Theorem genpelv 9822

Description: Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
genp.1	|- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )
genp.2	|- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )

Assertion

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

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

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

Proof of Theorem genpelv

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		genp.1	. . . 4 |- F = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y G z ) } )
2		genp.2	. . . 4 |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )
3	1, 2	genpv 9821	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) = { f | E. g e. A E. h e. B f = ( g G h ) } )
4	3	eleq2d 2687	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( A F B ) <-> C e. { f | E. g e. A E. h e. B f = ( g G h ) } ) )
5		id 22	. . . . . 6 |- ( C = ( g G h ) -> C = ( g G h ) )
6		ovex 6678	. . . . . 6 |- ( g G h ) e. _V
7	5, 6	syl6eqel 2709	. . . . 5 |- ( C = ( g G h ) -> C e. _V )
8	7	rexlimivw 3029	. . . 4 |- ( E. h e. B C = ( g G h ) -> C e. _V )
9	8	rexlimivw 3029	. . 3 |- ( E. g e. A E. h e. B C = ( g G h ) -> C e. _V )
10		eqeq1 2626	. . . 4 |- ( f = C -> ( f = ( g G h ) <-> C = ( g G h ) ) )
11	10	2rexbidv 3057	. . 3 |- ( f = C -> ( E. g e. A E. h e. B f = ( g G h ) <-> E. g e. A E. h e. B C = ( g G h ) ) )
12	9, 11	elab3 3358	. 2 |- ( C e. { f | E. g e. A E. h e. B f = ( g G h ) } <-> E. g e. A E. h e. B C = ( g G h ) )
13	4, 12	syl6bb 276	1 |- ( ( A e. P. /\ B e. P. ) -> ( C e. ( A F B ) <-> E. g e. A E. h e. B C = ( g G h ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200  (class class class)co 6650   |-> cmpt2 6652   Q.cnq 9674   P.cnp 9681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-ni 9694  df-nq 9734  df-np 9803

This theorem is referenced by:  genpprecl  9823  genpss  9826  genpnnp  9827  genpcd  9828  genpnmax  9829  genpass  9831  distrlem1pr  9847  distrlem5pr  9849  1idpr  9851  ltexprlem6  9863  reclem3pr  9871  reclem4pr  9872