Theorem qqhval2 26265

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 2971	. . . 4 |- ( R e. DivRing -> R e. _V )
2	1	adantr 462	. . 3 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> R e. _V )
3		qqhval2.1	. . . 4 |- ./ = (/r ` R )
4		eqid 2433	. . . 4 |- ( 1r ` R ) = ( 1r ` R )
5		qqhval2.2	. . . 4 |- L = ( ZRHom ` R )
6	3, 4, 5	qqhval 26257	. . 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 16	. 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 2434	. . . 4 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ZZ = ZZ )
9		qqhval2.0	. . . . 5 |- B = ( Base ` R )
10		eqid 2433	. . . . 5 |- ( 0g ` R ) = ( 0g ` R )
11	9, 5, 10	zrhunitpreima 26261	. . . 4 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( `' L " (Unit ` R ) ) = ( ZZ \ { 0 } ) )
12		mpt2eq12 6135	. . . 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 654	. . 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 5054	. 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 1672	. . . 4 |- F/ e ( R e. DivRing /\ (chr ` R ) = 0 )
16		nfab1 2571	. . . 4 |- F/_ e { e | E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. }
17		nfcv 2569	. . . 4 |- F/_ e { <. q , s >. | ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) }
18		simpr 458	. . . . . . . . . 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 10948	. . . . . . . . . . . 12 |- ZZ C_ QQ
20		simplrl 752	. . . . . . . . . . . 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 3342	. . . . . . . . . . 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 753	. . . . . . . . . . . . 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 3328	. . . . . . . . . . . 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 3342	. . . . . . . . . . 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 3329	. . . . . . . . . . . 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 3879	. . . . . . . . . . . . 13 |- ( y e. { 0 } <-> y = 0 )
27	26	necon3bbii 2629	. . . . . . . . . . . 12 |- ( -. y e. { 0 } <-> y =/= 0 )
28	25, 27	sylib 196	. . . . . . . . . . 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 10962	. . . . . . . . . . 11 |- ( ( x e. QQ /\ y e. QQ /\ y =/= 0 ) -> ( x / y ) e. QQ )
30	21, 24, 28, 29	syl3anc 1211	. . . . . . . . . 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 750	. . . . . . . . . . 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 751	. . . . . . . . . . 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 26264	. . . . . . . . . . . 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 2438	. . . . . . . . . . 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 1218	. . . . . . . . . 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 6105	. . . . . . . . . . 11 |- ( x / y ) e. _V
37		ovex 6105	. . . . . . . . . . 11 |- ( ( L ` x ) ./ ( L ` y ) ) e. _V
38		opeq12 4049	. . . . . . . . . . . . 13 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> <. q , s >. = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. )
39	38	eqeq2d 2444	. . . . . . . . . . . 12 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( e = <. q , s >. <-> e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )
40		simpl 454	. . . . . . . . . . . . . 14 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> q = ( x / y ) )
41	40	eleq1d 2499	. . . . . . . . . . . . 13 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( q e. QQ <-> ( x / y ) e. QQ ) )
42		simpr 458	. . . . . . . . . . . . . 14 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> s = ( ( L ` x ) ./ ( L ` y ) ) )
43	40	fveq2d 5683	. . . . . . . . . . . . . . . 16 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> (numer ` q ) = (numer ` ( x / y ) ) )
44	43	fveq2d 5683	. . . . . . . . . . . . . . 15 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( L ` (numer ` q ) ) = ( L ` (numer ` ( x / y ) ) ) )
45	40	fveq2d 5683	. . . . . . . . . . . . . . . 16 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> (denom ` q ) = (denom ` ( x / y ) ) )
46	45	fveq2d 5683	. . . . . . . . . . . . . . 15 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( L ` (denom ` q ) ) = ( L ` (denom ` ( x / y ) ) ) )
47	44, 46	oveq12d 6098	. . . . . . . . . . . . . 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 2447	. . . . . . . . . . . . 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 703	. . . . . . . . . . . 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 703	. . . . . . . . . . 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 3054	. . . . . . . . . 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 1209	. . . . . . . . 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 434	. . . . . . . 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 2838	. . . . . . 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 429	. . . . . 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 1924	. . . . . . 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 756	. . . . . . . . . 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 13801	. . . . . . . . . 10 |- ( q e. QQ -> (numer ` q ) e. ZZ )
59	57, 58	syl 16	. . . . . . . . 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 13802	. . . . . . . . . . . 12 |- ( q e. QQ -> (denom ` q ) e. NN )
61	57, 60	syl 16	. . . . . . . . . . 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 10734	. . . . . . . . . 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 10342	. . . . . . . . . . 11 |- ( (denom ` q ) e. NN -> (denom ` q ) =/= 0 )
64		elsni 3890	. . . . . . . . . . . 12 |- ( (denom ` q ) e. { 0 } -> (denom ` q ) = 0 )
65	64	necon3ai 2641	. . . . . . . . . . 11 |- ( (denom ` q ) =/= 0 -> -. (denom ` q ) e. { 0 } )
66	61, 63, 65	3syl 20	. . . . . . . . . 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 3327	. . . . . . . . 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 748	. . . . . . . . . 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 13807	. . . . . . . . . . . 12 |- ( q e. QQ -> q = ( (numer ` q ) / (denom ` q ) ) )
70	57, 69	syl 16	. . . . . . . . . . 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 757	. . . . . . . . . . 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 4055	. . . . . . . . . 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 2465	. . . . . . . . 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 6087	. . . . . . . . . . . 12 |- ( x = (numer ` q ) -> ( x / y ) = ( (numer ` q ) / y ) )
75		fveq2 5679	. . . . . . . . . . . . 13 |- ( x = (numer ` q ) -> ( L ` x ) = ( L ` (numer ` q ) ) )
76	75	oveq1d 6095	. . . . . . . . . . . 12 |- ( x = (numer ` q ) -> ( ( L ` x ) ./ ( L ` y ) ) = ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) )
77	74, 76	opeq12d 4055	. . . . . . . . . . 11 |- ( x = (numer ` q ) -> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. = <. ( (numer ` q ) / y ) , ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) >. )
78	77	eqeq2d 2444	. . . . . . . . . 10 |- ( x = (numer ` q ) -> ( e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. <-> e = <. ( (numer ` q ) / y ) , ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) >. ) )
79		oveq2 6088	. . . . . . . . . . . 12 |- ( y = (denom ` q ) -> ( (numer ` q ) / y ) = ( (numer ` q ) / (denom ` q ) ) )
80		fveq2 5679	. . . . . . . . . . . . 13 |- ( y = (denom ` q ) -> ( L ` y ) = ( L ` (denom ` q ) ) )
81	80	oveq2d 6096	. . . . . . . . . . . 12 |- ( y = (denom ` q ) -> ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) )
82	79, 81	opeq12d 4055	. . . . . . . . . . 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 2444	. . . . . . . . . 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 3070	. . . . . . . . 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 1211	. . . . . . . 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 1688	. . . . . . 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 213	. . . . . 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 821	. . . . 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 2421	. . . . 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 4585	. . . . 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 288	. . . 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 3362	. . 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 2433	. . . 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 6189	. . 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 4340	. . 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 2490	. 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 2469	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 369   /\ w3a 958   = wceq 1362   E.wex 1589   e. wcel 1755   {cab 2419   =/= wne 2596   E.wrex 2706   _Vcvv 2962   \ cdif 3313   {csn 3865   <.cop 3871   {copab 4337   e. cmpt 4338   `'ccnv 4826   ran crn 4828   "cima 4830   ` cfv 5406  (class class class)co 6080   e. cmpt2 6082   0cc0 9270   / cdiv 9981   NNcn 10310   ZZcz 10634   QQcq 10941  numercnumer 13794  denomcdenom 13795   Basecbs 14157   0gc0g 14361   1rcur 16579  Unitcui 16665  /_rcdvr 16708   DivRingcdr 16756   ZRHomczrh 17773  chrcchr 17775  QQHomcqqh 26255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-tpos 6734  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-fz 11425  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-dvds 13519  df-gcd 13674  df-numer 13796  df-denom 13797  df-gz 13974  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-0g 14363  df-mnd 15398  df-mhm 15447  df-grp 15525  df-minusg 15526  df-sbg 15527  df-mulg 15528  df-subg 15658  df-ghm 15725  df-od 16012  df-cmn 16259  df-mgp 16566  df-rng 16580  df-cring 16581  df-ur 16582  df-oppr 16649  df-dvdsr 16667  df-unit 16668  df-invr 16698  df-dvr 16709  df-rnghom 16740  df-drng 16758  df-subrg 16787  df-cnfld 17663  df-zring 17726  df-zrh 17777  df-chr 17779  df-qqh 26256

This theorem is referenced by:  qqhvval  26266  qqhf  26269