Theorem qqhrhm 26421

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 22871	. 2 |- QQ = ( Base ` Q )
3	1	qrng1 22874	. 2 |- 1 = ( 1r ` Q )
4		eqid 2443	. 2 |- ( 1r ` R ) = ( 1r ` R )
5		qex 10968	. . 3 |- QQ e. _V
6		cnfldmul 17827	. . . 4 |- x. = ( .r ` ℂfld )
7	1, 6	ressmulr 14294	. . 3 |- ( QQ e. _V -> x. = ( .r ` Q ) )
8	5, 7	ax-mp 5	. 2 |- x. = ( .r ` Q )
9		eqid 2443	. 2 |- ( .r ` R ) = ( .r ` R )
10	1	qdrng 22872	. . 3 |- Q e. DivRing
11		drngrng 16842	. . 3 |- ( Q e. DivRing -> Q e. Ring )
12	10, 11	mp1i 12	. 2 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> Q e. Ring )
13		isfld 16844	. . . . 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 16842	. . 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 26417	. . 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 2443	. . . 4 |- (Unit ` R ) = (Unit ` R )
24		eqid 2443	. . . 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 17946	. . . . . . 7 |- ( R e. Ring -> L e. (ℤring RingHom R ) )
28		zringbas 17892	. . . . . . . 8 |- ZZ = ( Base ` ℤring )
29	28, 18	rhmf 16819	. . . . . . 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 13821	. . . . . 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 5847	. . . 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 13822	. . . . . . 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 10749	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. ZZ )
41	39	nnne0d 10369	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) =/= 0 )
42		eqid 2443	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
43	18, 20, 42	elzrhunit 26411	. . . . 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 1216	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (denom ` x ) ) e. (Unit ` R ) )
45		qnumcl 13821	. . . . . 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 5847	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( L ` (numer ` y ) ) e. B )
48		qdencl 13822	. . . . . . 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 10749	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) e. ZZ )
51	49	nnne0d 10369	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` y ) =/= 0 )
52	18, 20, 42	elzrhunit 26411	. . . . 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 1216	. . . 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 26262	. . 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 13827	. . . . . . 7 |- ( x e. QQ -> x = ( (numer ` x ) / (denom ` x ) ) )
56	55	fveq2d 5698	. . . . . 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 26419	. . . . . 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 1220	. . . . 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 2475	. . . 4 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (QQHom ` R ) ` x ) = ( ( L ` (numer ` x ) ) ./ ( L ` (denom ` x ) ) ) )
61		qeqnumdivden 13827	. . . . . . 7 |- ( y e. QQ -> y = ( (numer ` y ) / (denom ` y ) ) )
62	61	fveq2d 5698	. . . . . 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 26419	. . . . . 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 1220	. . . . 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 2475	. . . 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 6112	. . 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 6112	. . . . . 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 10751	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` x ) e. CC )
72	40	zcnd 10751	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (denom ` x ) e. CC )
73	46	zcnd 10751	. . . . . . 7 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> (numer ` y ) e. CC )
74	50	zcnd 10751	. . . . . . 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 10151	. . . . . 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 2475	. . . . 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 5698	. . . 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 10756	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` x ) x. (numer ` y ) ) e. ZZ )
79	40, 50	zmulcld 10756	. . . . 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 9991	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (denom ` x ) x. (denom ` y ) ) =/= 0 )
81	18, 19, 20	qqhvq 26419	. . . . 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 1220	. . . 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 17895	. . . . . . 7 |- x. = ( .r ` ℤring )
86	28, 85, 9	rhmmul 16820	. . . . . 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 1218	. . . . 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 16820	. . . . . 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 1218	. . . . 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 6112	. . . 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 2479	. . 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 2486	. 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 17826	. . . 4 |- + = ( +g ` ℂfld )
94	1, 93	ressplusg 14283	. . 3 |- ( QQ e. _V -> + = ( +g ` Q ) )
95	5, 94	ax-mp 5	. 2 |- + = ( +g ` Q )
96	18, 19, 20	qqhf 26418	. . 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 10756	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` x ) x. (denom ` y ) ) e. ZZ )
99	31, 98	ffvelrnd 5847	. . . 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 10756	. . . . 5 |- ( ( ( R e. Field /\ (chr ` R ) = 0 ) /\ ( x e. QQ /\ y e. QQ ) ) -> ( (numer ` y ) x. (denom ` x ) ) e. ZZ )
101	31, 100	ffvelrnd 5847	. . . 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 16759	. . . . . 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 1218	. . . . 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 2517	. . . 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 26261	. . . 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 1220	. . 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 6112	. . . . . 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 10154	. . . . . 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 2475	. . . . 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 5698	. . . 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 10754	. . . . 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 26419	. . . . 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 1220	. . . 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 16818	. . . . . 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 17893	. . . . . . 7 |- + = ( +g ` ℤring )
117	28, 116, 24	ghmlin 15755	. . . . . 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 6109	. . . . 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 1218	. . . 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 2479	. . 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 26292	. . . . . 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 1236	. . . . 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 2479	. . . 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 26292	. . . . . . 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 1236	. . . . . 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 9410	. . . . . . . 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 5698	. . . . . . 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 6110	. . . . . 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 2485	. . . . 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 2475	. . . 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 6112	. . 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 2485	. 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 16822	1 |- ( ( R e. Field /\ (chr ` R ) = 0 ) -> (QQHom ` R ) e. ( Q RingHom R ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2609   _Vcvv 2975   -->wf 5417   ` cfv 5421  (class class class)co 6094   0cc0 9285   1c1 9286   + caddc 9288   x. cmul 9290   / cdiv 9996   NNcn 10325   ZZcz 10649   QQcq 10956  numercnumer 13814  denomcdenom 13815   Basecbs 14177   ↾_s cress 14178   +g cplusg 14241   .rcmulr 14242   0gc0g 14381   GrpHom cghm 15747   1rcur 16606   Ringcrg 16648   CRingccrg 16649  Unitcui 16734  /_rcdvr 16777   RingHom crh 16807   DivRingcdr 16835  Fieldcfield 16836  ℂ_fldccnfld 17821  ℤ_ringzring 17886   ZRHomczrh 17934  chrcchr 17936  QQHomcqqh 26404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363  ax-addf 9364  ax-mulf 9365

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-tpos 6748  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-map 7219  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-9 10390  df-10 10391  df-n0 10583  df-z 10650  df-dec 10759  df-uz 10865  df-q 10957  df-rp 10995  df-fz 11441  df-fl 11645  df-mod 11712  df-seq 11810  df-exp 11869  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-dvds 13539  df-gcd 13694  df-numer 13816  df-denom 13817  df-gz 13994  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-mulr 14255  df-starv 14256  df-tset 14260  df-ple 14261  df-ds 14263  df-unif 14264  df-0g 14383  df-mnd 15418  df-mhm 15467  df-grp 15548  df-minusg 15549  df-sbg 15550  df-mulg 15551  df-subg 15681  df-ghm 15748  df-od 16035  df-cmn 16282  df-mgp 16595  df-ur 16607  df-rng 16650  df-cring 16651  df-oppr 16718  df-dvdsr 16736  df-unit 16737  df-invr 16767  df-dvr 16778  df-rnghom 16809  df-drng 16837  df-field 16838  df-subrg 16866  df-cnfld 17822  df-zring 17887  df-zrh 17938  df-chr 17940  df-qqh 26405

This theorem is referenced by: (None)