Theorem genpml 7832

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 7790	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prml 7792	. . . 4 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. f e. Q. f e. ( 1st ` A ) )
3		rexex 2588	. . . 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 7790	. . . . 5 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
7		prml 7792	. . . . 5 |- ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. -> E. g e. Q. g e. ( 1st ` B ) )
8		rexex 2588	. . . . 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 489	. . 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 7829	. . . . . 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 7796	. . . . . . . . . 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 7796	. . . . . . . . . 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 592	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 1st ` A ) /\ g e. ( 1st ` B ) ) ) -> ( f e. Q. /\ g e. Q. ) )
21	12	caovcl 6209	. . . . . . 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 2301	. . . . . 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 2925	. . . . 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 400	. . 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 1948	. 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 1948	1 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A F B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2203   E.wrex 2521   {crab 2524   <.cop 3692   ` cfv 5352  (class class class)co 6050   e. cmpo 6052   1stc1st 6332   2ndc2nd 6333   Q.cnq 7595   P.cnp 7606

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-qs 6773  df-ni 7619  df-nqqs 7663  df-inp 7781

This theorem is referenced by:  addclpr  7852  mulclpr  7887