Theorem genpelxp 7443

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 3222	. . . . 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 7295	. . . . . 6 |- Q. e. _V
3	2	elpw2 4130	. . . . 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 145	. . . 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 3222	. . . . 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 4130	. . . . 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 145	. . . 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 4630	. . . 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 423	. . 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 5480	. . . . . . . . 9 |- ( w = A -> ( 1st ` w ) = ( 1st ` A ) )
11	10	eleq2d 2234	. . . . . . . 8 |- ( w = A -> ( y e. ( 1st ` w ) <-> y e. ( 1st ` A ) ) )
12	11	3anbi1d 1305	. . . . . . 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 2489	. . . . . 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 2710	. . . . 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 5480	. . . . . . . . 9 |- ( w = A -> ( 2nd ` w ) = ( 2nd ` A ) )
16	15	eleq2d 2234	. . . . . . . 8 |- ( w = A -> ( y e. ( 2nd ` w ) <-> y e. ( 2nd ` A ) ) )
17	16	3anbi1d 1305	. . . . . . 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 2489	. . . . . 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 2710	. . . . 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 3760	. . . 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 5480	. . . . . . . . 9 |- ( v = B -> ( 1st ` v ) = ( 1st ` B ) )
22	21	eleq2d 2234	. . . . . . . 8 |- ( v = B -> ( z e. ( 1st ` v ) <-> z e. ( 1st ` B ) ) )
23	22	3anbi2d 1306	. . . . . . 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 2489	. . . . . 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 2710	. . . . 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 5480	. . . . . . . . 9 |- ( v = B -> ( 2nd ` v ) = ( 2nd ` B ) )
27	26	eleq2d 2234	. . . . . . . 8 |- ( v = B -> ( z e. ( 2nd ` v ) <-> z e. ( 2nd ` B ) ) )
28	27	3anbi2d 1306	. . . . . . 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 2489	. . . . . 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 2710	. . . . 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 3760	. . . 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 5967	. . 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 1315	. 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 2255	1 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. ( ~P Q. X. ~P Q. ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 967   = wceq 1342   e. wcel 2135   E.wrex 2443   {crab 2446   C_ wss 3111   ~Pcpw 3553   <.cop 3573   X. cxp 4596   ` cfv 5182  (class class class)co 5836   e. cmpo 5838   1stc1st 6098   2ndc2nd 6099   Q.cnq 7212   P.cnp 7223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-qs 6498  df-ni 7236  df-nqqs 7280

This theorem is referenced by:  addclpr  7469  mulclpr  7504