Theorem dfmpq2 7111

Description: Alternate definition of pre-multiplication on positive fractions. (Contributed by Jim Kingdon, 13-Sep-2019.)

Assertion

Ref	Expression
dfmpq2	|- .pQ = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) ) }

Distinct variable group:   x, y, z, w, v, u, f

Proof of Theorem dfmpq2

Step	Hyp	Ref	Expression
1		df-mpo 5733	. 2 |- ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ z = <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) }
2		df-mpq 7101	. 2 |- .pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
3		1st2nd2 6027	. . . . . . . . . 10 |- ( x e. ( N. X. N. ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
4	3	eqeq1d 2123	. . . . . . . . 9 |- ( x e. ( N. X. N. ) -> ( x = <. w , v >. <-> <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. ) )
5		1st2nd2 6027	. . . . . . . . . 10 |- ( y e. ( N. X. N. ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
6	5	eqeq1d 2123	. . . . . . . . 9 |- ( y e. ( N. X. N. ) -> ( y = <. u , f >. <-> <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) )
7	4, 6	bi2anan9 578	. . . . . . . 8 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( ( x = <. w , v >. /\ y = <. u , f >. ) <-> ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. /\ <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) ) )
8	7	anbi1d 458	. . . . . . 7 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) <-> ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. /\ <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) ) )
9	8	bicomd 140	. . . . . 6 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. /\ <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) <-> ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) ) )
10	9	4exbidv 1824	. . . . 5 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( E. w E. v E. u E. f ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. /\ <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) <-> E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) ) )
11		xp1st 6017	. . . . . . 7 |- ( x e. ( N. X. N. ) -> ( 1st ` x ) e. N. )
12		xp2nd 6018	. . . . . . 7 |- ( x e. ( N. X. N. ) -> ( 2nd ` x ) e. N. )
13	11, 12	jca 302	. . . . . 6 |- ( x e. ( N. X. N. ) -> ( ( 1st ` x ) e. N. /\ ( 2nd ` x ) e. N. ) )
14		xp1st 6017	. . . . . . 7 |- ( y e. ( N. X. N. ) -> ( 1st ` y ) e. N. )
15		xp2nd 6018	. . . . . . 7 |- ( y e. ( N. X. N. ) -> ( 2nd ` y ) e. N. )
16	14, 15	jca 302	. . . . . 6 |- ( y e. ( N. X. N. ) -> ( ( 1st ` y ) e. N. /\ ( 2nd ` y ) e. N. ) )
17		simpll 501	. . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> w = ( 1st ` x ) )
18		simprl 503	. . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> u = ( 1st ` y ) )
19	17, 18	oveq12d 5746	. . . . . . . . 9 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( w .N u ) = ( ( 1st ` x ) .N ( 1st ` y ) ) )
20		simplr 502	. . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> v = ( 2nd ` x ) )
21		simprr 504	. . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> f = ( 2nd ` y ) )
22	20, 21	oveq12d 5746	. . . . . . . . 9 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( v .N f ) = ( ( 2nd ` x ) .N ( 2nd ` y ) ) )
23	19, 22	opeq12d 3679	. . . . . . . 8 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> <. ( w .N u ) , ( v .N f ) >. = <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
24	23	eqeq2d 2126	. . . . . . 7 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( z = <. ( w .N u ) , ( v .N f ) >. <-> z = <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) )
25	24	copsex4g 4129	. . . . . 6 |- ( ( ( ( 1st ` x ) e. N. /\ ( 2nd ` x ) e. N. ) /\ ( ( 1st ` y ) e. N. /\ ( 2nd ` y ) e. N. ) ) -> ( E. w E. v E. u E. f ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. /\ <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) <-> z = <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) )
26	13, 16, 25	syl2an 285	. . . . 5 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( E. w E. v E. u E. f ( ( <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. /\ <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) <-> z = <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) )
27	10, 26	bitr3d 189	. . . 4 |- ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) -> ( E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) <-> z = <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) )
28	27	pm5.32i 447	. . 3 |- ( ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) ) <-> ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ z = <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) )
29	28	oprabbii 5780	. 2 |- { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) ) } = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ z = <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. ) }
30	1, 2, 29	3eqtr4i 2145	1 |- .pQ = { <. <. x , y >. , z >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w .N u ) , ( v .N f ) >. ) ) }

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1314   E.wex 1451   e. wcel 1463   <.cop 3496   X. cxp 4497   ` cfv 5081  (class class class)co 5728   {coprab 5729   e. cmpo 5730   1stc1st 5990   2ndc2nd 5991   N.cnpi 7028   .N cmi 7030   .pQ cmpq 7033

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-iota 5046  df-fun 5083  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-mpq 7101

This theorem is referenced by:  mulpipqqs  7129