Theorem genpdf 7788

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 7755	. . . . . . 7 |- ( w e. P. -> <. ( 1st ` w ) , ( 2nd ` w ) >. e. P. )
3		elprnql 7761	. . . . . . 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 7755	. . . . . . 7 |- ( v e. P. -> <. ( 1st ` v ) , ( 2nd ` v ) >. e. P. )
7		elprnql 7761	. . . . . . 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 7787	. . . 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 7762	. . . . . . 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 7762	. . . . . . 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 7787	. . . 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 3875	. . 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 6096	. 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 2252	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 1005   = wceq 1398   e. wcel 2202   E.wrex 2512   {crab 2515   <.cop 3676   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   1stc1st 6310   2ndc2nd 6311   Q.cnq 7560   P.cnp 7571

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-qs 6751  df-ni 7584  df-nqqs 7628  df-inp 7746

This theorem is referenced by:  genipv  7789