Theorem genpmu 7585

Description: The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-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
genpmu	|- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 2nd ` ( A F B ) ) )

Distinct variable groups:   x, y, z, w, v, q, A   x, B, y, z, w, v, q   x, G, y, z, w, v, q   F, q

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

Proof of Theorem genpmu

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7542	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prmu 7545	. . . 4 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. f e. Q. f e. ( 2nd ` A ) )
3		rexex 2543	. . . 4 |- ( E. f e. Q. f e. ( 2nd ` A ) -> E. f f e. ( 2nd ` A ) )
4	1, 2, 3	3syl 17	. . 3 |- ( A e. P. -> E. f f e. ( 2nd ` A ) )
5	4	adantr 276	. 2 |- ( ( A e. P. /\ B e. P. ) -> E. f f e. ( 2nd ` A ) )
6		prop 7542	. . . . 5 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
7		prmu 7545	. . . . 5 |- ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. -> E. g e. Q. g e. ( 2nd ` B ) )
8		rexex 2543	. . . . 5 |- ( E. g e. Q. g e. ( 2nd ` B ) -> E. g g e. ( 2nd ` B ) )
9	6, 7, 8	3syl 17	. . . 4 |- ( B e. P. -> E. g g e. ( 2nd ` B ) )
10	9	ad2antlr 489	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ f e. ( 2nd ` A ) ) -> E. g g e. ( 2nd ` B ) )
11		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 ) ) } >. )
12		genpelvl.2	. . . . . . 7 |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )
13	11, 12	genppreclu 7582	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( f e. ( 2nd ` A ) /\ g e. ( 2nd ` B ) ) -> ( f G g ) e. ( 2nd ` ( A F B ) ) ) )
14	13	imp 124	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 2nd ` A ) /\ g e. ( 2nd ` B ) ) ) -> ( f G g ) e. ( 2nd ` ( A F B ) ) )
15		elprnqu 7549	. . . . . . . . . 10 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
16	1, 15	sylan 283	. . . . . . . . 9 |- ( ( A e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
17		elprnqu 7549	. . . . . . . . . 10 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ g e. ( 2nd ` B ) ) -> g e. Q. )
18	6, 17	sylan 283	. . . . . . . . 9 |- ( ( B e. P. /\ g e. ( 2nd ` B ) ) -> g e. Q. )
19	16, 18	anim12i 338	. . . . . . . 8 |- ( ( ( A e. P. /\ f e. ( 2nd ` A ) ) /\ ( B e. P. /\ g e. ( 2nd ` B ) ) ) -> ( f e. Q. /\ g e. Q. ) )
20	19	an4s 588	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 2nd ` A ) /\ g e. ( 2nd ` B ) ) ) -> ( f e. Q. /\ g e. Q. ) )
21	12	caovcl 6078	. . . . . . 7 |- ( ( f e. Q. /\ g e. Q. ) -> ( f G g ) e. Q. )
22	20, 21	syl 14	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 2nd ` A ) /\ g e. ( 2nd ` B ) ) ) -> ( f G g ) e. Q. )
23		simpr 110	. . . . . . 7 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 2nd ` A ) /\ g e. ( 2nd ` B ) ) ) /\ q = ( f G g ) ) -> q = ( f G g ) )
24	23	eleq1d 2265	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 2nd ` A ) /\ g e. ( 2nd ` B ) ) ) /\ q = ( f G g ) ) -> ( q e. ( 2nd ` ( A F B ) ) <-> ( f G g ) e. ( 2nd ` ( A F B ) ) ) )
25	22, 24	rspcedv 2872	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 2nd ` A ) /\ g e. ( 2nd ` B ) ) ) -> ( ( f G g ) e. ( 2nd ` ( A F B ) ) -> E. q e. Q. q e. ( 2nd ` ( A F B ) ) ) )
26	14, 25	mpd 13	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 2nd ` A ) /\ g e. ( 2nd ` B ) ) ) -> E. q e. Q. q e. ( 2nd ` ( A F B ) ) )
27	26	anassrs 400	. . 3 |- ( ( ( ( A e. P. /\ B e. P. ) /\ f e. ( 2nd ` A ) ) /\ g e. ( 2nd ` B ) ) -> E. q e. Q. q e. ( 2nd ` ( A F B ) ) )
28	10, 27	exlimddv 1913	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ f e. ( 2nd ` A ) ) -> E. q e. Q. q e. ( 2nd ` ( A F B ) ) )
29	5, 28	exlimddv 1913	1 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 2nd ` ( A F B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   E.wex 1506   e. wcel 2167   E.wrex 2476   {crab 2479   <.cop 3625   ` cfv 5258  (class class class)co 5922   e. cmpo 5924   1stc1st 6196   2ndc2nd 6197   Q.cnq 7347   P.cnp 7358

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-qs 6598  df-ni 7371  df-nqqs 7415  df-inp 7533

This theorem is referenced by:  addclpr  7604  mulclpr  7639