Theorem qqhval2 28798

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 3056	. . . 4 |- ( R e. DivRing -> R e. _V )
2	1	adantr 467	. . 3 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> R e. _V )
3		qqhval2.1	. . . 4 |- ./ = (/r ` R )
4		eqid 2453	. . . 4 |- ( 1r ` R ) = ( 1r ` R )
5		qqhval2.2	. . . 4 |- L = ( ZRHom ` R )
6	3, 4, 5	qqhval 28790	. . 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 2454	. . . 4 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ZZ = ZZ )
9		qqhval2.0	. . . . 5 |- B = ( Base ` R )
10		eqid 2453	. . . . 5 |- ( 0g ` R ) = ( 0g ` R )
11	9, 5, 10	zrhunitpreima 28794	. . . 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 667	. . 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 5065	. 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 1763	. . . 4 |- F/ e ( R e. DivRing /\ (chr ` R ) = 0 )
16		nfab1 2596	. . . 4 |- F/_ e { e | E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. }
17		nfcv 2594	. . . 4 |- F/_ e { <. q , s >. | ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) }
18		simpr 463	. . . . . . . . . 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 11278	. . . . . . . . . . . 12 |- ZZ C_ QQ
20		simplrl 771	. . . . . . . . . . . 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 3432	. . . . . . . . . . 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 772	. . . . . . . . . . . . 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 3418	. . . . . . . . . . . 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 3432	. . . . . . . . . . 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 3419	. . . . . . . . . . . 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 3984	. . . . . . . . . . . . 13 |- ( y e. { 0 } <-> y = 0 )
27	26	necon3bbii 2673	. . . . . . . . . . . 12 |- ( -. y e. { 0 } <-> y =/= 0 )
28	25, 27	sylib 200	. . . . . . . . . . 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 11292	. . . . . . . . . . 11 |- ( ( x e. QQ /\ y e. QQ /\ y =/= 0 ) -> ( x / y ) e. QQ )
30	21, 24, 28, 29	syl3anc 1269	. . . . . . . . . 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 769	. . . . . . . . . . 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 770	. . . . . . . . . . 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 28797	. . . . . . . . . . . 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 2459	. . . . . . . . . . 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 1276	. . . . . . . . . 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 6323	. . . . . . . . . . 11 |- ( x / y ) e. _V
37		ovex 6323	. . . . . . . . . . 11 |- ( ( L ` x ) ./ ( L ` y ) ) e. _V
38		opeq12 4171	. . . . . . . . . . . . 13 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> <. q , s >. = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. )
39	38	eqeq2d 2463	. . . . . . . . . . . 12 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( e = <. q , s >. <-> e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )
40		simpl 459	. . . . . . . . . . . . . 14 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> q = ( x / y ) )
41	40	eleq1d 2515	. . . . . . . . . . . . 13 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( q e. QQ <-> ( x / y ) e. QQ ) )
42		simpr 463	. . . . . . . . . . . . . 14 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> s = ( ( L ` x ) ./ ( L ` y ) ) )
43	40	fveq2d 5874	. . . . . . . . . . . . . . . 16 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> (numer ` q ) = (numer ` ( x / y ) ) )
44	43	fveq2d 5874	. . . . . . . . . . . . . . 15 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( L ` (numer ` q ) ) = ( L ` (numer ` ( x / y ) ) ) )
45	40	fveq2d 5874	. . . . . . . . . . . . . . . 16 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> (denom ` q ) = (denom ` ( x / y ) ) )
46	45	fveq2d 5874	. . . . . . . . . . . . . . 15 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( L ` (denom ` q ) ) = ( L ` (denom ` ( x / y ) ) ) )
47	44, 46	oveq12d 6313	. . . . . . . . . . . . . 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 2468	. . . . . . . . . . . . 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 718	. . . . . . . . . . . 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 718	. . . . . . . . . . 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 3144	. . . . . . . . . 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 1267	. . . . . . . . 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 436	. . . . . . . 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 2888	. . . . . . 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 431	. . . . . 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 1838	. . . . . . 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 775	. . . . . . . . . 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 14701	. . . . . . . . . 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 14702	. . . . . . . . . . . 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 11046	. . . . . . . . . 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 10649	. . . . . . . . . . 11 |- ( (denom ` q ) e. NN -> (denom ` q ) =/= 0 )
64		elsni 3995	. . . . . . . . . . . 12 |- ( (denom ` q ) e. { 0 } -> (denom ` q ) = 0 )
65	64	necon3ai 2651	. . . . . . . . . . 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 3417	. . . . . . . . 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 765	. . . . . . . . . 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 14707	. . . . . . . . . . . 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 776	. . . . . . . . . . 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 4177	. . . . . . . . . 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 2487	. . . . . . . . 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 6302	. . . . . . . . . . . 12 |- ( x = (numer ` q ) -> ( x / y ) = ( (numer ` q ) / y ) )
75		fveq2 5870	. . . . . . . . . . . . 13 |- ( x = (numer ` q ) -> ( L ` x ) = ( L ` (numer ` q ) ) )
76	75	oveq1d 6310	. . . . . . . . . . . 12 |- ( x = (numer ` q ) -> ( ( L ` x ) ./ ( L ` y ) ) = ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) )
77	74, 76	opeq12d 4177	. . . . . . . . . . 11 |- ( x = (numer ` q ) -> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. = <. ( (numer ` q ) / y ) , ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) >. )
78	77	eqeq2d 2463	. . . . . . . . . 10 |- ( x = (numer ` q ) -> ( e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. <-> e = <. ( (numer ` q ) / y ) , ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) >. ) )
79		oveq2 6303	. . . . . . . . . . . 12 |- ( y = (denom ` q ) -> ( (numer ` q ) / y ) = ( (numer ` q ) / (denom ` q ) ) )
80		fveq2 5870	. . . . . . . . . . . . 13 |- ( y = (denom ` q ) -> ( L ` y ) = ( L ` (denom ` q ) ) )
81	80	oveq2d 6311	. . . . . . . . . . . 12 |- ( y = (denom ` q ) -> ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) )
82	79, 81	opeq12d 4177	. . . . . . . . . . 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 2463	. . . . . . . . . 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 3163	. . . . . . . . 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 1269	. . . . . . . 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 1780	. . . . . . 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 217	. . . . . 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 844	. . . . 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 2441	. . . . 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 4712	. . . . 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 292	. . . 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 3452	. . 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 2453	. . . 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 4466	. . 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 2512	. 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 2491	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 371   /\ w3a 986   = wceq 1446   E.wex 1665   e. wcel 1889   {cab 2439   =/= wne 2624   E.wrex 2740   _Vcvv 3047   \ cdif 3403   {csn 3970   <.cop 3976   {copab 4463   |-> cmpt 4464   `'ccnv 4836   ran crn 4838   "cima 4840   ` cfv 5585  (class class class)co 6295   |-> cmpt2 6297   0cc0 9544   / cdiv 10276   NNcn 10616   ZZcz 10944   QQcq 11271  numercnumer 14694  denomcdenom 14695   Basecbs 15133   0gc0g 15350   1rcur 17747  Unitcui 17879  /_rcdvr 17922   DivRingcdr 17987   ZRHomczrh 19083  chrcchr 19085  QQHomcqqh 28788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622  ax-addf 9623  ax-mulf 9624

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-tpos 6978  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-inf 7962  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-fz 11792  df-fl 12035  df-mod 12104  df-seq 12221  df-exp 12280  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-dvds 14318  df-gcd 14481  df-numer 14696  df-denom 14697  df-gz 14886  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-starv 15217  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-0g 15352  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-mhm 16594  df-grp 16685  df-minusg 16686  df-sbg 16687  df-mulg 16688  df-subg 16826  df-ghm 16893  df-od 17184  df-cmn 17444  df-mgp 17736  df-ur 17748  df-ring 17794  df-cring 17795  df-oppr 17863  df-dvdsr 17881  df-unit 17882  df-invr 17912  df-dvr 17923  df-rnghom 17955  df-drng 17989  df-subrg 18018  df-cnfld 18983  df-zring 19052  df-zrh 19087  df-chr 19089  df-qqh 28789

This theorem is referenced by:  qqhvval  28799  qqhf  28802