Theorem genpdf 6698

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 6665	. . . . . . 7 |- ( w e. P. -> <. ( 1st ` w ) , ( 2nd ` w ) >. e. P. )
3		elprnql 6671	. . . . . . 7 |- ( ( <. ( 1st ` w ) , ( 2nd ` w ) >. e. P. /\ r e. ( 1st ` w ) ) -> r e. Q. )
4	2, 3	sylan 277	. . . . . 6 |- ( ( w e. P. /\ r e. ( 1st ` w ) ) -> r e. Q. )
5	4	adantlr 460	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ r e. ( 1st ` w ) ) -> r e. Q. )
6		prop 6665	. . . . . . 7 |- ( v e. P. -> <. ( 1st ` v ) , ( 2nd ` v ) >. e. P. )
7		elprnql 6671	. . . . . . 7 |- ( ( <. ( 1st ` v ) , ( 2nd ` v ) >. e. P. /\ s e. ( 1st ` v ) ) -> s e. Q. )
8	6, 7	sylan 277	. . . . . 6 |- ( ( v e. P. /\ s e. ( 1st ` v ) ) -> s e. Q. )
9	8	adantll 459	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ s e. ( 1st ` v ) ) -> s e. Q. )
10	5, 9	genpdflem 6697	. . . 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 6672	. . . . . . 7 |- ( ( <. ( 1st ` w ) , ( 2nd ` w ) >. e. P. /\ r e. ( 2nd ` w ) ) -> r e. Q. )
12	2, 11	sylan 277	. . . . . 6 |- ( ( w e. P. /\ r e. ( 2nd ` w ) ) -> r e. Q. )
13	12	adantlr 460	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ r e. ( 2nd ` w ) ) -> r e. Q. )
14		elprnqu 6672	. . . . . . 7 |- ( ( <. ( 1st ` v ) , ( 2nd ` v ) >. e. P. /\ s e. ( 2nd ` v ) ) -> s e. Q. )
15	6, 14	sylan 277	. . . . . 6 |- ( ( v e. P. /\ s e. ( 2nd ` v ) ) -> s e. Q. )
16	15	adantll 459	. . . . 5 |- ( ( ( w e. P. /\ v e. P. ) /\ s e. ( 2nd ` v ) ) -> s e. Q. )
17	13, 16	genpdflem 6697	. . . 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 3578	. . 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	mpt2eq3ia 5590	. 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 2101	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 102   /\ w3a 919   = wceq 1284   e. wcel 1433   E.wrex 2349   {crab 2352   <.cop 3401   ` cfv 4922  (class class class)co 5532   |-> cmpt2 5534   1stc1st 5785   2ndc2nd 5786   Q.cnq 6470   P.cnp 6481

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-qs 6135  df-ni 6494  df-nqqs 6538  df-inp 6656

This theorem is referenced by:  genipv  6699