Theorem genpml 7890

Description: The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-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
genpml	|- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( 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 genpml

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

Step	Hyp	Ref	Expression
1		prop 7848	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prml 7850	. . . 4 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. f e. Q. f e. ( 1st ` A ) )
3		rexex 2596	. . . 4 |- ( E. f e. Q. f e. ( 1st ` A ) -> E. f f e. ( 1st ` A ) )
4	1, 2, 3	3syl 17	. . 3 |- ( A e. P. -> E. f f e. ( 1st ` A ) )
5	4	adantr 276	. 2 |- ( ( A e. P. /\ B e. P. ) -> E. f f e. ( 1st ` A ) )
6		prop 7848	. . . . 5 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
7		prml 7850	. . . . 5 |- ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. -> E. g e. Q. g e. ( 1st ` B ) )
8		rexex 2596	. . . . 5 |- ( E. g e. Q. g e. ( 1st ` B ) -> E. g g e. ( 1st ` B ) )
9	6, 7, 8	3syl 17	. . . 4 |- ( B e. P. -> E. g g e. ( 1st ` B ) )
10	9	ad2antlr 493	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ f e. ( 1st ` A ) ) -> E. g g e. ( 1st ` 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	genpprecll 7887	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( f e. ( 1st ` A ) /\ g e. ( 1st ` B ) ) -> ( f G g ) e. ( 1st ` ( A F B ) ) ) )
14	13	imp 124	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 1st ` A ) /\ g e. ( 1st ` B ) ) ) -> ( f G g ) e. ( 1st ` ( A F B ) ) )
15		elprnql 7854	. . . . . . . . . 10 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ f e. ( 1st ` A ) ) -> f e. Q. )
16	1, 15	sylan 283	. . . . . . . . 9 |- ( ( A e. P. /\ f e. ( 1st ` A ) ) -> f e. Q. )
17		elprnql 7854	. . . . . . . . . 10 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ g e. ( 1st ` B ) ) -> g e. Q. )
18	6, 17	sylan 283	. . . . . . . . 9 |- ( ( B e. P. /\ g e. ( 1st ` B ) ) -> g e. Q. )
19	16, 18	anim12i 338	. . . . . . . 8 |- ( ( ( A e. P. /\ f e. ( 1st ` A ) ) /\ ( B e. P. /\ g e. ( 1st ` B ) ) ) -> ( f e. Q. /\ g e. Q. ) )
20	19	an4s 596	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 1st ` A ) /\ g e. ( 1st ` B ) ) ) -> ( f e. Q. /\ g e. Q. ) )
21	12	caovcl 6247	. . . . . . 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. ( 1st ` A ) /\ g e. ( 1st ` B ) ) ) -> ( f G g ) e. Q. )
23		simpr 110	. . . . . . 7 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 1st ` A ) /\ g e. ( 1st ` B ) ) ) /\ q = ( f G g ) ) -> q = ( f G g ) )
24	23	eleq1d 2307	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 1st ` A ) /\ g e. ( 1st ` B ) ) ) /\ q = ( f G g ) ) -> ( q e. ( 1st ` ( A F B ) ) <-> ( f G g ) e. ( 1st ` ( A F B ) ) ) )
25	22, 24	rspcedv 2933	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 1st ` A ) /\ g e. ( 1st ` B ) ) ) -> ( ( f G g ) e. ( 1st ` ( A F B ) ) -> E. q e. Q. q e. ( 1st ` ( A F B ) ) ) )
26	14, 25	mpd 13	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 1st ` A ) /\ g e. ( 1st ` B ) ) ) -> E. q e. Q. q e. ( 1st ` ( A F B ) ) )
27	26	anassrs 404	. . 3 |- ( ( ( ( A e. P. /\ B e. P. ) /\ f e. ( 1st ` A ) ) /\ g e. ( 1st ` B ) ) -> E. q e. Q. q e. ( 1st ` ( A F B ) ) )
28	10, 27	exlimddv 1954	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ f e. ( 1st ` A ) ) -> E. q e. Q. q e. ( 1st ` ( A F B ) ) )
29	5, 28	exlimddv 1954	1 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A F B ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   {crab 2532   <.cop 3712   ` cfv 5380  (class class class)co 6088   e. cmpo 6090   1stc1st 6376   2ndc2nd 6377   Q.cnq 7653   P.cnp 7664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-qs 6817  df-ni 7677  df-nqqs 7721  df-inp 7839

This theorem is used by:  addclpr  7910  mulclpr  7945