =/= 0 )
52	18, 20, 42	elzrhunit 27785	. . . . 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 1226	. . . 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 27606	. . 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 14155	. . . . . . 7 |- ( x e. QQ -> x = ( (numer ` x ) / (denom ` x ) ) )
56	55	fveq2d 5876	. . . . . 6 |- ( x e. QQ -> ( (QQHom ` R ) ` x ) = ( (QQHom ` R ) ` ( (numer ` x ) / (denom ` x ) ) ) )
57	56	ad2antrl 727	. . . . 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 27793	. . . . . 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 1230	. . . . 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 2508	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` x ) = ( ( L ` (numer ` x ) ) ./ ( L ` (denom ` x ) ) ) )
61		qeqnumdivden 14155	. . . . . . 7 |- ( y e. QQ -> y = ( (numer ` y ) / (denom ` y ) ) )
62	61	fveq2d 5876	. . . . . 6 |- ( y e. QQ -> ( (QQHom ` R ) ` y ) = ( (QQHom ` R ) ` ( (numer ` y ) / (denom ` y ) ) ) )
63	62	ad2antll 728	. . . . 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 27793	. . . . . 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 1230	. . . . 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 2508	. . . 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 6313	. . 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 727	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> x = ( (numer ` x ) / (denom ` x ) ) )
69	61	ad2antll 728	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> y = ( (numer ` y ) / (denom ` y ) ) )
70	68, 69	oveq12d 6313	. . . . . 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 10979	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` x ) e. CC )
72	40	zcnd 10979	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. CC )
73	46	zcnd 10979	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` y ) e. CC )
74	50	zcnd 10979	. . . . . . 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 10373	. . . . . 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 2508	. . . . 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 5876	. . . 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 10984	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` x ) x. (numer ` y ) ) e. ZZ )
79	40, 50	zmulcld 10984	. . . . 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 10213	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (denom ` x ) x. (denom ` y ) ) =/= 0 )
81	18, 19, 20	qqhvq 27793	. . . . 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 1230	. . . 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 16	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> R e. Ring )
84	83, 27	syl 16	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> L e. (ℤring RingHom R ) )
85		zringmulr 18367	. . . . . . 7 |- x. = ( .r ` ℤring )
86	28, 85, 9	rhmmul 17248	. . . . . 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 1228	. . . . 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 17248	. . . . . 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 1228	. . . . 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 6313	. . . 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 2512	. . 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 2519	. 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 18295	. . . 4 |- + = ( +g ` ℂfld )
94	1, 93	ressplusg 14614	. . 3 |- ( QQ e. _V -> + = ( +g ` Q ) )
95	5, 94	ax-mp 5	. 2 |- + = ( +g ` Q )
96	18, 19, 20	qqhf 27792	. . 3 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> B )
97	14, 96	sylan 471	. 2 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> B )
98	33, 50	zmulcld 10984	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` x ) x. (denom ` y ) ) e. ZZ )
99	31, 98	ffvelrnd 6033	. . . 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 10984	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` y ) x. (denom ` x ) ) e. ZZ )
101	31, 100	ffvelrnd 6033	. . . 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 17185	. . . . . 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 1228	. . . . 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 2555	. . . 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 27605	. . . 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 1230	. . 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 6313	. . . . . 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 10376	. . . . . 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 2508	. . . . 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 5876	. . . 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 10982	. . . . 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 27793	. . . . 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 1230	. . . 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 17246	. . . . . 6 |- ( L e. (ℤring RingHom R ) -> L e. (ℤring GrpHom R ) )
115	84, 114	syl 16	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> L e. (ℤring GrpHom R ) )
116		zringplusg 18365	. . . . . . 7 |- + = ( +g ` ℤring )
117	28, 116, 24	ghmlin 16144	. . . . . 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 6310	. . . . 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 1228	. . . 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 2512	. . 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 27636	. . . . . 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 1246	. . . . 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 2512	. . . 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 27636	. . . . . . 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 1246	. . . . . 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 9629	. . . . . . . 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 5876	. . . . . . 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 6311	. . . . . 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 2518	. . . . 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 2508	. . . 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 6313	. . 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 2518	. 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 17250	1 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q RingHom R ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   _Vcvv 3118   -->wf 5590   ` cfv 5594  (class class class)co 6295   0cc0 9504   1c1 9505   + caddc 9507   x. cmul 9509   / cdiv 10218   NNcn 10548   ZZcz 10876   QQcq 11194  numercnumer 14142  denomcdenom 14143   Basecbs 14507   ↾_s cress 14508   +g cplusg 14572   .rcmulr 14573   0gc0g 14712   GrpHom cghm 16136   1rcur 17025   Ringcrg 17070   CRingccrg 17071  Unitcui 17160  /_rcdvr 17203   RingHom crh 17233   DivRingcdr 17267  Fieldcfield 17268  ℂ_fldccnfld 18290  ℤ_ringzring 18358   ZRHomczrh 18406  chrcchr 18408  QQHomcqqh 27778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-fz 11685  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-numer 14144  df-denom 14145  df-gz 14324  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-ghm 16137  df-od 16426  df-cmn 16673  df-mgp 17014  df-ur 17026  df-ring 17072  df-cring 17073  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-rnghom 17236  df-drng 17269  df-field 17270  df-subrg 17298  df-cnfld 18291  df-zring 18359  df-zrh 18410  df-chr 18412  df-qqh 27779

This theorem is referenced by: (None)