Theorem genpdf 6491

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 6458	. . . . . . 7 |- (w e. P. -> <.( 1st ` w) , ( 2nd ` w) >. e. P.)
3		elprnql 6464	. . . . . . 7 |- (( <.( 1st ` w) , ( 2nd ` w) >. e. P. /\ r e. ( 1st ` w)) -> r e. Q.)
4	2, 3	sylan 267	. . . . . 6 |- ((w e. P. /\ r e. ( 1st ` w)) -> r e. Q.)
5	4	adantlr 446	. . . . 5 |- (((w e. P. /\ v e. P.) /\ r e. ( 1st ` w)) -> r e. Q.)
6		prop 6458	. . . . . . 7 |- (v e. P. -> <.( 1st ` v) , ( 2nd ` v) >. e. P.)
7		elprnql 6464	. . . . . . 7 |- (( <.( 1st ` v) , ( 2nd ` v) >. e. P. /\ s e. ( 1st ` v)) -> s e. Q.)
8	6, 7	sylan 267	. . . . . 6 |- ((v e. P. /\ s e. ( 1st ` v)) -> s e. Q.)
9	8	adantll 445	. . . . 5 |- (((w e. P. /\ v e. P.) /\ s e. ( 1st ` v)) -> s e. Q.)
10	5, 9	genpdflem 6490	. . . 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 6465	. . . . . . 7 |- (( <.( 1st ` w) , ( 2nd ` w) >. e. P. /\ r e. ( 2nd ` w)) -> r e. Q.)
12	2, 11	sylan 267	. . . . . 6 |- ((w e. P. /\ r e. ( 2nd ` w)) -> r e. Q.)
13	12	adantlr 446	. . . . 5 |- (((w e. P. /\ v e. P.) /\ r e. ( 2nd ` w)) -> r e. Q.)
14		elprnqu 6465	. . . . . . 7 |- (( <.( 1st ` v) , ( 2nd ` v) >. e. P. /\ s e. ( 2nd ` v)) -> s e. Q.)
15	6, 14	sylan 267	. . . . . 6 |- ((v e. P. /\ s e. ( 2nd ` v)) -> s e. Q.)
16	15	adantll 445	. . . . 5 |- (((w e. P. /\ v e. P.) /\ s e. ( 2nd ` v)) -> s e. Q.)
17	13, 16	genpdflem 6490	. . . 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 3548	. . 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 5512	. 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 2057	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 97   /\ w3a 884   = wceq 1242   e. wcel 1390  E.wrex 2301   {crab 2304   <.cop 3370   ` cfv 4845  (class class class)co 5455   |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264   P.cnp 6275

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-qs 6048  df-ni 6288  df-nqqs 6332  df-inp 6449

This theorem is referenced by:  genipv  6492