Theorem qqhrhm 28131

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 23930	. 2 |- QQ = ( Base ` Q )
3	1	qrng1 23933	. 2 |- 1 = ( 1r ` Q )
4		eqid 2457	. 2 |- ( 1r ` R ) = ( 1r ` R )
5		qex 11219	. . 3 |- QQ e. _V
6		cnfldmul 18553	. . . 4 |- x. = ( .r ` ℂfld )
7	1, 6	ressmulr 14769	. . 3 |- ( QQ e. _V -> x. = ( .r ` Q ) )
8	5, 7	ax-mp 5	. 2 |- x. = ( .r ` Q )
9		eqid 2457	. 2 |- ( .r ` R ) = ( .r ` R )
10	1	qdrng 23931	. . 3 |- Q e. DivRing
11		drngring 17530	. . 3 |- ( Q e. DivRing -> Q e. Ring )
12	10, 11	mp1i 12	. 2 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> Q e. Ring )
13		isfld 17532	. . . . 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		drngring 17530	. . 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 28127	. . 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 2457	. . . 4 |- (Unit ` R ) = (Unit ` R )
24		eqid 2457	. . . 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 18676	. . . . . . 7 |- ( R e. Ring -> L e. (ℤring RingHom R ) )
28		zringbas 18621	. . . . . . . 8 |- ZZ = ( Base ` ℤring )
29	28, 18	rhmf 17502	. . . . . . 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 14285	. . . . . 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 755	. . . . . 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 14286	. . . . . . 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 10989	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. ZZ )
41	39	nnne0d 10601	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) =/= 0 )
42		eqid 2457	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
43	18, 20, 42	elzrhunit 28121	. . . . 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 14285	. . . . . 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 14286	. . . . . . 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 10989	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) e. ZZ )
51	49	nnne0d 10601	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) =/= 0 )
52	18, 20, 42	elzrhunit 28121	. . . . 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 27942	. . 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 14291	. . . . . . 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 28129	. . . . . 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 2498	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` x ) = ( ( L ` (numer ` x ) ) ./ ( L ` (denom ` x ) ) ) )
61		qeqnumdivden 14291	. . . . . . 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 28129	. . . . . 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 2498	. . . 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 6314	. . 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 6314	. . . . . 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 10991	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` x ) e. CC )
72	40	zcnd 10991	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. CC )
73	46	zcnd 10991	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` y ) e. CC )
74	50	zcnd 10991	. . . . . . 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 10382	. . . . . 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 2498	. . . . 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 10996	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` x ) x. (numer ` y ) ) e. ZZ )
79	40, 50	zmulcld 10996	. . . . 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 10222	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (denom ` x ) x. (denom ` y ) ) =/= 0 )
81	18, 19, 20	qqhvq 28129	. . . . 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 18624	. . . . . . 7 |- x. = ( .r ` ℤring )
86	28, 85, 9	rhmmul 17503	. . . . . 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 17503	. . . . . 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 6314	. . . 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 2502	. . 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 2509	. 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 18552	. . . 4 |- + = ( +g ` ℂfld )
94	1, 93	ressplusg 14758	. . 3 |- ( QQ e. _V -> + = ( +g ` Q ) )
95	5, 94	ax-mp 5	. 2 |- + = ( +g ` Q )
96	18, 19, 20	qqhf 28128	. . 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 10996	. . . . 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 10996	. . . . 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 17440	. . . . . 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 2545	. . . 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 27941	. . . 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 6314	. . . . . 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 10385	. . . . . 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 2498	. . . . 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 10994	. . . . 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 28129	. . . . 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 17501	. . . . . 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 18622	. . . . . . 7 |- + = ( +g ` ℤring )
117	28, 116, 24	ghmlin 16399	. . . . . 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 6311	. . . . 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 2502	. . 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 27972	. . . . . 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 2502	. . . 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 27972	. . . . . . 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 9634	. . . . . . . 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 6312	. . . . . 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 2508	. . . . 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 2498	. . . 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 6314	. . 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 2508	. 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 17505	1 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q RingHom R ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   =/= wne 2652   _Vcvv 3109   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510   + caddc 9512   x. cmul 9514   / cdiv 10227   NNcn 10556   ZZcz 10885   QQcq 11207  numercnumer 14278  denomcdenom 14279   Basecbs 14644   ↾_s cress 14645   +g cplusg 14712   .rcmulr 14713   0gc0g 14857   GrpHom cghm 16391   1rcur 17280   Ringcrg 17325   CRingccrg 17326  Unitcui 17415  /_rcdvr 17458   RingHom crh 17488   DivRingcdr 17523  Fieldcfield 17524  ℂ_fldccnfld 18547  ℤ_ringzring 18615   ZRHomczrh 18664  chrcchr 18666  QQHomcqqh 28114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-fz 11698  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-dvds 13999  df-gcd 14157  df-numer 14280  df-denom 14281  df-gz 14460  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mulg 16187  df-subg 16325  df-ghm 16392  df-od 16680  df-cmn 16927  df-mgp 17269  df-ur 17281  df-ring 17327  df-cring 17328  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-dvr 17459  df-rnghom 17491  df-drng 17525  df-field 17526  df-subrg 17554  df-cnfld 18548  df-zring 18616  df-zrh 18668  df-chr 18670  df-qqh 28115

This theorem is referenced by: (None)