Theorem genpelvu 7464

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 7460	. . . . . 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 5498	. . . . 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 7314	. . . . . . 7 |- Q. e. _V
6	5	rabex 4131	. . . . . 6 |- { f e. Q. | E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) } e. _V
7	5	rabex 4131	. . . . . 6 |- { f e. Q. | E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) } e. _V
8	6, 7	op2nd 6124	. . . . 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 2219	. . . 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 2240	. . 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 2883	. . 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 162	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 2nd ` ( A F B ) ) -> C e. Q. ) )
13		prop 7426	. . . . . . 7 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
14		elprnqu 7433	. . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ g e. ( 2nd ` A ) ) -> g e. Q. )
15	13, 14	sylan 281	. . . . . 6 |- ( ( A e. P. /\ g e. ( 2nd ` A ) ) -> g e. Q. )
16		prop 7426	. . . . . . 7 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
17		elprnqu 7433	. . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ h e. ( 2nd ` B ) ) -> h e. Q. )
18	16, 17	sylan 281	. . . . . 6 |- ( ( B e. P. /\ h e. ( 2nd ` B ) ) -> h e. Q. )
19	2	caovcl 6005	. . . . . 6 |- ( ( g e. Q. /\ h e. Q. ) -> ( g G h ) e. Q. )
20	15, 18, 19	syl2an 287	. . . . 5 |- ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) -> ( g G h ) e. Q. )
21	20	an4s 583	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( g e. ( 2nd ` A ) /\ h e. ( 2nd ` B ) ) ) -> ( g G h ) e. Q. )
22		eleq1 2233	. . . 4 |- ( C = ( g G h ) -> ( C e. Q. <-> ( g G h ) e. Q. ) )
23	21, 22	syl5ibrcom 156	. . 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 2595	. 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 2177	. . . . . 6 |- ( f = C -> ( f = ( g G h ) <-> C = ( g G h ) ) )
26	25	2rexbidv 2495	. . . . 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 2887	. . . 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 459	. . 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 114	. 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 700	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 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   E.wrex 2449   {crab 2452   <.cop 3584   ` cfv 5196  (class class class)co 5851   e. cmpo 5853   1stc1st 6115   2ndc2nd 6116   Q.cnq 7231   P.cnp 7242

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-qs 6516  df-ni 7255  df-nqqs 7299  df-inp 7417

This theorem is referenced by:  genppreclu  7466  genpcuu  7471  genprndu  7473  genpdisj  7474  genpassu  7476  addnqprlemru  7509  mulnqprlemru  7525  distrlem1pru  7534  distrlem5pru  7538  1idpru  7542  ltexprlemfu  7562  recexprlem1ssu  7585  recexprlemss1u  7587  cauappcvgprlemladdfu  7605  caucvgprlemladdfu  7628