Theorem qqhval2 24319

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 2924	. . . 4 |- ( R e. DivRing -> R e. _V )
2	1	adantr 452	. . 3 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> R e. _V )
3		qqhval2.1	. . . 4 |- ./ = (/r ` R )
4		eqid 2404	. . . 4 |- ( 1r ` R ) = ( 1r ` R )
5		qqhval2.2	. . . 4 |- L = ( ZRHom ` R )
6	3, 4, 5	qqhval 24311	. . 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 2405	. . . 4 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ZZ = ZZ )
9		qqhval2.0	. . . . 5 |- B = ( Base ` R )
10		eqid 2404	. . . . 5 |- ( 0g ` R ) = ( 0g ` R )
11	9, 5, 10	zrhunitpreima 24315	. . . 4 |- ( ( R e. DivRing /\ (chr ` R ) = 0 ) -> ( `' L " (Unit ` R ) ) = ( ZZ \ { 0 } ) )
12		mpt2eq12 6093	. . . 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 643	. . 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 5056	. 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 1626	. . . 4 |- F/ e ( R e. DivRing /\ (chr ` R ) = 0 )
16		nfab1 2542	. . . 4 |- F/_ e { e | E. x e. ZZ E. y e. ( ZZ \ { 0 } ) e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. }
17		nfcv 2540	. . . 4 |- F/_ e { <. q , s >. | ( q e. QQ /\ s = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) ) }
18		simpr 448	. . . . . . . . . 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 10537	. . . . . . . . . . . 12 |- ZZ C_ QQ
20		simplrl 737	. . . . . . . . . . . 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 3306	. . . . . . . . . . 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 738	. . . . . . . . . . . . 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 3292	. . . . . . . . . . . 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 3306	. . . . . . . . . . 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 3293	. . . . . . . . . . . 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 3789	. . . . . . . . . . . . 13 |- ( y e. { 0 } <-> y = 0 )
27	26	necon3bbii 2598	. . . . . . . . . . . 12 |- ( -. y e. { 0 } <-> y =/= 0 )
28	25, 27	sylib 189	. . . . . . . . . . 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 10551	. . . . . . . . . . 11 |- ( ( x e. QQ /\ y e. QQ /\ y =/= 0 ) -> ( x / y ) e. QQ )
30	21, 24, 28, 29	syl3anc 1184	. . . . . . . . . 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 735	. . . . . . . . . . 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 736	. . . . . . . . . . 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 24318	. . . . . . . . . . . 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 2409	. . . . . . . . . . 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 1191	. . . . . . . . . 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 6065	. . . . . . . . . . 11 |- ( x / y ) e. _V
37		ovex 6065	. . . . . . . . . . 11 |- ( ( L ` x ) ./ ( L ` y ) ) e. _V
38		opeq12 3946	. . . . . . . . . . . . 13 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> <. q , s >. = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. )
39	38	eqeq2d 2415	. . . . . . . . . . . 12 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( e = <. q , s >. <-> e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )
40		simpl 444	. . . . . . . . . . . . . 14 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> q = ( x / y ) )
41	40	eleq1d 2470	. . . . . . . . . . . . 13 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( q e. QQ <-> ( x / y ) e. QQ ) )
42		simpr 448	. . . . . . . . . . . . . 14 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> s = ( ( L ` x ) ./ ( L ` y ) ) )
43	40	fveq2d 5691	. . . . . . . . . . . . . . . 16 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> (numer ` q ) = (numer ` ( x / y ) ) )
44	43	fveq2d 5691	. . . . . . . . . . . . . . 15 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( L ` (numer ` q ) ) = ( L ` (numer ` ( x / y ) ) ) )
45	40	fveq2d 5691	. . . . . . . . . . . . . . . 16 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> (denom ` q ) = (denom ` ( x / y ) ) )
46	45	fveq2d 5691	. . . . . . . . . . . . . . 15 |- ( ( q = ( x / y ) /\ s = ( ( L ` x ) ./ ( L ` y ) ) ) -> ( L ` (denom ` q ) ) = ( L ` (denom ` ( x / y ) ) ) )
47	44, 46	oveq12d 6058	. . . . . . . . . . . . . 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 2418	. . . . . . . . . . . . 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 692	. . . . . . . . . . . 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 692	. . . . . . . . . . 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 3004	. . . . . . . . . 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 1182	. . . . . . . . 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 424	. . . . . . . 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 2797	. . . . . . 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 419	. . . . . 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 741	. . . . . . . . . 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 13087	. . . . . . . . . 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 13088	. . . . . . . . . . . 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 10330	. . . . . . . . . 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 9988	. . . . . . . . . . 11 |- ( (denom ` q ) e. NN -> (denom ` q ) =/= 0 )
64		elsni 3798	. . . . . . . . . . . 12 |- ( (denom ` q ) e. { 0 } -> (denom ` q ) = 0 )
65	64	necon3ai 2607	. . . . . . . . . . 11 |- ( (denom ` q ) =/= 0 -> -. (denom ` q ) e. { 0 } )
66	61, 63, 65	3syl 19	. . . . . . . . . 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 3291	. . . . . . . . 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 733	. . . . . . . . . 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 13093	. . . . . . . . . . . 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 742	. . . . . . . . . . 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 3952	. . . . . . . . . 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 2436	. . . . . . . . 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 6047	. . . . . . . . . . . 12 |- ( x = (numer ` q ) -> ( x / y ) = ( (numer ` q ) / y ) )
75		fveq2 5687	. . . . . . . . . . . . 13 |- ( x = (numer ` q ) -> ( L ` x ) = ( L ` (numer ` q ) ) )
76	75	oveq1d 6055	. . . . . . . . . . . 12 |- ( x = (numer ` q ) -> ( ( L ` x ) ./ ( L ` y ) ) = ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) )
77	74, 76	opeq12d 3952	. . . . . . . . . . 11 |- ( x = (numer ` q ) -> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. = <. ( (numer ` q ) / y ) , ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) >. )
78	77	eqeq2d 2415	. . . . . . . . . 10 |- ( x = (numer ` q ) -> ( e = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. <-> e = <. ( (numer ` q ) / y ) , ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) >. ) )
79		oveq2 6048	. . . . . . . . . . . 12 |- ( y = (denom ` q ) -> ( (numer ` q ) / y ) = ( (numer ` q ) / (denom ` q ) ) )
80		fveq2 5687	. . . . . . . . . . . . 13 |- ( y = (denom ` q ) -> ( L ` y ) = ( L ` (denom ` q ) ) )
81	80	oveq2d 6056	. . . . . . . . . . . 12 |- ( y = (denom ` q ) -> ( ( L ` (numer ` q ) ) ./ ( L ` y ) ) = ( ( L ` (numer ` q ) ) ./ ( L ` (denom ` q ) ) ) )
82	79, 81	opeq12d 3952	. . . . . . . . . . 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 2415	. . . . . . . . . 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 3020	. . . . . . . . 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 1184	. . . . . . . 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 1642	. . . . . . 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 205	. . . . . 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 806	. . . . 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 2392	. . . . 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 4422	. . . . 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 280	. . . 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 3326	. . 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 2404	. . . 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 6139	. . 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 4228	. . 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 2461	. 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 2440	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 359   /\ w3a 936   E.wex 1547   = wceq 1649   e. wcel 1721   {cab 2390   =/= wne 2567   E.wrex 2667   _Vcvv 2916   \ cdif 3277   {csn 3774   <.cop 3777   {copab 4225   e. cmpt 4226   `'ccnv 4836   ran crn 4838   "cima 4840   ` cfv 5413  (class class class)co 6040   e. cmpt2 6042   0cc0 8946   / cdiv 9633   NNcn 9956   ZZcz 10238   QQcq 10530  numercnumer 13080  denomcdenom 13081   Basecbs 13424   0gc0g 13678   1rcur 15617  Unitcui 15699  /_rcdvr 15742   DivRingcdr 15790   ZRHomczrh 16733  chrcchr 16735  QQHomcqqh 24309

This theorem is referenced by:  qqhvval  24320  qqhf  24323

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-numer 13082  df-denom 13083  df-gz 13253  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-mnd 14645  df-mhm 14693  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-od 15122  df-cmn 15369  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-rnghom 15774  df-drng 15792  df-subrg 15821  df-cnfld 16659  df-zrh 16737  df-chr 16739  df-qqh 24310