Theorem genpmu 7519

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 7476	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prmu 7479	. . . 4 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. f e. Q. f e. ( 2nd ` A ) )
3		rexex 2523	. . . 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 7476	. . . . 5 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
7		prmu 7479	. . . . 5 |- ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. -> E. g e. Q. g e. ( 2nd ` B ) )
8		rexex 2523	. . . . 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 7516	. . . . . 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 7483	. . . . . . . . . 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 7483	. . . . . . . . . 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 6031	. . . . . . 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 2246	. . . . . 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 2847	. . . . 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 1898	. 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 1898	1 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 2nd ` ( A F B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   E.wex 1492   e. wcel 2148   E.wrex 2456   {crab 2459   <.cop 3597   ` cfv 5218  (class class class)co 5877   e. cmpo 5879   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281   P.cnp 7292

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-qs 6543  df-ni 7305  df-nqqs 7349  df-inp 7467

This theorem is referenced by:  addclpr  7538  mulclpr  7573