Theorem genpelvl 7574

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 7571	. . . . . 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 5559	. . . . 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 7425	. . . . . . 7 |- Q. e. _V
6	5	rabex 4174	. . . . . 6 |- { f e. Q. | E. g e. ( 1st ` A ) E. h e. ( 1st ` B ) f = ( g G h ) } e. _V
7	5	rabex 4174	. . . . . 6 |- { f e. Q. | E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) } e. _V
8	6, 7	op1st 6201	. . . . 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 2242	. . . 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 2263	. . 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 2914	. . 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 7537	. . . . . . 7 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
14		elprnql 7543	. . . . . . 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 7537	. . . . . . 7 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
17		elprnql 7543	. . . . . . 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 6075	. . . . . 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 588	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( g e. ( 1st ` A ) /\ h e. ( 1st ` B ) ) ) -> ( g G h ) e. Q. )
22		eleq1 2256	. . . 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 2619	. 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 2200	. . . . . 6 |- ( f = C -> ( f = ( g G h ) <-> C = ( g G h ) ) )
26	25	2rexbidv 2519	. . . . 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 2918	. . . 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 706	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 980   = wceq 1364   e. wcel 2164   E.wrex 2473   {crab 2476   <.cop 3622   ` cfv 5255  (class class class)co 5919   e. cmpo 5921   1stc1st 6193   2ndc2nd 6194   Q.cnq 7342   P.cnp 7353

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-qs 6595  df-ni 7366  df-nqqs 7410  df-inp 7528

This theorem is referenced by:  genpprecll  7576  genpcdl  7581  genprndl  7583  genpdisj  7585  genpassl  7586  addnqprlemrl  7619  mulnqprlemrl  7635  distrlem1prl  7644  distrlem5prl  7648  1idprl  7652  ltexprlemfl  7671  recexprlem1ssl  7695  recexprlemss1l  7697  cauappcvgprlemladdfl  7717