Theorem qqhval2 26433

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 3002	. . . 4 |- ( R e. DivRing -> R e. _V )
2	1	adantr 465	. . 3 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> R e. _V )
3		qqhval2.1	. . . 4 |- ./ = (/r ` R )
4		eqid 2443	. . . 4 |- ( 1r ` R ) = ( 1r ` R )
5		qqhval2.2	. . . 4 |- L = ( ZRHom ` R )
6	3, 4, 5	qqhval 26425	. . 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 2444	. . . 4 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ZZ = ZZ )
9		qqhval2.0	. . . . 5 |- B = ( Base ` R )
10		eqid 2443	. . . . 5 |- ( 0g ` R ) = ( 0g ` R )
11	9, 5, 10	zrhunitpreima 26429	. . . 4 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( `' L " (Unit ` R ) ) = ( ZZ \ { 0 } ) )
12		mpt2eq12 6167	. . . 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 661	. . 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 5088	. 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 1673	. . . 4 |- F/ e ( R e. DivRing /\ (chr ` R ) = 0 )
16		nfab1 2591	. . . 4 |- F/_ e { e | E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. }
17		nfcv 2589	. . . 4 |- F/_ e { <. q , s >. | ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) }
18		simpr 461	. . . . . . . . . 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 10981	. . . . . . . . . . . 12 |- ZZ C_ QQ
20		simplrl 759	. . . . . . . . . . . 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 3375	. . . . . . . . . . 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 760	. . . . . . . . . . . . 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 3361	. . . . . . . . . . . 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 3375	. . . . . . . . . . 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 3362	. . . . . . . . . . . 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 3912	. . . . . . . . . . . . 13 |- ( y e. { 0 } <-> y = 0 )
27	26	necon3bbii 2633	. . . . . . . . . . . 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 10995	. . . . . . . . . . 11 |- ( ( x e. QQ /\ y e. QQ /\ y =/= 0 ) -> ( x / y ) e. QQ )
30	21, 24, 28, 29	syl3anc 1218	. . . . . . . . . 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 757	. . . . . . . . . . 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 758	. . . . . . . . . . 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 26432	. . . . . . . . . . . 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 2448	. . . . . . . . . . 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 1225	. . . . . . . . . 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 6137	. . . . . . . . . . 11 |- ( x / y ) e. _V
37		ovex 6137	. . . . . . . . . . 11 |- ( ( L ` x ) ./ ( L ` y ) ) e. _V
38		opeq12 4082	. . . . . . . . . . . . 13 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> <. q , s >. = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. )
39	38	eqeq2d 2454	. . . . . . . . . . . 12 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( e = <. q , s >. <-> e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )
40		simpl 457	. . . . . . . . . . . . . 14 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> q = ( x / y ) )
41	40	eleq1d 2509	. . . . . . . . . . . . 13 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( q e. QQ <-> ( x / y ) e. QQ ) )
42		simpr 461	. . . . . . . . . . . . . 14 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> s = ( ( L ` x ) ./ ( L ` y ) ) )
43	40	fveq2d 5716	. . . . . . . . . . . . . . . 16 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> (numer ` q ) = (numer ` ( x / y ) ) )
44	43	fveq2d 5716	. . . . . . . . . . . . . . 15 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( L ` (numer ` q ) ) = ( L ` (numer ` ( x / y ) ) ) )
45	40	fveq2d 5716	. . . . . . . . . . . . . . . 16 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> (denom ` q ) = (denom ` ( x / y ) ) )
46	45	fveq2d 5716	. . . . . . . . . . . . . . 15 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( L ` (denom ` q ) ) = ( L ` (denom ` ( x / y ) ) ) )
47	44, 46	oveq12d 6130	. . . . . . . . . . . . . 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 2457	. . . . . . . . . . . . 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 710	. . . . . . . . . . . 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 710	. . . . . . . . . . 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 3086	. . . . . . . . . 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 1216	. . . . . . . . 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 2869	. . . . . . 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 1926	. . . . . . 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 763	. . . . . . . . . 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 13839	. . . . . . . . . 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 13840	. . . . . . . . . . . 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 10767	. . . . . . . . . 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 10375	. . . . . . . . . . 11 |- ( (denom ` q ) e. NN -> (denom ` q ) =/= 0 )
64		elsni 3923	. . . . . . . . . . . 12 |- ( (denom ` q ) e. { 0 } -> (denom ` q ) = 0 )
65	64	necon3ai 2675	. . . . . . . . . . 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 3360	. . . . . . . . 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 755	. . . . . . . . . 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 13845	. . . . . . . . . . . 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 764	. . . . . . . . . . 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 4088	. . . . . . . . . 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 2475	. . . . . . . . 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 6119	. . . . . . . . . . . 12 |- ( x = (numer ` q ) -> ( x / y ) = ( (numer ` q ) / y ) )
75		fveq2 5712	. . . . . . . . . . . . 13 |- ( x = (numer ` q ) -> ( L ` x ) = ( L ` (numer ` q ) ) )
76	75	oveq1d 6127	. . . . . . . . . . . 12 |- ( x = (numer ` q ) -> ( ( L ` x ) ./ ( L ` y ) ) = ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) )
77	74, 76	opeq12d 4088	. . . . . . . . . . 11 |- ( x = (numer ` q ) -> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. = <. ( (numer ` q ) / y ) , ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) >. )
78	77	eqeq2d 2454	. . . . . . . . . 10 |- ( x = (numer ` q ) -> ( e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. <-> e = <. ( (numer ` q ) / y ) , ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) >. ) )
79		oveq2 6120	. . . . . . . . . . . 12 |- ( y = (denom ` q ) -> ( (numer ` q ) / y ) = ( (numer ` q ) / (denom ` q ) ) )
80		fveq2 5712	. . . . . . . . . . . . 13 |- ( y = (denom ` q ) -> ( L ` y ) = ( L ` (denom ` q ) ) )
81	80	oveq2d 6128	. . . . . . . . . . . 12 |- ( y = (denom ` q ) -> ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) )
82	79, 81	opeq12d 4088	. . . . . . . . . . 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 2454	. . . . . . . . . 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 3102	. . . . . . . . 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 1218	. . . . . . . 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 1689	. . . . . . 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 828	. . . . 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 2431	. . . . 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 4618	. . . . 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 3395	. . 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 2443	. . . 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 6221	. . 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 4373	. . 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 2500	. 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 2479	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 965   = wceq 1369   E.wex 1586   e. wcel 1756   {cab 2429   =/= wne 2620   E.wrex 2737   _Vcvv 2993   \ cdif 3346   {csn 3898   <.cop 3904   {copab 4370   e. cmpt 4371   `'ccnv 4860   ran crn 4862   "cima 4864   ` cfv 5439  (class class class)co 6112   e. cmpt2 6114   0cc0 9303   / cdiv 10014   NNcn 10343   ZZcz 10667   QQcq 10974  numercnumer 13832  denomcdenom 13833   Basecbs 14195   0gc0g 14399   1rcur 16625  Unitcui 16753  /_rcdvr 16796   DivRingcdr 16854   ZRHomczrh 17953  chrcchr 17955  QQHomcqqh 26423

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 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383

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 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-fz 11459  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-dvds 13557  df-gcd 13712  df-numer 13834  df-denom 13835  df-gz 14012  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-starv 14274  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-0g 14401  df-mnd 15436  df-mhm 15485  df-grp 15566  df-minusg 15567  df-sbg 15568  df-mulg 15569  df-subg 15699  df-ghm 15766  df-od 16053  df-cmn 16300  df-mgp 16614  df-ur 16626  df-rng 16669  df-cring 16670  df-oppr 16737  df-dvdsr 16755  df-unit 16756  df-invr 16786  df-dvr 16797  df-rnghom 16828  df-drng 16856  df-subrg 16885  df-cnfld 17841  df-zring 17906  df-zrh 17957  df-chr 17959  df-qqh 26424

This theorem is referenced by:  qqhvval  26434  qqhf  26437