Theorem genpdf 7695

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 7662	. . . . . . 7 |- ( w e. P. -> <. ( 1st ` w ) , ( 2nd ` w ) >. e. P. )
3		elprnql 7668	. . . . . . 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 477	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ r e. ( 1st ` w ) ) -> r e. Q. )
6		prop 7662	. . . . . . 7 |- ( v e. P. -> <. ( 1st ` v ) , ( 2nd ` v ) >. e. P. )
7		elprnql 7668	. . . . . . 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 476	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ s e. ( 1st ` v ) ) -> s e. Q. )
10	5, 9	genpdflem 7694	. . . 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 7669	. . . . . . 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 477	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ r e. ( 2nd ` w ) ) -> r e. Q. )
14		elprnqu 7669	. . . . . . 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 476	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ s e. ( 2nd ` v ) ) -> s e. Q. )
17	13, 16	genpdflem 7694	. . . 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 3865	. . 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 6069	. 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 2250	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 ) } >. )

Syntax hints:   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   {crab 2512   <.cop 3669   ` cfv 5318  (class class class)co 6001   e. cmpo 6003   1stc1st 6284   2ndc2nd 6285   Q.cnq 7467   P.cnp 7478

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-qs 6686  df-ni 7491  df-nqqs 7535  df-inp 7653

This theorem is referenced by:  genipv  7696