Theorem genpmu 7473

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 7430	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prmu 7433	. . . 4 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. f e. Q. f e. ( 2nd ` A ) )
3		rexex 2516	. . . 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 274	. 2 |- ( ( A e. P. /\ B e. P. ) -> E. f f e. ( 2nd ` A ) )
6		prop 7430	. . . . 5 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
7		prmu 7433	. . . . 5 |- ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. -> E. g e. Q. g e. ( 2nd ` B ) )
8		rexex 2516	. . . . 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 486	. . 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 7470	. . . . . 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 123	. . . . 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 7437	. . . . . . . . . 10 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
16	1, 15	sylan 281	. . . . . . . . 9 |- ( ( A e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
17		elprnqu 7437	. . . . . . . . . 10 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ g e. ( 2nd ` B ) ) -> g e. Q. )
18	6, 17	sylan 281	. . . . . . . . 9 |- ( ( B e. P. /\ g e. ( 2nd ` B ) ) -> g e. Q. )
19	16, 18	anim12i 336	. . . . . . . 8 |- ( ( ( A e. P. /\ f e. ( 2nd ` A ) ) /\ ( B e. P. /\ g e. ( 2nd ` B ) ) ) -> ( f e. Q. /\ g e. Q. ) )
20	19	an4s 583	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ ( f e. ( 2nd ` A ) /\ g e. ( 2nd ` B ) ) ) -> ( f e. Q. /\ g e. Q. ) )
21	12	caovcl 6005	. . . . . . 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 109	. . . . . . 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 2239	. . . . . 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 2838	. . . . 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 398	. . 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 1891	. 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 1891	1 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 2nd ` ( A F B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   E.wex 1485   e. wcel 2141   E.wrex 2449   {crab 2452   <.cop 3584   ` cfv 5196  (class class class)co 5851   e. cmpo 5853   1stc1st 6115   2ndc2nd 6116   Q.cnq 7235   P.cnp 7246

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-qs 6517  df-ni 7259  df-nqqs 7303  df-inp 7421

This theorem is referenced by:  addclpr  7492  mulclpr  7527