Theorem genpmu 7728

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 7685	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prmu 7688	. . . 4 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. f e. Q. f e. ( 2nd ` A ) )
3		rexex 2576	. . . 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 7685	. . . . 5 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
7		prmu 7688	. . . . 5 |- ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. -> E. g e. Q. g e. ( 2nd ` B ) )
8		rexex 2576	. . . . 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 7725	. . . . . 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 7692	. . . . . . . . . 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 7692	. . . . . . . . . 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 590	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 2nd ` A ) /\ g e. ( 2nd ` B ) ) ) -> ( f e. Q. /\ g e. Q. ) )
21	12	caovcl 6172	. . . . . . 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 2298	. . . . . 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 2912	. . . . 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 1945	. 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 1945	1 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 2nd ` ( A F B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   E.wrex 2509   {crab 2512   <.cop 3670   ` cfv 5324  (class class class)co 6013   e. cmpo 6015   1stc1st 6296   2ndc2nd 6297   Q.cnq 7490   P.cnp 7501

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-qs 6703  df-ni 7514  df-nqqs 7558  df-inp 7676

This theorem is referenced by:  addclpr  7747  mulclpr  7782