Theorem genpelxp 7659

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 3286	. . . . 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 7511	. . . . . 6 |- Q. e. _V
3	2	elpw2 4217	. . . . 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 3286	. . . . 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 4217	. . . . 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 4725	. . . 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 5599	. . . . . . . . 9 |- ( w = A -> ( 1st ` w ) = ( 1st ` A ) )
11	10	eleq2d 2277	. . . . . . . 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 2533	. . . . . 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 2765	. . . . 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 5599	. . . . . . . . 9 |- ( w = A -> ( 2nd ` w ) = ( 2nd ` A ) )
16	15	eleq2d 2277	. . . . . . . 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 2533	. . . . . 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 2765	. . . . 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 3841	. . . 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 5599	. . . . . . . . 9 |- ( v = B -> ( 1st ` v ) = ( 1st ` B ) )
22	21	eleq2d 2277	. . . . . . . 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 2533	. . . . . 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 2765	. . . . 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 5599	. . . . . . . . 9 |- ( v = B -> ( 2nd ` v ) = ( 2nd ` B ) )
27	26	eleq2d 2277	. . . . . . . 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 2533	. . . . . 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 2765	. . . . 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 3841	. . . 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 6103	. . 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 2298	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 2178   E.wrex 2487   {crab 2490   C_ wss 3174   ~Pcpw 3626   <.cop 3646   X. cxp 4691   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   1stc1st 6247   2ndc2nd 6248   Q.cnq 7428   P.cnp 7439

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-qs 6649  df-ni 7452  df-nqqs 7496

This theorem is referenced by:  addclpr  7685  mulclpr  7720