Theorem dfmpq2 6339

Description: Alternative 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.wE.vE.uE.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-mpt2 5460	. 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 6329	. 2 |- .pQ = (x e. ( N. X. N.) , y e. ( N. X. N.) |-> <.(( 1st ` x) .N ( 1st ` y)) , (( 2nd ` x) .N ( 2nd ` y)) >.)
3		1st2nd2 5743	. . . . . . . . . 10 |- (x e. ( N. X. N.) -> x = <.( 1st ` x) , ( 2nd ` x) >.)
4	3	eqeq1d 2045	. . . . . . . . 9 |- (x e. ( N. X. N.) -> (x = <.w , v >. <-> <.( 1st ` x) , ( 2nd ` x) >. = <.w , v >.))
5		1st2nd2 5743	. . . . . . . . . 10 |- (y e. ( N. X. N.) -> y = <.( 1st ` y) , ( 2nd ` y) >.)
6	5	eqeq1d 2045	. . . . . . . . 9 |- (y e. ( N. X. N.) -> (y = <.u , f >. <-> <.( 1st ` y) , ( 2nd ` y) >. = <.u , f >.))
7	4, 6	bi2anan9 538	. . . . . . . 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 438	. . . . . . 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 129	. . . . . 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 1747	. . . . 5 |- ((x e. ( N. X. N.) /\ y e. ( N. X. N.)) -> (E.wE.vE.uE.f(( <.( 1st ` x) , ( 2nd ` x) >. = <.w , v >. /\ <.( 1st ` y) , ( 2nd ` y) >. = <.u , f >.) /\ z = <.(w .N u) , (v .N f) >.) <-> E.wE.vE.uE.f((x = <.w , v >. /\ y = <.u , f >.) /\ z = <.(w .N u) , (v .N f) >.)))
11		xp1st 5734	. . . . . . 7 |- (x e. ( N. X. N.) -> ( 1st ` x) e. N.)
12		xp2nd 5735	. . . . . . 7 |- (x e. ( N. X. N.) -> ( 2nd ` x) e. N.)
13	11, 12	jca 290	. . . . . 6 |- (x e. ( N. X. N.) -> (( 1st ` x) e. N. /\ ( 2nd ` x) e. N.))
14		xp1st 5734	. . . . . . 7 |- (y e. ( N. X. N.) -> ( 1st ` y) e. N.)
15		xp2nd 5735	. . . . . . 7 |- (y e. ( N. X. N.) -> ( 2nd ` y) e. N.)
16	14, 15	jca 290	. . . . . 6 |- (y e. ( N. X. N.) -> (( 1st ` y) e. N. /\ ( 2nd ` y) e. N.))
17		simpll 481	. . . . . . . . . 10 |- (((w = ( 1st ` x) /\ v = ( 2nd ` x)) /\ (u = ( 1st ` y) /\ f = ( 2nd ` y))) -> w = ( 1st ` x))
18		simprl 483	. . . . . . . . . 10 |- (((w = ( 1st ` x) /\ v = ( 2nd ` x)) /\ (u = ( 1st ` y) /\ f = ( 2nd ` y))) -> u = ( 1st ` y))
19	17, 18	oveq12d 5473	. . . . . . . . 9 |- (((w = ( 1st ` x) /\ v = ( 2nd ` x)) /\ (u = ( 1st ` y) /\ f = ( 2nd ` y))) -> (w .N u) = (( 1st ` x) .N ( 1st ` y)))
20		simplr 482	. . . . . . . . . 10 |- (((w = ( 1st ` x) /\ v = ( 2nd ` x)) /\ (u = ( 1st ` y) /\ f = ( 2nd ` y))) -> v = ( 2nd ` x))
21		simprr 484	. . . . . . . . . 10 |- (((w = ( 1st ` x) /\ v = ( 2nd ` x)) /\ (u = ( 1st ` y) /\ f = ( 2nd ` y))) -> f = ( 2nd ` y))
22	20, 21	oveq12d 5473	. . . . . . . . 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 3548	. . . . . . . 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 2048	. . . . . . 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 3975	. . . . . 6 |- (((( 1st ` x) e. N. /\ ( 2nd ` x) e. N.) /\ (( 1st ` y) e. N. /\ ( 2nd ` y) e. N.)) -> (E.wE.vE.uE.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 273	. . . . 5 |- ((x e. ( N. X. N.) /\ y e. ( N. X. N.)) -> (E.wE.vE.uE.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 179	. . . 4 |- ((x e. ( N. X. N.) /\ y e. ( N. X. N.)) -> (E.wE.vE.uE.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 427	. . 3 |- (((x e. ( N. X. N.) /\ y e. ( N. X. N.)) /\ E.wE.vE.uE.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 5502	. 2 |- { <. <.x , y >. , z >. | ((x e. ( N. X. N.) /\ y e. ( N. X. N.)) /\ E.wE.vE.uE.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 2067	1 |- .pQ = { <. <.x , y >. , z >. | ((x e. ( N. X. N.) /\ y e. ( N. X. N.)) /\ E.wE.vE.uE.f((x = <.w , v >. /\ y = <.u , f >.) /\ z = <.(w .N u) , (v .N f) >.)) }

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1242  E.wex 1378   e. wcel 1390   <.cop 3370   X. cxp 4286   ` cfv 4845  (class class class)co 5455   {coprab 5456   |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708   N.cnpi 6256   .N cmi 6258   .pQ cmpq 6261

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-mpq 6329

This theorem is referenced by:  mulpipqqs  6357