Theorem genpdf 7875

Description: Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.)

Hypothesis

Ref	Expression
genpdf.1	|- F = ( w e. P. , v e. P. |-> <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` w ) /\ s e. ( 1st ` v ) /\ q = ( r G s ) ) } , { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` w ) /\ s e. ( 2nd ` v ) /\ q = ( r G s ) ) } >. )

Assertion

Ref	Expression
genpdf	|- F = ( w e. P. , v e. P. |-> <. { q e. Q. | E. r e. ( 1st ` w ) E. s e. ( 1st ` v ) q = ( r G s ) } , { q e. Q. | E. r e. ( 2nd ` w ) E. s e. ( 2nd ` v ) q = ( r G s ) } >. )

Distinct variable group:   r, q, s, v, w

Allowed substitution hints:   F( w, v, s, r, q)   G( w, v, s, r, q)

Proof of Theorem genpdf

Step	Hyp	Ref	Expression
1		genpdf.1	. 2 |- F = ( w e. P. , v e. P. |-> <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` w ) /\ s e. ( 1st ` v ) /\ q = ( r G s ) ) } , { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` w ) /\ s e. ( 2nd ` v ) /\ q = ( r G s ) ) } >. )
2		prop 7842	. . . . . . 7 |- ( w e. P. -> <. ( 1st ` w ) , ( 2nd ` w ) >. e. P. )
3		elprnql 7848	. . . . . . 7 |- ( ( <. ( 1st ` w ) , ( 2nd ` w ) >. e. P. /\ r e. ( 1st ` w ) ) -> r e. Q. )
4	2, 3	sylan 283	. . . . . 6 |- ( ( w e. P. /\ r e. ( 1st ` w ) ) -> r e. Q. )
5	4	adantlr 481	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ r e. ( 1st ` w ) ) -> r e. Q. )
6		prop 7842	. . . . . . 7 |- ( v e. P. -> <. ( 1st ` v ) , ( 2nd ` v ) >. e. P. )
7		elprnql 7848	. . . . . . 7 |- ( ( <. ( 1st ` v ) , ( 2nd ` v ) >. e. P. /\ s e. ( 1st ` v ) ) -> s e. Q. )
8	6, 7	sylan 283	. . . . . 6 |- ( ( v e. P. /\ s e. ( 1st ` v ) ) -> s e. Q. )
9	8	adantll 480	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ s e. ( 1st ` v ) ) -> s e. Q. )
10	5, 9	genpdflem 7874	. . . 4 |- ( ( w e. P. /\ v e. P. ) -> { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` w ) /\ s e. ( 1st ` v ) /\ q = ( r G s ) ) } = { q e. Q. | E. r e. ( 1st ` w ) E. s e. ( 1st ` v ) q = ( r G s ) } )
11		elprnqu 7849	. . . . . . 7 |- ( ( <. ( 1st ` w ) , ( 2nd ` w ) >. e. P. /\ r e. ( 2nd ` w ) ) -> r e. Q. )
12	2, 11	sylan 283	. . . . . 6 |- ( ( w e. P. /\ r e. ( 2nd ` w ) ) -> r e. Q. )
13	12	adantlr 481	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ r e. ( 2nd ` w ) ) -> r e. Q. )
14		elprnqu 7849	. . . . . . 7 |- ( ( <. ( 1st ` v ) , ( 2nd ` v ) >. e. P. /\ s e. ( 2nd ` v ) ) -> s e. Q. )
15	6, 14	sylan 283	. . . . . 6 |- ( ( v e. P. /\ s e. ( 2nd ` v ) ) -> s e. Q. )
16	15	adantll 480	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ s e. ( 2nd ` v ) ) -> s e. Q. )
17	13, 16	genpdflem 7874	. . . 4 |- ( ( w e. P. /\ v e. P. ) -> { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` w ) /\ s e. ( 2nd ` v ) /\ q = ( r G s ) ) } = { q e. Q. | E. r e. ( 2nd ` w ) E. s e. ( 2nd ` v ) q = ( r G s ) } )
18	10, 17	opeq12d 3912	. . 3 |- ( ( w e. P. /\ v e. P. ) -> <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` w ) /\ s e. ( 1st ` v ) /\ q = ( r G s ) ) } , { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` w ) /\ s e. ( 2nd ` v ) /\ q = ( r G s ) ) } >. = <. { q e. Q. | E. r e. ( 1st ` w ) E. s e. ( 1st ` v ) q = ( r G s ) } , { q e. Q. | E. r e. ( 2nd ` w ) E. s e. ( 2nd ` v ) q = ( r G s ) } >. )
19	18	mpoeq3ia 6153	. 2 |- ( w e. P. , v e. P. |-> <. { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 1st ` w ) /\ s e. ( 1st ` v ) /\ q = ( r G s ) ) } , { q e. Q. | E. r e. Q. E. s e. Q. ( r e. ( 2nd ` w ) /\ s e. ( 2nd ` v ) /\ q = ( r G s ) ) } >. ) = ( w e. P. , v e. P. |-> <. { q e. Q. | E. r e. ( 1st ` w ) E. s e. ( 1st ` v ) q = ( r G s ) } , { q e. Q. | E. r e. ( 2nd ` w ) E. s e. ( 2nd ` v ) q = ( r G s ) } >. )
20	1, 19	eqtri 2259	1 |- F = ( w e. P. , v e. P. |-> <. { q e. Q. | E. r e. ( 1st ` w ) E. s e. ( 1st ` v ) q = ( r G s ) } , { q e. Q. | E. r e. ( 2nd ` w ) E. s e. ( 2nd ` v ) q = ( r G s ) } >. )

This proof depends on syntax axioms:   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   {crab 2532   <.cop 3712   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   1stc1st 6372   2ndc2nd 6373   Q.cnq 7647   P.cnp 7658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  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-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-iinf 4735

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-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  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-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-qs 6813  df-ni 7671  df-nqqs 7715  df-inp 7833

This theorem is used by:  genipv  7876