Theorem qqhval2 28651

Description: Value of the canonical homormorphism from the rational number when the target ring is a division ring. (Contributed by Thierry Arnoux, 26-Oct-2017.)

Hypotheses

Ref	Expression
qqhval2.0	|- B = ( Base ` R )
qqhval2.1	|- ./ = (/r ` R )
qqhval2.2	|- L = ( ZRHom ` R )

Assertion

Ref	Expression
qqhval2	|- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) = ( q e. QQ |-> ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) )

Distinct variable groups:   ./ , q   B, q   L, q   R, q

Proof of Theorem qqhval2

Dummy variables x y e s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3087	. . . 4 |- ( R e. DivRing -> R e. _V )
2	1	adantr 466	. . 3 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> R e. _V )
3		qqhval2.1	. . . 4 |- ./ = (/r ` R )
4		eqid 2420	. . . 4 |- ( 1r ` R ) = ( 1r ` R )
5		qqhval2.2	. . . 4 |- L = ( ZRHom ` R )
6	3, 4, 5	qqhval 28643	. . 3 |- ( R e. _V -> (QQHom ` R ) = ran ( x e. ZZ , y e. ( `' L " (Unit ` R ) ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )
7	2, 6	syl 17	. 2 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) = ran ( x e. ZZ , y e. ( `' L " (Unit ` R ) ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )
8		eqidd 2421	. . . 4 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ZZ = ZZ )
9		qqhval2.0	. . . . 5 |- B = ( Base ` R )
10		eqid 2420	. . . . 5 |- ( 0g ` R ) = ( 0g ` R )
11	9, 5, 10	zrhunitpreima 28647	. . . 4 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( `' L " (Unit ` R ) ) = ( ZZ \ { 0 } ) )
12		mpt2eq12 6356	. . . 4 |- ( ( ZZ = ZZ /\ ( `' L " (Unit ` R ) ) = ( ZZ \ { 0 } ) ) -> ( x e. ZZ , y e. ( `' L " (Unit ` R ) ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) = ( x e. ZZ , y e. ( ZZ \ { 0 } ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )
13	8, 11, 12	syl2anc 665	. . 3 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( x e. ZZ , y e. ( `' L " (Unit ` R ) ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) = ( x e. ZZ , y e. ( ZZ \ { 0 } ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )
14	13	rneqd 5073	. 2 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ran ( x e. ZZ , y e. ( `' L " (Unit ` R ) ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) = ran ( x e. ZZ , y e. ( ZZ \ { 0 } ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )
15		nfv 1751	. . . 4 |- F/ e ( R e. DivRing /\ (chr ` R ) = 0 )
16		nfab1 2584	. . . 4 |- F/_ e { e | E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. }
17		nfcv 2582	. . . 4 |- F/_ e { <. q , s >. | ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) }
18		simpr 462	. . . . . . . . . 10 |- ( ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) /\ e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. )
19		zssq 11260	. . . . . . . . . . . 12 |- ZZ C_ QQ
20		simplrl 768	. . . . . . . . . . . 12 |- ( ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) /\ e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> x e. ZZ )
21	19, 20	sseldi 3459	. . . . . . . . . . 11 |- ( ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) /\ e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> x e. QQ )
22		simplrr 769	. . . . . . . . . . . . 13 |- ( ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) /\ e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> y e. ( ZZ \ { 0 } ) )
23	22	eldifad 3445	. . . . . . . . . . . 12 |- ( ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) /\ e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> y e. ZZ )
24	19, 23	sseldi 3459	. . . . . . . . . . 11 |- ( ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) /\ e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> y e. QQ )
25	22	eldifbd 3446	. . . . . . . . . . . 12 |- ( ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) /\ e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> -. y e. { 0 } )
26		elsn 4007	. . . . . . . . . . . . 13 |- ( y e. { 0 } <-> y = 0 )
27	26	necon3bbii 2683	. . . . . . . . . . . 12 |- ( -. y e. { 0 } <-> y =/= 0 )
28	25, 27	sylib 199	. . . . . . . . . . 11 |- ( ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) /\ e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> y =/= 0 )
29		qdivcl 11274	. . . . . . . . . . 11 |- ( ( x e. QQ /\ y e. QQ /\ y =/= 0 ) -> ( x / y ) e. QQ )
30	21, 24, 28, 29	syl3anc 1264	. . . . . . . . . 10 |- ( ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) /\ e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> ( x / y ) e. QQ )
31		simplll 766	. . . . . . . . . . 11 |- ( ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) /\ e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> R e. DivRing )
32		simpllr 767	. . . . . . . . . . 11 |- ( ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) /\ e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> (chr ` R ) = 0 )
33	9, 3, 5	qqhval2lem 28650	. . . . . . . . . . . 12 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ /\ y =/= 0 ) ) -> ( ( L ` (numer ` ( x / y ) ) ) ./ ( L ` (denom ` ( x / y ) ) ) ) = ( ( L ` x ) ./ ( L ` y ) ) )
34	33	eqcomd 2428	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ /\ y =/= 0 ) ) -> ( ( L ` x ) ./ ( L ` y ) ) = ( ( L ` (numer ` ( x / y ) ) ) ./ ( L ` (denom ` ( x / y ) ) ) ) )
35	31, 32, 20, 23, 28, 34	syl23anc 1271	. . . . . . . . . 10 |- ( ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) /\ e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> ( ( L ` x ) ./ ( L ` y ) ) = ( ( L ` (numer ` ( x / y ) ) ) ./ ( L ` (denom ` ( x / y ) ) ) ) )
36		ovex 6324	. . . . . . . . . . 11 |- ( x / y ) e. _V
37		ovex 6324	. . . . . . . . . . 11 |- ( ( L ` x ) ./ ( L ` y ) ) e. _V
38		opeq12 4183	. . . . . . . . . . . . 13 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> <. q , s >. = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. )
39	38	eqeq2d 2434	. . . . . . . . . . . 12 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( e = <. q , s >. <-> e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )
40		simpl 458	. . . . . . . . . . . . . 14 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> q = ( x / y ) )
41	40	eleq1d 2489	. . . . . . . . . . . . 13 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( q e. QQ <-> ( x / y ) e. QQ ) )
42		simpr 462	. . . . . . . . . . . . . 14 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> s = ( ( L ` x ) ./ ( L ` y ) ) )
43	40	fveq2d 5876	. . . . . . . . . . . . . . . 16 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> (numer ` q ) = (numer ` ( x / y ) ) )
44	43	fveq2d 5876	. . . . . . . . . . . . . . 15 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( L ` (numer ` q ) ) = ( L ` (numer ` ( x / y ) ) ) )
45	40	fveq2d 5876	. . . . . . . . . . . . . . . 16 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> (denom ` q ) = (denom ` ( x / y ) ) )
46	45	fveq2d 5876	. . . . . . . . . . . . . . 15 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( L ` (denom ` q ) ) = ( L ` (denom ` ( x / y ) ) ) )
47	44, 46	oveq12d 6314	. . . . . . . . . . . . . 14 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) = ( ( L ` (numer ` ( x / y ) ) ) ./ ( L ` (denom ` ( x / y ) ) ) ) )
48	42, 47	eqeq12d 2442	. . . . . . . . . . . . 13 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) <-> ( ( L ` x ) ./ ( L ` y ) ) = ( ( L ` (numer ` ( x / y ) ) ) ./ ( L ` (denom ` ( x / y ) ) ) ) ) )
49	41, 48	anbi12d 715	. . . . . . . . . . . 12 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) <-> ( ( x / y ) e. QQ /\ ( ( L ` x ) ./ ( L ` y ) ) = ( ( L ` (numer ` ( x / y ) ) ) ./ ( L ` (denom ` ( x / y ) ) ) ) ) ) )
50	39, 49	anbi12d 715	. . . . . . . . . . 11 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) <-> ( e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. /\ ( ( x / y ) e. QQ /\ ( ( L ` x ) ./ ( L ` y ) ) = ( ( L ` (numer ` ( x / y ) ) ) ./ ( L ` (denom ` ( x / y ) ) ) ) ) ) ) )
51	36, 37, 50	spc2ev 3171	. . . . . . . . . 10 |- ( ( e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. /\ ( ( x / y ) e. QQ /\ ( ( L ` x ) ./ ( L ` y ) ) = ( ( L ` (numer ` ( x / y ) ) ) ./ ( L ` (denom ` ( x / y ) ) ) ) ) ) -> E. q E. s ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) )
52	18, 30, 35, 51	syl12anc 1262	. . . . . . . . 9 |- ( ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) /\ e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> E. q E. s ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) )
53	52	ex 435	. . . . . . . 8 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( x e. ZZ /\ y e. ( ZZ \ { 0 } ) ) ) -> ( e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. -> E. q E. s ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) )
54	53	rexlimdvva 2922	. . . . . . 7 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. -> E. q E. s ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) )
55	54	imp 430	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) -> E. q E. s ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) )
56		19.42vv 1825	. . . . . . 7 |- ( E. q E. s ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) <-> ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ E. q E. s ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) )
57		simprrl 772	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> q e. QQ )
58		qnumcl 14649	. . . . . . . . . 10 |- ( q e. QQ -> (numer ` q ) e. ZZ )
59	57, 58	syl 17	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> (numer ` q ) e. ZZ )
60		qdencl 14650	. . . . . . . . . . . 12 |- ( q e. QQ -> (denom ` q ) e. NN )
61	57, 60	syl 17	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> (denom ` q ) e. NN )
62	61	nnzd 11028	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> (denom ` q ) e. ZZ )
63		nnne0 10631	. . . . . . . . . . 11 |- ( (denom ` q ) e. NN -> (denom ` q ) =/= 0 )
64		elsni 4018	. . . . . . . . . . . 12 |- ( (denom ` q ) e. { 0 } -> (denom ` q ) = 0 )
65	64	necon3ai 2650	. . . . . . . . . . 11 |- ( (denom ` q ) =/= 0 -> -. (denom ` q ) e. { 0 } )
66	61, 63, 65	3syl 18	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> -. (denom ` q ) e. { 0 } )
67	62, 66	eldifd 3444	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> (denom ` q ) e. ( ZZ \ { 0 } ) )
68		simprl 762	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> e = <. q , s >. )
69		qeqnumdivden 14655	. . . . . . . . . . . 12 |- ( q e. QQ -> q = ( (numer ` q ) / (denom ` q ) ) )
70	57, 69	syl 17	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> q = ( (numer ` q ) / (denom ` q ) ) )
71		simprrr 773	. . . . . . . . . . 11 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) )
72	70, 71	opeq12d 4189	. . . . . . . . . 10 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> <. q , s >. = <. ( (numer ` q ) / (denom ` q ) ) , ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) >. )
73	68, 72	eqtrd 2461	. . . . . . . . 9 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> e = <. ( (numer ` q ) / (denom ` q ) ) , ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) >. )
74		oveq1 6303	. . . . . . . . . . . 12 |- ( x = (numer ` q ) -> ( x / y ) = ( (numer ` q ) / y ) )
75		fveq2 5872	. . . . . . . . . . . . 13 |- ( x = (numer ` q ) -> ( L ` x ) = ( L ` (numer ` q ) ) )
76	75	oveq1d 6311	. . . . . . . . . . . 12 |- ( x = (numer ` q ) -> ( ( L ` x ) ./ ( L ` y ) ) = ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) )
77	74, 76	opeq12d 4189	. . . . . . . . . . 11 |- ( x = (numer ` q ) -> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. = <. ( (numer ` q ) / y ) , ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) >. )
78	77	eqeq2d 2434	. . . . . . . . . 10 |- ( x = (numer ` q ) -> ( e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. <-> e = <. ( (numer ` q ) / y ) , ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) >. ) )
79		oveq2 6304	. . . . . . . . . . . 12 |- ( y = (denom ` q ) -> ( (numer ` q ) / y ) = ( (numer ` q ) / (denom ` q ) ) )
80		fveq2 5872	. . . . . . . . . . . . 13 |- ( y = (denom ` q ) -> ( L ` y ) = ( L ` (denom ` q ) ) )
81	80	oveq2d 6312	. . . . . . . . . . . 12 |- ( y = (denom ` q ) -> ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) )
82	79, 81	opeq12d 4189	. . . . . . . . . . 11 |- ( y = (denom ` q ) -> <. ( (numer ` q ) / y ) , ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) >. = <. ( (numer ` q ) / (denom ` q ) ) , ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) >. )
83	82	eqeq2d 2434	. . . . . . . . . 10 |- ( y = (denom ` q ) -> ( e = <. ( (numer ` q ) / y ) , ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) >. <-> e = <. ( (numer ` q ) / (denom ` q ) ) , ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) >. ) )
84	78, 83	rspc2ev 3190	. . . . . . . . 9 |- ( ( (numer ` q ) e. ZZ /\ (denom ` q ) e. ( ZZ \ { 0 } ) /\ e = <. ( (numer ` q ) / (denom ` q ) ) , ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) >. ) -> E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. )
85	59, 67, 73, 84	syl3anc 1264	. . . . . . . 8 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. )
86	85	exlimivv 1767	. . . . . . 7 |- ( E. q E. s ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. )
87	56, 86	sylbir 216	. . . . . 6 |- ( ( ( R e. DivRing /\ (chr ` R ) = 0 ) /\ E. q E. s ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) -> E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. )
88	55, 87	impbida 840	. . . . 5 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. <-> E. q E. s ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) ) )
89		abid 2407	. . . . 5 |- ( e e. { e | E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. } <-> E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. )
90		elopab 4720	. . . . 5 |- ( e e. { <. q , s >. | ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) } <-> E. q E. s ( e = <. q , s >. /\ ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) ) )
91	88, 89, 90	3bitr4g 291	. . . 4 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( e e. { e | E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. } <-> e e. { <. q , s >. | ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) } ) )
92	15, 16, 17, 91	eqrd 3479	. . 3 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> { e | E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. } = { <. q , s >. | ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) } )
93		eqid 2420	. . . 4 |- ( x e. ZZ , y e. ( ZZ \ { 0 } ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) = ( x e. ZZ , y e. ( ZZ \ { 0 } ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. )
94	93	rnmpt2 6411	. . 3 |- ran ( x e. ZZ , y e. ( ZZ \ { 0 } ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) = { e | E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. }
95		df-mpt 4477	. . 3 |- ( q e. QQ |-> ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) = { <. q , s >. | ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) }
96	92, 94, 95	3eqtr4g 2486	. 2 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ran ( x e. ZZ , y e. ( ZZ \ { 0 } ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) = ( q e. QQ |-> ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) )
97	7, 14, 96	3eqtrd 2465	1 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> (QQHom ` R ) = ( q e. QQ |-> ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   /\ w3a 982   = wceq 1437   E.wex 1659   e. wcel 1867   {cab 2405   =/= wne 2616   E.wrex 2774   _Vcvv 3078   \ cdif 3430   {csn 3993   <.cop 3999   {copab 4474   |-> cmpt 4475   `'ccnv 4844   ran crn 4846   "cima 4848   ` cfv 5592  (class class class)co 6296   |-> cmpt2 6298   0cc0 9528   / cdiv 10258   NNcn 10598   ZZcz 10926   QQcq 11253  numercnumer 14642  denomcdenom 14643   Basecbs 15073   0gc0g 15290   1rcur 17663  Unitcui 17795  /_rcdvr 17838   DivRingcdr 17903   ZRHomczrh 18995  chrcchr 18997  QQHomcqqh 28641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607  ax-mulf 9608

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-tpos 6972  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-map 7473  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-inf 7954  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-uz 11149  df-q 11254  df-rp 11292  df-fz 11772  df-fl 12014  df-mod 12083  df-seq 12200  df-exp 12259  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-dvds 14273  df-gcd 14432  df-numer 14644  df-denom 14645  df-gz 14826  df-struct 15075  df-ndx 15076  df-slot 15077  df-base 15078  df-sets 15079  df-ress 15080  df-plusg 15155  df-mulr 15156  df-starv 15157  df-tset 15161  df-ple 15162  df-ds 15164  df-unif 15165  df-0g 15292  df-mgm 16432  df-sgrp 16471  df-mnd 16481  df-mhm 16526  df-grp 16617  df-minusg 16618  df-sbg 16619  df-mulg 16620  df-subg 16758  df-ghm 16825  df-od 17113  df-cmn 17360  df-mgp 17652  df-ur 17664  df-ring 17710  df-cring 17711  df-oppr 17779  df-dvdsr 17797  df-unit 17798  df-invr 17828  df-dvr 17839  df-rnghom 17871  df-drng 17905  df-subrg 17934  df-cnfld 18899  df-zring 18967  df-zrh 18999  df-chr 19001  df-qqh 28642

This theorem is referenced by:  qqhvval  28652  qqhf  28655