Theorem genpcl 9830

Description: Closure of an operation on reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (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. )
genpcl.3	|- ( h e. Q. -> ( f <Q g <-> ( h G f ) <Q ( h G g ) ) )
genpcl.4	|- ( x G y ) = ( y G x )
genpcl.5	|- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x <Q ( g G h ) -> x e. ( A F B ) ) )

Assertion

Ref	Expression
genpcl	|- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. P. )

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

Allowed substitution hint:   F( z)

Proof of Theorem genpcl

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	genpn0 9825	. . 3 |- ( ( A e. P. /\ B e. P. ) -> (/) C. ( A F B ) )
4	1, 2	genpss 9826	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) C_ Q. )
5		vex 3203	. . . . . 6 |- x e. _V
6		vex 3203	. . . . . 6 |- y e. _V
7		genpcl.3	. . . . . 6 |- ( h e. Q. -> ( f <Q g <-> ( h G f ) <Q ( h G g ) ) )
8	5, 6, 7	caovord 6845	. . . . 5 |- ( z e. Q. -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )
9		genpcl.4	. . . . 5 |- ( x G y ) = ( y G x )
10	1, 2, 8, 9	genpnnp 9827	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> -. ( A F B ) = Q. )
11		dfpss2 3692	. . . 4 |- ( ( A F B ) C. Q. <-> ( ( A F B ) C_ Q. /\ -. ( A F B ) = Q. ) )
12	4, 10, 11	sylanbrc 698	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) C. Q. )
13		genpcl.5	. . . . . . 7 |- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x <Q ( g G h ) -> x e. ( A F B ) ) )
14	1, 2, 13	genpcd 9828	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( A F B ) -> ( x <Q f -> x e. ( A F B ) ) ) )
15	14	alrimdv 1857	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( A F B ) -> A. x ( x <Q f -> x e. ( A F B ) ) ) )
16		vex 3203	. . . . . . 7 |- z e. _V
17		vex 3203	. . . . . . 7 |- w e. _V
18	16, 17, 7	caovord 6845	. . . . . 6 |- ( v e. Q. -> ( z <Q w <-> ( v G z ) <Q ( v G w ) ) )
19	16, 17, 9	caovcom 6831	. . . . . 6 |- ( z G w ) = ( w G z )
20	1, 2, 18, 19	genpnmax 9829	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( A F B ) -> E. x e. ( A F B ) f <Q x ) )
21	15, 20	jcad 555	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( A F B ) -> ( A. x ( x <Q f -> x e. ( A F B ) ) /\ E. x e. ( A F B ) f <Q x ) ) )
22	21	ralrimiv 2965	. . 3 |- ( ( A e. P. /\ B e. P. ) -> A. f e. ( A F B ) ( A. x ( x <Q f -> x e. ( A F B ) ) /\ E. x e. ( A F B ) f <Q x ) )
23	3, 12, 22	jca31 557	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( ( (/) C. ( A F B ) /\ ( A F B ) C. Q. ) /\ A. f e. ( A F B ) ( A. x ( x <Q f -> x e. ( A F B ) ) /\ E. x e. ( A F B ) f <Q x ) ) )
24		elnp 9809	. 2 |- ( ( A F B ) e. P. <-> ( ( (/) C. ( A F B ) /\ ( A F B ) C. Q. ) /\ A. f e. ( A F B ) ( A. x ( x <Q f -> x e. ( A F B ) ) /\ E. x e. ( A F B ) f <Q x ) ) )
25	23, 24	sylibr 224	1 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. P. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   C_ wss 3574   C. wpss 3575   (/)c0 3915   class class class wbr 4653  (class class class)co 6650   |-> cmpt2 6652   Q.cnq 9674   <Q cltq 9680   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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-mi 9696  df-lti 9697  df-ltpq 9732  df-enq 9733  df-nq 9734  df-ltnq 9740  df-np 9803

This theorem is referenced by:  addclpr  9840  mulclpr  9842