Theorem qqhrhm 26354

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 22827	. 2 |- QQ = ( Base ` Q )
3	1	qrng1 22830	. 2 |- 1 = ( 1r ` Q )
4		eqid 2441	. 2 |- ( 1r ` R ) = ( 1r ` R )
5		qex 10961	. . 3 |- QQ e. _V
6		cnfldmul 17783	. . . 4 |- x. = ( .r ` ℂfld )
7	1, 6	ressmulr 14287	. . 3 |- ( QQ e. _V -> x. = ( .r ` Q ) )
8	5, 7	ax-mp 5	. 2 |- x. = ( .r ` Q )
9		eqid 2441	. 2 |- ( .r ` R ) = ( .r ` R )
10	1	qdrng 22828	. . 3 |- Q e. DivRing
11		drngrng 16819	. . 3 |- ( Q e. DivRing -> Q e. Ring )
12	10, 11	mp1i 12	. 2 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> Q e. Ring )
13		isfld 16821	. . . . 5 |- ( R e. Field <-> ( R e. DivRing /\ R e. CRing ) )
14	13	simplbi 457	. . . 4 |- ( R e. Field -> R e. DivRing )
15	14	adantr 462	. . 3 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> R e. DivRing )
16		drngrng 16819	. . 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 26350	. . 3 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( (QQHom ` R ) ` 1 ) = ( 1r ` R ) )
22	14, 21	sylan 468	. 2 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> ( (QQHom ` R ) ` 1 ) = ( 1r ` R ) )
23		eqid 2441	. . . 4 |- (Unit ` R ) = (Unit ` R )
24		eqid 2441	. . . 4 |- ( +g ` R ) = ( +g ` R )
25	13	simprbi 461	. . . . 5 |- ( R e. Field -> R e. CRing )
26	25	ad2antrr 720	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> R e. CRing )
27	20	zrhrhm 17902	. . . . . . 7 |- ( R e. Ring -> L e. (ℤring RingHom R ) )
28		zringbas 17848	. . . . . . . 8 |- ZZ = ( Base ` ℤring )
29	28, 18	rhmf 16804	. . . . . . 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 462	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> L : ZZ --> B )
32		qnumcl 13814	. . . . . 6 |- ( x e. QQ -> (numer ` x ) e. ZZ )
33	32	ad2antrl 722	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` x ) e. ZZ )
34	31, 33	ffvelrnd 5841	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (numer ` x ) ) e. B )
35	14	ad2antrr 720	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> R e. DivRing )
36		simplr 749	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (chr ` R ) = 0 )
37	35, 36	jca 529	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( R e. DivRing /\ (chr ` R ) = 0 ) )
38		qdencl 13815	. . . . . . 7 |- ( x e. QQ -> (denom ` x ) e. NN )
39	38	ad2antrl 722	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. NN )
40	39	nnzd 10742	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. ZZ )
41	39	nnne0d 10362	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) =/= 0 )
42		eqid 2441	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
43	18, 20, 42	elzrhunit 26344	. . . . 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 1211	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (denom ` x ) ) e. (Unit ` R ) )
45		qnumcl 13814	. . . . . 6 |- ( y e. QQ -> (numer ` y ) e. ZZ )
46	45	ad2antll 723	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` y ) e. ZZ )
47	31, 46	ffvelrnd 5841	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (numer ` y ) ) e. B )
48		qdencl 13815	. . . . . . 7 |- ( y e. QQ -> (denom ` y ) e. NN )
49	48	ad2antll 723	. . . . . 6 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) e. NN )
50	49	nnzd 10742	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) e. ZZ )
51	49	nnne0d 10362	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) =/= 0 )
52	18, 20, 42	elzrhunit 26344	. . . . 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 1211	. . . 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 26194	. . 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 13820	. . . . . . 7 |- ( x e. QQ -> x = ( (numer ` x ) / (denom ` x ) ) )
56	55	fveq2d 5692	. . . . . 6 |- ( x e. QQ -> ( (QQHom ` R ) ` x ) = ( (QQHom ` R ) ` ( (numer ` x ) / (denom ` x ) ) ) )
57	56	ad2antrl 722	. . . . 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 26352	. . . . . 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 1215	. . . . 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 2473	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` x ) = ( ( L ` (numer ` x ) ) ./ ( L ` (denom ` x ) ) ) )
61		qeqnumdivden 13820	. . . . . . 7 |- ( y e. QQ -> y = ( (numer ` y ) / (denom ` y ) ) )
62	61	fveq2d 5692	. . . . . 6 |- ( y e. QQ -> ( (QQHom ` R ) ` y ) = ( (QQHom ` R ) ` ( (numer ` y ) / (denom ` y ) ) ) )
63	62	ad2antll 723	. . . . 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 26352	. . . . . 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 1215	. . . . 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 2473	. . . 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 6108	. . 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 722	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> x = ( (numer ` x ) / (denom ` x ) ) )
69	61	ad2antll 723	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> y = ( (numer ` y ) / (denom ` y ) ) )
70	68, 69	oveq12d 6108	. . . . . 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 10744	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` x ) e. CC )
72	40	zcnd 10744	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. CC )
73	46	zcnd 10744	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` y ) e. CC )
74	50	zcnd 10744	. . . . . . 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 10144	. . . . . 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 2473	. . . . 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 5692	. . . 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 10749	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` x ) x. (numer ` y ) ) e. ZZ )
79	40, 50	zmulcld 10749	. . . . 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 9984	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (denom ` x ) x. (denom ` y ) ) =/= 0 )
81	18, 19, 20	qqhvq 26352	. . . . 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 1215	. . . 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 17851	. . . . . . 7 |- x. = ( .r ` ℤring )
86	28, 85, 9	rhmmul 16805	. . . . . 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 1213	. . . . 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 16805	. . . . . 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 1213	. . . . 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 6108	. . . 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 2477	. . 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 2484	. 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 17782	. . . 4 |- + = ( +g ` ℂfld )
94	1, 93	ressplusg 14276	. . 3 |- ( QQ e. _V -> + = ( +g ` Q ) )
95	5, 94	ax-mp 5	. 2 |- + = ( +g ` Q )
96	18, 19, 20	qqhf 26351	. . 3 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> B )
97	14, 96	sylan 468	. 2 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> (QQHom ` R ) : QQ --> B )
98	33, 50	zmulcld 10749	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` x ) x. (denom ` y ) ) e. ZZ )
99	31, 98	ffvelrnd 5841	. . . 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 10749	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` y ) x. (denom ` x ) ) e. ZZ )
101	31, 100	ffvelrnd 5841	. . . 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 16746	. . . . . 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 1213	. . . . 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 2515	. . . 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 26193	. . . 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 1215	. . 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 6108	. . . . . 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 10147	. . . . . 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 2473	. . . . 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 5692	. . . 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 10747	. . . . 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 26352	. . . . 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 1215	. . . 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 16803	. . . . . 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 17849	. . . . . . 7 |- + = ( +g ` ℤring )
117	28, 116, 24	ghmlin 15745	. . . . . 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 6105	. . . . 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 1213	. . . 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 2477	. . 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 26224	. . . . . 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 1231	. . . . 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 2477	. . . 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 26224	. . . . . . 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 1231	. . . . . 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 9403	. . . . . . . 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 5692	. . . . . . 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 6106	. . . . . 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 2483	. . . . 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 2473	. . . 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 6108	. . 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 2483	. 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 16807	1 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q RingHom R ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 960   = wceq 1364   e. wcel 1761   =/= wne 2604   _Vcvv 2970   -->wf 5411   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279   + caddc 9281   x. cmul 9283   / cdiv 9989   NNcn 10318   ZZcz 10642   QQcq 10949  numercnumer 13807  denomcdenom 13808   Basecbs 14170   ↾_s cress 14171   +g cplusg 14234   .rcmulr 14235   0gc0g 14374   GrpHom cghm 15737   1rcur 16593   Ringcrg 16635   CRingccrg 16636  Unitcui 16721  /_rcdvr 16764   RingHom crh 16794   DivRingcdr 16812  Fieldcfield 16813  ℂ_fldccnfld 17777  ℤ_ringzring 17842   ZRHomczrh 17890  chrcchr 17892  QQHomcqqh 26337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-fz 11434  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-dvds 13532  df-gcd 13687  df-numer 13809  df-denom 13810  df-gz 13987  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-0g 14376  df-mnd 15411  df-mhm 15460  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-ghm 15738  df-od 16025  df-cmn 16272  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-rnghom 16796  df-drng 16814  df-field 16815  df-subrg 16843  df-cnfld 17778  df-zring 17843  df-zrh 17894  df-chr 17896  df-qqh 26338

This theorem is referenced by: (None)