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Theorem dfmpq2 7503
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
StepHypRef Expression
1 df-mpo 5972 . 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 7493 . 2 |- .pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
3 1st2nd2 6284 . . . . . . . . . 10 |- ( x e. ( N. X. N. ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
43eqeq1d 2216 . . . . . . . . 9 |- ( x e. ( N. X. N. ) -> ( x = <. w , v >. <-> <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. ) )
5 1st2nd2 6284 . . . . . . . . . 10 |- ( y e. ( N. X. N. ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
65eqeq1d 2216 . . . . . . . . 9 |- ( y e. ( N. X. N. ) -> ( y = <. u , f >. <-> <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) )
74, 6bi2anan9 606 . . . . . . . 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 >. ) ) )
87anbi1d 465 . . . . . . 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 ) >. ) ) )
98bicomd 141 . . . . . 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 ) >. ) ) )
1094exbidv 1894 . . . . 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 6274 . . . . . . 7 |- ( x e. ( N. X. N. ) -> ( 1st ` x ) e. N. )
12 xp2nd 6275 . . . . . . 7 |- ( x e. ( N. X. N. ) -> ( 2nd ` x ) e. N. )
1311, 12jca 306 . . . . . 6 |- ( x e. ( N. X. N. ) -> ( ( 1st ` x ) e. N. /\ ( 2nd ` x ) e. N. ) )
14 xp1st 6274 . . . . . . 7 |- ( y e. ( N. X. N. ) -> ( 1st ` y ) e. N. )
15 xp2nd 6275 . . . . . . 7 |- ( y e. ( N. X. N. ) -> ( 2nd ` y ) e. N. )
1614, 15jca 306 . . . . . 6 |- ( y e. ( N. X. N. ) -> ( ( 1st ` y ) e. N. /\ ( 2nd ` y ) e. N. ) )
17 simpll 527 . . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> w = ( 1st ` x ) )
18 simprl 529 . . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> u = ( 1st ` y ) )
1917, 18oveq12d 5985 . . . . . . . . 9 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( w .N u ) = ( ( 1st ` x ) .N ( 1st ` y ) ) )
20 simplr 528 . . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> v = ( 2nd ` x ) )
21 simprr 531 . . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> f = ( 2nd ` y ) )
2220, 21oveq12d 5985 . . . . . . . . 9 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( v .N f ) = ( ( 2nd ` x ) .N ( 2nd ` y ) ) )
2319, 22opeq12d 3841 . . . . . . . 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 ) ) >. )
2423eqeq2d 2219 . . . . . . 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 ) ) >. ) )
2524copsex4g 4309 . . . . . 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 ) ) >. ) )
2613, 16, 25syl2an 289 . . . . 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 ) ) >. ) )
2710, 26bitr3d 190 . . . 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 ) ) >. ) )
2827pm5.32i 454 . . 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 ) ) >. ) )
2928oprabbii 6023 . 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 ) ) >. ) }
301, 2, 293eqtr4i 2238 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 ) >. ) ) }
Colors of variables: wff set class
Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2178   <.cop 3646   X. cxp 4691   ` cfv 5290  (class class class)co 5967   {coprab 5968   e. cmpo 5969   1stc1st 6247   2ndc2nd 6248   N.cnpi 7420   .N cmi 7422   .pQ cmpq 7425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-mpq 7493
This theorem is referenced by:  mulpipqqs  7521
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