Theorem genpdf 7309

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 7276	. . . . . . 7 |- ( w e. P. -> <. ( 1st ` w ) , ( 2nd ` w ) >. e. P. )
3		elprnql 7282	. . . . . . 7 |- ( ( <. ( 1st ` w ) , ( 2nd ` w ) >. e. P. /\ r e. ( 1st ` w ) ) -> r e. Q. )
4	2, 3	sylan 281	. . . . . 6 |- ( ( w e. P. /\ r e. ( 1st ` w ) ) -> r e. Q. )
5	4	adantlr 468	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ r e. ( 1st ` w ) ) -> r e. Q. )
6		prop 7276	. . . . . . 7 |- ( v e. P. -> <. ( 1st ` v ) , ( 2nd ` v ) >. e. P. )
7		elprnql 7282	. . . . . . 7 |- ( ( <. ( 1st ` v ) , ( 2nd ` v ) >. e. P. /\ s e. ( 1st ` v ) ) -> s e. Q. )
8	6, 7	sylan 281	. . . . . 6 |- ( ( v e. P. /\ s e. ( 1st ` v ) ) -> s e. Q. )
9	8	adantll 467	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ s e. ( 1st ` v ) ) -> s e. Q. )
10	5, 9	genpdflem 7308	. . . 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 7283	. . . . . . 7 |- ( ( <. ( 1st ` w ) , ( 2nd ` w ) >. e. P. /\ r e. ( 2nd ` w ) ) -> r e. Q. )
12	2, 11	sylan 281	. . . . . 6 |- ( ( w e. P. /\ r e. ( 2nd ` w ) ) -> r e. Q. )
13	12	adantlr 468	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ r e. ( 2nd ` w ) ) -> r e. Q. )
14		elprnqu 7283	. . . . . . 7 |- ( ( <. ( 1st ` v ) , ( 2nd ` v ) >. e. P. /\ s e. ( 2nd ` v ) ) -> s e. Q. )
15	6, 14	sylan 281	. . . . . 6 |- ( ( v e. P. /\ s e. ( 2nd ` v ) ) -> s e. Q. )
16	15	adantll 467	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ s e. ( 2nd ` v ) ) -> s e. Q. )
17	13, 16	genpdflem 7308	. . . 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 3708	. . 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 5829	. 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 2158	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 103   /\ w3a 962   = wceq 1331   e. wcel 1480   E.wrex 2415   {crab 2418   <.cop 3525   ` cfv 5118  (class class class)co 5767   e. cmpo 5769   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081   P.cnp 7092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-qs 6428  df-ni 7105  df-nqqs 7149  df-inp 7267

This theorem is referenced by:  genipv  7310