Theorem genpelxp 7791

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 3313	. . . . 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 7643	. . . . . 6 |- Q. e. _V
3	2	elpw2 4252	. . . . 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 3313	. . . . 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 4252	. . . . 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 4763	. . . 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 5648	. . . . . . . . 9 |- ( w = A -> ( 1st ` w ) = ( 1st ` A ) )
11	10	eleq2d 2301	. . . . . . . 8 |- ( w = A -> ( y e. ( 1st ` w ) <-> y e. ( 1st ` A ) ) )
12	11	3anbi1d 1353	. . . . . . 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 2558	. . . . . 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 2792	. . . . 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 5648	. . . . . . . . 9 |- ( w = A -> ( 2nd ` w ) = ( 2nd ` A ) )
16	15	eleq2d 2301	. . . . . . . 8 |- ( w = A -> ( y e. ( 2nd ` w ) <-> y e. ( 2nd ` A ) ) )
17	16	3anbi1d 1353	. . . . . . 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 2558	. . . . . 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 2792	. . . . 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 3875	. . . 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 5648	. . . . . . . . 9 |- ( v = B -> ( 1st ` v ) = ( 1st ` B ) )
22	21	eleq2d 2301	. . . . . . . 8 |- ( v = B -> ( z e. ( 1st ` v ) <-> z e. ( 1st ` B ) ) )
23	22	3anbi2d 1354	. . . . . . 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 2558	. . . . . 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 2792	. . . . 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 5648	. . . . . . . . 9 |- ( v = B -> ( 2nd ` v ) = ( 2nd ` B ) )
27	26	eleq2d 2301	. . . . . . . 8 |- ( v = B -> ( z e. ( 2nd ` v ) <-> z e. ( 2nd ` B ) ) )
28	27	3anbi2d 1354	. . . . . . 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 2558	. . . . . 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 2792	. . . . 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 3875	. . . 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 6166	. . 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 1363	. 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 2322	1 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. ( ~P Q. X. ~P Q. ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2202   E.wrex 2512   {crab 2515   C_ wss 3201   ~Pcpw 3656   <.cop 3676   X. cxp 4729   ` 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-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  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-ne 2404  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-ov 6031  df-oprab 6032  df-mpo 6033  df-qs 6751  df-ni 7584  df-nqqs 7628

This theorem is referenced by:  addclpr  7817  mulclpr  7852