Theorem genpml 7325

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 7283	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prml 7285	. . . 4 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. f e. Q. f e. ( 1st ` A ) )
3		rexex 2479	. . . 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 274	. 2 |- ( ( A e. P. /\ B e. P. ) -> E. f f e. ( 1st ` A ) )
6		prop 7283	. . . . 5 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
7		prml 7285	. . . . 5 |- ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. -> E. g e. Q. g e. ( 1st ` B ) )
8		rexex 2479	. . . . 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 480	. . 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 7322	. . . . . 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 123	. . . . 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 7289	. . . . . . . . . 10 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ f e. ( 1st ` A ) ) -> f e. Q. )
16	1, 15	sylan 281	. . . . . . . . 9 |- ( ( A e. P. /\ f e. ( 1st ` A ) ) -> f e. Q. )
17		elprnql 7289	. . . . . . . . . 10 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ g e. ( 1st ` B ) ) -> g e. Q. )
18	6, 17	sylan 281	. . . . . . . . 9 |- ( ( B e. P. /\ g e. ( 1st ` B ) ) -> g e. Q. )
19	16, 18	anim12i 336	. . . . . . . 8 |- ( ( ( A e. P. /\ f e. ( 1st ` A ) ) /\ ( B e. P. /\ g e. ( 1st ` B ) ) ) -> ( f e. Q. /\ g e. Q. ) )
20	19	an4s 577	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 1st ` A ) /\ g e. ( 1st ` B ) ) ) -> ( f e. Q. /\ g e. Q. ) )
21	12	caovcl 5925	. . . . . . 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 109	. . . . . . 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 2208	. . . . . 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 2793	. . . . 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 397	. . 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 1870	. 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 1870	1 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A F B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   E.wex 1468   e. wcel 1480   E.wrex 2417   {crab 2420   <.cop 3530   ` cfv 5123  (class class class)co 5774   e. cmpo 5776   1stc1st 6036   2ndc2nd 6037   Q.cnq 7088   P.cnp 7099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-qs 6435  df-ni 7112  df-nqqs 7156  df-inp 7274

This theorem is referenced by:  addclpr  7345  mulclpr  7380