Theorem genpcuu 7663

Description: Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-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. )
genpcuu.2	|- ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) /\ x e. Q. ) -> ( ( g G h ) <Q x -> x e. ( 2nd ` ( A F B ) ) ) )

Assertion

Ref	Expression
genpcuu	|- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 2nd ` ( A F B ) ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) )

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

Allowed substitution hints:   F( x, y, z, w, v)

Proof of Theorem genpcuu

Step	Hyp	Ref	Expression
1		ltrelnq 7508	. . . . . . 7 |- <Q C_ ( Q. X. Q. )
2	1	brel 4740	. . . . . 6 |- ( f <Q x -> ( f e. Q. /\ x e. Q. ) )
3	2	simprd 114	. . . . 5 |- ( f <Q x -> x e. Q. )
4		genpelvl.1	. . . . . . . . 9 |- 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 ) ) } >. )
5		genpelvl.2	. . . . . . . . 9 |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )
6	4, 5	genpelvu 7656	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 2nd ` ( A F B ) ) <-> E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) ) )
7	6	adantr 276	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ x e. Q. ) -> ( f e. ( 2nd ` ( A F B ) ) <-> E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) ) )
8		breq1 4057	. . . . . . . . . . . . 13 |- ( f = ( g G h ) -> ( f <Q x <-> ( g G h ) <Q x ) )
9	8	biimpd 144	. . . . . . . . . . . 12 |- ( f = ( g G h ) -> ( f <Q x -> ( g G h ) <Q x ) )
10		genpcuu.2	. . . . . . . . . . . 12 |- ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) /\ x e. Q. ) -> ( ( g G h ) <Q x -> x e. ( 2nd ` ( A F B ) ) ) )
11	9, 10	sylan9r 410	. . . . . . . . . . 11 |- ( ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) /\ x e. Q. ) /\ f = ( g G h ) ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) )
12	11	exp31 364	. . . . . . . . . 10 |- ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) -> ( x e. Q. -> ( f = ( g G h ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) ) )
13	12	an4s 588	. . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. ) /\ ( g e. ( 2nd ` A ) /\ h e. ( 2nd ` B ) ) ) -> ( x e. Q. -> ( f = ( g G h ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) ) )
14	13	impancom 260	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. ) /\ x e. Q. ) -> ( ( g e. ( 2nd ` A ) /\ h e. ( 2nd ` B ) ) -> ( f = ( g G h ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) ) )
15	14	rexlimdvv 2631	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ x e. Q. ) -> ( E. g e. ( 2nd ` A ) E. h e. ( 2nd ` B ) f = ( g G h ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) )
16	7, 15	sylbid 150	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ x e. Q. ) -> ( f e. ( 2nd ` ( A F B ) ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) )
17	16	ex 115	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( x e. Q. -> ( f e. ( 2nd ` ( A F B ) ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) ) )
18	3, 17	syl5 32	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( f <Q x -> ( f e. ( 2nd ` ( A F B ) ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) ) )
19	18	com34 83	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( f <Q x -> ( f <Q x -> ( f e. ( 2nd ` ( A F B ) ) -> x e. ( 2nd ` ( A F B ) ) ) ) ) )
20	19	pm2.43d 50	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( f <Q x -> ( f e. ( 2nd ` ( A F B ) ) -> x e. ( 2nd ` ( A F B ) ) ) ) )
21	20	com23 78	1 |- ( ( A e. P. /\ B e. P. ) -> ( f e. ( 2nd ` ( A F B ) ) -> ( f <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2177   E.wrex 2486   {crab 2489   <.cop 3641   class class class wbr 4054   ` cfv 5285  (class class class)co 5962   e. cmpo 5964   1stc1st 6242   2ndc2nd 6243   Q.cnq 7423   <Q cltq 7428   P.cnp 7434

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-qs 6644  df-ni 7447  df-nqqs 7491  df-ltnqqs 7496  df-inp 7609

This theorem is referenced by:  genprndu  7665