Theorem qqhval 30018

Description: Value of the canonical homormorphism from the rational number to a field. (Contributed by Thierry Arnoux, 22-Oct-2017.)

Hypotheses

Ref	Expression
qqhval.1	|- ./ = (/r ` R )
qqhval.2	|- .1. = ( 1r ` R )
qqhval.3	|- L = ( ZRHom ` R )

Assertion

Ref	Expression
qqhval	|- ( R e. _V -> (QQHom ` R ) = ran ( x e. ZZ , y e. ( `' L " (Unit ` R ) ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )

Distinct variable groups:   x, y, R   y, L

Allowed substitution hints:   ./ ( x, y)   .1. ( x, y)   L( x)

Proof of Theorem qqhval

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2623	. . . 4 |- ( f = R -> ZZ = ZZ )
2		fveq2 6191	. . . . . . 7 |- ( f = R -> ( ZRHom ` f ) = ( ZRHom ` R ) )
3		qqhval.3	. . . . . . 7 |- L = ( ZRHom ` R )
4	2, 3	syl6eqr 2674	. . . . . 6 |- ( f = R -> ( ZRHom ` f ) = L )
5	4	cnveqd 5298	. . . . 5 |- ( f = R -> `' ( ZRHom ` f ) = `' L )
6		fveq2 6191	. . . . 5 |- ( f = R -> (Unit ` f ) = (Unit ` R ) )
7	5, 6	imaeq12d 5467	. . . 4 |- ( f = R -> ( `' ( ZRHom ` f ) " (Unit ` f ) ) = ( `' L " (Unit ` R ) ) )
8		fveq2 6191	. . . . . . 7 |- ( f = R -> (/r ` f ) = (/r ` R ) )
9		qqhval.1	. . . . . . 7 |- ./ = (/r ` R )
10	8, 9	syl6eqr 2674	. . . . . 6 |- ( f = R -> (/r ` f ) = ./ )
11	4	fveq1d 6193	. . . . . 6 |- ( f = R -> ( ( ZRHom ` f ) ` x ) = ( L ` x ) )
12	4	fveq1d 6193	. . . . . 6 |- ( f = R -> ( ( ZRHom ` f ) ` y ) = ( L ` y ) )
13	10, 11, 12	oveq123d 6671	. . . . 5 |- ( f = R -> ( ( ( ZRHom ` f ) ` x ) (/r ` f ) ( ( ZRHom ` f ) ` y ) ) = ( ( L ` x ) ./ ( L ` y ) ) )
14	13	opeq2d 4409	. . . 4 |- ( f = R -> <. ( x / y ) , ( ( ( ZRHom ` f ) ` x ) (/r ` f ) ( ( ZRHom ` f ) ` y ) ) >. = <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. )
15	1, 7, 14	mpt2eq123dv 6717	. . 3 |- ( f = R -> ( x e. ZZ , y e. ( `' ( ZRHom ` f ) " (Unit ` f ) ) |-> <. ( x / y ) , ( ( ( ZRHom ` f ) ` x ) (/r ` f ) ( ( ZRHom ` f ) ` y ) ) >. ) = ( x e. ZZ , y e. ( `' L " (Unit ` R ) ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )
16	15	rneqd 5353	. 2 |- ( f = R -> ran ( x e. ZZ , y e. ( `' ( ZRHom ` f ) " (Unit ` f ) ) |-> <. ( x / y ) , ( ( ( ZRHom ` f ) ` x ) (/r ` f ) ( ( ZRHom ` f ) ` y ) ) >. ) = ran ( x e. ZZ , y e. ( `' L " (Unit ` R ) ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )
17		df-qqh 30017	. 2 |- QQHom = ( f e. _V |-> ran ( x e. ZZ , y e. ( `' ( ZRHom ` f ) " (Unit ` f ) ) |-> <. ( x / y ) , ( ( ( ZRHom ` f ) ` x ) (/r ` f ) ( ( ZRHom ` f ) ` y ) ) >. ) )
18		zex 11386	. . . 4 |- ZZ e. _V
19		fvex 6201	. . . . . . 7 |- ( ZRHom ` R ) e. _V
20	3, 19	eqeltri 2697	. . . . . 6 |- L e. _V
21	20	cnvex 7113	. . . . 5 |- `' L e. _V
22		imaexg 7103	. . . . 5 |- ( `' L e. _V -> ( `' L " (Unit ` R ) ) e. _V )
23	21, 22	ax-mp 5	. . . 4 |- ( `' L " (Unit ` R ) ) e. _V
24	18, 23	mpt2ex 7247	. . 3 |- ( x e. ZZ , y e. ( `' L " (Unit ` R ) ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) e. _V
25	24	rnex 7100	. 2 |- ran ( x e. ZZ , y e. ( `' L " (Unit ` R ) ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) e. _V
26	16, 17, 25	fvmpt 6282	1 |- ( R e. _V -> (QQHom ` R ) = ran ( x e. ZZ , y e. ( `' L " (Unit ` R ) ) |-> <. ( x / y ) , ( ( L ` x ) ./ ( L ` y ) ) >. ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   `'ccnv 5113   ran crn 5115   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   / cdiv 10684   ZZcz 11377   1rcur 18501  Unitcui 18639  /_rcdvr 18682   ZRHomczrh 19848  QQHomcqqh 30016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-neg 10269  df-z 11378  df-qqh 30017

This theorem is referenced by:  qqhval2  30026