Theorem genpelvl 7879

Description: Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-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
genpelvl	|- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 1st ` ( A F B ) ) <-> E. g e. ( 1st ` A ) E. h e. ( 1st ` 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 genpelvl

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 7876	. . . . . 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 5699	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( 1st ` ( A F B ) ) = ( 1st ` <. { 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 7730	. . . . . . 7 |- Q. e. _V
6	5	rabex 4280	. . . . . 6 |- { f e. Q. | E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) } e. _V
7	5	rabex 4280	. . . . . 6 |- { f e. Q. | E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) } e. _V
8	6, 7	op1st 6380	. . . . 5 |- ( 1st ` <. { 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. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) }
9	4, 8	eqtrdi 2287	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( 1st ` ( A F B ) ) = { f e. Q. | E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) } )
10	9	eleq2d 2308	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 1st ` ( A F B ) ) <-> C e. { f e. Q. | E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) } ) )
11		elrabi 2979	. . 3 |- ( C e. { f e. Q. | E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) } -> C e. Q. )
12	10, 11	biimtrdi 163	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 1st ` ( A F B ) ) -> C e. Q. ) )
13		prop 7842	. . . . . . 7 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
14		elprnql 7848	. . . . . . 7 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ g e. ( 1st ` A ) ) -> g e. Q. )
15	13, 14	sylan 283	. . . . . 6 |- ( ( A e. P. /\ g e. ( 1st ` A ) ) -> g e. Q. )
16		prop 7842	. . . . . . 7 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
17		elprnql 7848	. . . . . . 7 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ h e. ( 1st ` B ) ) -> h e. Q. )
18	16, 17	sylan 283	. . . . . 6 |- ( ( B e. P. /\ h e. ( 1st ` B ) ) -> h e. Q. )
19	2	caovcl 6244	. . . . . 6 |- ( ( g e. Q. /\ h e. Q. ) -> ( g G h ) e. Q. )
20	15, 18, 19	syl2an 289	. . . . 5 |- ( ( ( A e. P. /\ g e. ( 1st ` A ) ) /\ ( B e. P. /\ h e. ( 1st ` B ) ) ) -> ( g G h ) e. Q. )
21	20	an4s 596	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( g e. ( 1st ` A ) /\ h e. ( 1st ` B ) ) ) -> ( g G h ) e. Q. )
22		eleq1 2301	. . . 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. ( 1st ` A ) /\ h e. ( 1st ` B ) ) ) -> ( C = ( g G h ) -> C e. Q. ) )
24	23	rexlimdvva 2676	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) C = ( g G h ) -> C e. Q. ) )
25		eqeq1 2245	. . . . . 6 |- ( f = C -> ( f = ( g G h ) <-> C = ( g G h ) ) )
26	25	2rexbidv 2575	. . . . 5 |- ( f = C -> ( E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) <-> E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) C = ( g G h ) ) )
27	26	elrab3 2983	. . . 4 |- ( C e. Q. -> ( C e. { f e. Q. | E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) } <-> E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) C = ( g G h ) ) )
28	10, 27	sylan9bb 466	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ C e. Q. ) -> ( C e. ( 1st ` ( A F B ) ) <-> E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) C = ( g G h ) ) )
29	28	ex 115	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( C e. Q. -> ( C e. ( 1st ` ( A F B ) ) <-> E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) C = ( g G h ) ) ) )
30	12, 24, 29	pm5.21ndd 717	1 |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( 1st ` ( A F B ) ) <-> E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) C = ( g G h ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   {crab 2532   <.cop 3712   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   1stc1st 6372   2ndc2nd 6373   Q.cnq 7647   P.cnp 7658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-qs 6813  df-ni 7671  df-nqqs 7715  df-inp 7833

This theorem is used by:  genpprecll  7881  genpcdl  7886  genprndl  7888  genpdisj  7890  genpassl  7891  addnqprlemrl  7924  mulnqprlemrl  7940  distrlem1prl  7949  distrlem5prl  7953  1idprl  7957  ltexprlemfl  7976  recexprlem1ssl  8000  recexprlemss1l  8002  cauappcvgprlemladdfl  8022