Theorem genpelxp 7571

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 3264	. . . . 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 7423	. . . . . 6 |- Q. e. _V
3	2	elpw2 4186	. . . . 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 3264	. . . . 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 4186	. . . . 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 4691	. . . 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 5554	. . . . . . . . 9 |- ( w = A -> ( 1st ` w ) = ( 1st ` A ) )
11	10	eleq2d 2263	. . . . . . . 8 |- ( w = A -> ( y e. ( 1st ` w ) <-> y e. ( 1st ` A ) ) )
12	11	3anbi1d 1327	. . . . . . 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 2519	. . . . . 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 2749	. . . . 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 5554	. . . . . . . . 9 |- ( w = A -> ( 2nd ` w ) = ( 2nd ` A ) )
16	15	eleq2d 2263	. . . . . . . 8 |- ( w = A -> ( y e. ( 2nd ` w ) <-> y e. ( 2nd ` A ) ) )
17	16	3anbi1d 1327	. . . . . . 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 2519	. . . . . 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 2749	. . . . 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 3812	. . . 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 5554	. . . . . . . . 9 |- ( v = B -> ( 1st ` v ) = ( 1st ` B ) )
22	21	eleq2d 2263	. . . . . . . 8 |- ( v = B -> ( z e. ( 1st ` v ) <-> z e. ( 1st ` B ) ) )
23	22	3anbi2d 1328	. . . . . . 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 2519	. . . . . 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 2749	. . . . 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 5554	. . . . . . . . 9 |- ( v = B -> ( 2nd ` v ) = ( 2nd ` B ) )
27	26	eleq2d 2263	. . . . . . . 8 |- ( v = B -> ( z e. ( 2nd ` v ) <-> z e. ( 2nd ` B ) ) )
28	27	3anbi2d 1328	. . . . . . 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 2519	. . . . . 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 2749	. . . . 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 3812	. . . 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 6053	. . 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 1337	. 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 2284	1 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. ( ~P Q. X. ~P Q. ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   E.wrex 2473   {crab 2476   C_ wss 3153   ~Pcpw 3601   <.cop 3621   X. cxp 4657   ` cfv 5254  (class class class)co 5918   e. cmpo 5920   1stc1st 6191   2ndc2nd 6192   Q.cnq 7340   P.cnp 7351

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-qs 6593  df-ni 7364  df-nqqs 7408

This theorem is referenced by:  addclpr  7597  mulclpr  7632