Theorem genpml 7546

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 7504	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prml 7506	. . . 4 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. f e. Q. f e. ( 1st ` A ) )
3		rexex 2536	. . . 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 7504	. . . . 5 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
7		prml 7506	. . . . 5 |- ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. -> E. g e. Q. g e. ( 1st ` B ) )
8		rexex 2536	. . . . 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 7543	. . . . . 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 7510	. . . . . . . . . 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 7510	. . . . . . . . . 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 588	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 1st ` A ) /\ g e. ( 1st ` B ) ) ) -> ( f e. Q. /\ g e. Q. ) )
21	12	caovcl 6051	. . . . . . 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 2258	. . . . . 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 2860	. . . . 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 1910	. 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 1910	1 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A F B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2160   E.wrex 2469   {crab 2472   <.cop 3610   ` cfv 5235  (class class class)co 5896   e. cmpo 5898   1stc1st 6163   2ndc2nd 6164   Q.cnq 7309   P.cnp 7320

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-qs 6565  df-ni 7333  df-nqqs 7377  df-inp 7495

This theorem is referenced by:  addclpr  7566  mulclpr  7601