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Theorem dfmpq2 7722
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 6090 . 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 7712 . 2 |- .pQ = ( x e. ( N. X. N. ) , y e. ( N. X. N. ) |-> <. ( ( 1st ` x ) .N ( 1st ` y ) ) , ( ( 2nd ` x ) .N ( 2nd ` y ) ) >. )
3 1st2nd2 6409 . . . . . . . . . 10 |- ( x e. ( N. X. N. ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
43eqeq1d 2247 . . . . . . . . 9 |- ( x e. ( N. X. N. ) -> ( x = <. w , v >. <-> <. ( 1st ` x ) , ( 2nd ` x ) >. = <. w , v >. ) )
5 1st2nd2 6409 . . . . . . . . . 10 |- ( y e. ( N. X. N. ) -> y = <. ( 1st ` y ) , ( 2nd ` y ) >. )
65eqeq1d 2247 . . . . . . . . 9 |- ( y e. ( N. X. N. ) -> ( y = <. u , f >. <-> <. ( 1st ` y ) , ( 2nd ` y ) >. = <. u , f >. ) )
74, 6bi2anan9 614 . . . . . . . 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 469 . . . . . . 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 1923 . . . . 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 6399 . . . . . . 7 |- ( x e. ( N. X. N. ) -> ( 1st ` x ) e. N. )
12 xp2nd 6400 . . . . . . 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 6399 . . . . . . 7 |- ( y e. ( N. X. N. ) -> ( 1st ` y ) e. N. )
15 xp2nd 6400 . . . . . . 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 531 . . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> w = ( 1st ` x ) )
18 simprl 535 . . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> u = ( 1st ` y ) )
1917, 18oveq12d 6103 . . . . . . . . 9 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( w .N u ) = ( ( 1st ` x ) .N ( 1st ` y ) ) )
20 simplr 533 . . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> v = ( 2nd ` x ) )
21 simprr 537 . . . . . . . . . 10 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> f = ( 2nd ` y ) )
2220, 21oveq12d 6103 . . . . . . . . 9 |- ( ( ( w = ( 1st ` x ) /\ v = ( 2nd ` x ) ) /\ ( u = ( 1st ` y ) /\ f = ( 2nd ` y ) ) ) -> ( v .N f ) = ( ( 2nd ` x ) .N ( 2nd ` y ) ) )
2319, 22opeq12d 3912 . . . . . . . 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 2250 . . . . . . 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 4387 . . . . . 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 458 . . 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 6143 . 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 2269 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
This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   <.cop 3712   X. cxp 4772   ` cfv 5377  (class class class)co 6085   {coprab 6086   e. cmpo 6087   1stc1st 6372   2ndc2nd 6373   N.cnpi 7639   .N cmi 7641   .pQ cmpq 7644
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578
This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-mpq 7712
This theorem is used by:  mulpipqqs  7740
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