Theorem genpelvu 7511

Description: Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.)

Hypotheses

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 ) ) } >. )
genpelvl.2	|- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )

Assertion

Ref	Expression
genpelvu	|- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 2nd ` ( A F B ) ) <-> E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) C = ( g G h ) ) )

Distinct variable groups:   x, y, z, g, h, w, v, A   x, B, y, z, g, h, w, v   x, G, y, z, g, h, w, v   g, F   C, g, h

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

Proof of Theorem genpelvu

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		genpelvl.1	. . . . . . 7 |- 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 ) ) } >. )
2		genpelvl.2	. . . . . . 7 |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )
3	1, 2	genipv 7507	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) = <. { f e. Q. | E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) } , { f e. Q. | E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) } >. )
4	3	fveq2d 5519	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( 2nd ` ( A F B ) ) = ( 2nd ` <. { f e. Q. | E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) } , { f e. Q. | E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) } >. ) )
5		nqex 7361	. . . . . . 7 |- Q. e. _V
6	5	rabex 4147	. . . . . 6 |- { f e. Q. | E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) } e. _V
7	5	rabex 4147	. . . . . 6 |- { f e. Q. | E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) } e. _V
8	6, 7	op2nd 6147	. . . . 5 |- ( 2nd ` <. { f e. Q. | E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) } , { f e. Q. | E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) } >. ) = { f e. Q. | E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) }
9	4, 8	eqtrdi 2226	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( 2nd ` ( A F B ) ) = { f e. Q. | E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) } )
10	9	eleq2d 2247	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 2nd ` ( A F B ) ) <-> C e. { f e. Q. | E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) } ) )
11		elrabi 2890	. . 3 |- ( C e. { f e. Q. | E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) } -> C e. Q. )
12	10, 11	syl6bi 163	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 2nd ` ( A F B ) ) -> C e. Q. ) )
13		prop 7473	. . . . . . 7 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
14		elprnqu 7480	. . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ g e. ( 2nd ` A ) ) -> g e. Q. )
15	13, 14	sylan 283	. . . . . 6 |- ( ( A e. P. /\ g e. ( 2nd ` A ) ) -> g e. Q. )
16		prop 7473	. . . . . . 7 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
17		elprnqu 7480	. . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ h e. ( 2nd ` B ) ) -> h e. Q. )
18	16, 17	sylan 283	. . . . . 6 |- ( ( B e. P. /\ h e. ( 2nd ` B ) ) -> h e. Q. )
19	2	caovcl 6028	. . . . . 6 |- ( ( g e. Q. /\ h e. Q. ) -> ( g G h ) e. Q. )
20	15, 18, 19	syl2an 289	. . . . 5 |- ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) -> ( g G h ) e. Q. )
21	20	an4s 588	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( g e. ( 2nd ` A ) /\ h e. ( 2nd ` B ) ) ) -> ( g G h ) e. Q. )
22		eleq1 2240	. . . 4 |- ( C = ( g G h ) -> ( C e. Q. <-> ( g G h ) e. Q. ) )
23	21, 22	syl5ibrcom 157	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( g e. ( 2nd ` A ) /\ h e. ( 2nd ` B ) ) ) -> ( C = ( g G h ) -> C e. Q. ) )
24	23	rexlimdvva 2602	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) C = ( g G h ) -> C e. Q. ) )
25		eqeq1 2184	. . . . . 6 |- ( f = C -> ( f = ( g G h ) <-> C = ( g G h ) ) )
26	25	2rexbidv 2502	. . . . 5 |- ( f = C -> ( E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) <-> E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) C = ( g G h ) ) )
27	26	elrab3 2894	. . . 4 |- ( C e. Q. -> ( C e. { f e. Q. | E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) } <-> E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) C = ( g G h ) ) )
28	10, 27	sylan9bb 462	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ C e. Q. ) -> ( C e. ( 2nd ` ( A F B ) ) <-> E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) C = ( g G h ) ) )
29	28	ex 115	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( C e. Q. -> ( C e. ( 2nd ` ( A F B ) ) <-> E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) C = ( g G h ) ) ) )
30	12, 24, 29	pm5.21ndd 705	1 |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 2nd ` ( A F B ) ) <-> E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) C = ( g G h ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   E.wrex 2456   {crab 2459   <.cop 3595   ` cfv 5216  (class class class)co 5874   e. cmpo 5876   1stc1st 6138   2ndc2nd 6139   Q.cnq 7278   P.cnp 7289

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-qs 6540  df-ni 7302  df-nqqs 7346  df-inp 7464

This theorem is referenced by:  genppreclu  7513  genpcuu  7518  genprndu  7520  genpdisj  7521  genpassu  7523  addnqprlemru  7556  mulnqprlemru  7572  distrlem1pru  7581  distrlem5pru  7585  1idpru  7589  ltexprlemfu  7609  recexprlem1ssu  7632  recexprlemss1u  7634  cauappcvgprlemladdfu  7652  caucvgprlemladdfu  7675