Theorem genpelxp 7624

Description: Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.)

Hypothesis

Ref	Expression
genpelvl.1	|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )

Assertion

Ref	Expression
genpelxp	|- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. ( ~P Q. X. ~P Q. ) )

Distinct variable groups:   x, y, z, w, v, A   x, B, y, z, w, v   x, G, y, z, w, v

Allowed substitution hints:   F( x, y, z, w, v)

Proof of Theorem genpelxp

Step	Hyp	Ref	Expression
1		ssrab2 3278	. . . . 5 |- { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) } C_ Q.
2		nqex 7476	. . . . . 6 |- Q. e. _V
3	2	elpw2 4201	. . . . 5 |- ( { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) } e. ~P Q. <-> { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) } C_ Q. )
4	1, 3	mpbir 146	. . . 4 |- { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) } e. ~P Q.
5		ssrab2 3278	. . . . 5 |- { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) } C_ Q.
6	2	elpw2 4201	. . . . 5 |- ( { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) } e. ~P Q. <-> { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) } C_ Q. )
7	5, 6	mpbir 146	. . . 4 |- { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) } e. ~P Q.
8		opelxpi 4707	. . . 4 |- ( ( { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) } e. ~P Q. /\ { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) } e. ~P Q. ) -> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) } >. e. ( ~P Q. X. ~P Q. ) )
9	4, 7, 8	mp2an 426	. . 3 |- <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) } >. e. ( ~P Q. X. ~P Q. )
10		fveq2 5576	. . . . . . . . 9 |- ( w = A -> ( 1st ` w ) = ( 1st ` A ) )
11	10	eleq2d 2275	. . . . . . . 8 |- ( w = A -> ( y e. ( 1st ` w ) <-> y e. ( 1st ` A ) ) )
12	11	3anbi1d 1329	. . . . . . 7 |- ( w = A -> ( ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) <-> ( y e. ( 1st ` A ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) ) )
13	12	2rexbidv 2531	. . . . . 6 |- ( w = A -> ( E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) <-> E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) ) )
14	13	rabbidv 2761	. . . . 5 |- ( w = A -> { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } = { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } )
15		fveq2 5576	. . . . . . . . 9 |- ( w = A -> ( 2nd ` w ) = ( 2nd ` A ) )
16	15	eleq2d 2275	. . . . . . . 8 |- ( w = A -> ( y e. ( 2nd ` w ) <-> y e. ( 2nd ` A ) ) )
17	16	3anbi1d 1329	. . . . . . 7 |- ( w = A -> ( ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) <-> ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) ) )
18	17	2rexbidv 2531	. . . . . 6 |- ( w = A -> ( E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) <-> E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) ) )
19	18	rabbidv 2761	. . . . 5 |- ( w = A -> { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } = { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } )
20	14, 19	opeq12d 3827	. . . 4 |- ( w = A -> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. = <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )
21		fveq2 5576	. . . . . . . . 9 |- ( v = B -> ( 1st ` v ) = ( 1st ` B ) )
22	21	eleq2d 2275	. . . . . . . 8 |- ( v = B -> ( z e. ( 1st ` v ) <-> z e. ( 1st ` B ) ) )
23	22	3anbi2d 1330	. . . . . . 7 |- ( v = B -> ( ( y e. ( 1st ` A ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) <-> ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) ) )
24	23	2rexbidv 2531	. . . . . 6 |- ( v = B -> ( E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) <-> E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) ) )
25	24	rabbidv 2761	. . . . 5 |- ( v = B -> { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } = { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) } )
26		fveq2 5576	. . . . . . . . 9 |- ( v = B -> ( 2nd ` v ) = ( 2nd ` B ) )
27	26	eleq2d 2275	. . . . . . . 8 |- ( v = B -> ( z e. ( 2nd ` v ) <-> z e. ( 2nd ` B ) ) )
28	27	3anbi2d 1330	. . . . . . 7 |- ( v = B -> ( ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) <-> ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) ) )
29	28	2rexbidv 2531	. . . . . 6 |- ( v = B -> ( E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) <-> E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) ) )
30	29	rabbidv 2761	. . . . 5 |- ( v = B -> { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } = { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) } )
31	25, 30	opeq12d 3827	. . . 4 |- ( v = B -> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. = <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) } >. )
32		genpelvl.1	. . . 4 |- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )
33	20, 31, 32	ovmpog 6080	. . 3 |- ( ( A e. P. /\ B e. P. /\ <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) } >. e. ( ~P Q. X. ~P Q. ) ) -> ( A F B ) = <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) } >. )
34	9, 33	mp3an3 1339	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) = <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` A ) /\ z e. ( 1st ` B ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` A ) /\ z e. ( 2nd ` B ) /\ x = ( y G z ) ) } >. )
35	34, 9	eqeltrdi 2296	1 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. ( ~P Q. X. ~P Q. ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2176   E.wrex 2485   {crab 2488   C_ wss 3166   ~Pcpw 3616   <.cop 3636   X. cxp 4673   ` cfv 5271  (class class class)co 5944   e. cmpo 5946   1stc1st 6224   2ndc2nd 6225   Q.cnq 7393   P.cnp 7404

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-qs 6626  df-ni 7417  df-nqqs 7461

This theorem is referenced by:  addclpr  7650  mulclpr  7685