Theorem genpelvl 7731

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 7728	. . . . . 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 5643	. . . . 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 7582	. . . . . . 7 |- Q. e. _V
6	5	rabex 4234	. . . . . 6 |- { f e. Q. | E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) } e. _V
7	5	rabex 4234	. . . . . 6 |- { f e. Q. | E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) } e. _V
8	6, 7	op1st 6308	. . . . 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 2280	. . . 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 2301	. . 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 2959	. . 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 7694	. . . . . . 7 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
14		elprnql 7700	. . . . . . 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 7694	. . . . . . 7 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
17		elprnql 7700	. . . . . . 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 6176	. . . . . 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 592	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( g e. ( 1st ` A ) /\ h e. ( 1st ` B ) ) ) -> ( g G h ) e. Q. )
22		eleq1 2294	. . . 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 2658	. 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 2238	. . . . . 6 |- ( f = C -> ( f = ( g G h ) <-> C = ( g G h ) ) )
26	25	2rexbidv 2557	. . . . 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 2963	. . . 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 462	. . 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 712	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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   E.wrex 2511   {crab 2514   <.cop 3672   ` cfv 5326  (class class class)co 6017   e. cmpo 6019   1stc1st 6300   2ndc2nd 6301   Q.cnq 7499   P.cnp 7510

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-qs 6707  df-ni 7523  df-nqqs 7567  df-inp 7685

This theorem is referenced by:  genpprecll  7733  genpcdl  7738  genprndl  7740  genpdisj  7742  genpassl  7743  addnqprlemrl  7776  mulnqprlemrl  7792  distrlem1prl  7801  distrlem5prl  7805  1idprl  7809  ltexprlemfl  7828  recexprlem1ssl  7852  recexprlemss1l  7854  cauappcvgprlemladdfl  7874