Theorem genprndu 7354

Description: The upper cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-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. )
genprndu.ord	|- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )
genprndu.com	|- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) )
genprndu.upper	|- ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) /\ x e. Q. ) -> ( ( g G h ) <Q x -> x e. ( 2nd ` ( A F B ) ) ) )

Assertion

Ref	Expression
genprndu	|- ( ( A e. P. /\ B e. P. ) -> A. r e. Q. ( r e. ( 2nd ` ( A F B ) ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) ) )

Distinct variable groups:   x, y, z, g, h, w, v, q, A   x, B, y, z, g, h, w, v, q   x, G, y, z, g, h, w, v, q   g, F, q   A, r, q, v, w, x, y, z   B, r, g, h   h, F, r, v, w, x, y, z   G, r

Proof of Theorem genprndu

Dummy variables a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		genpelvl.1	. . . . . . . . . 10 |- 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 ) ) } >. )
2		genpelvl.2	. . . . . . . . . 10 |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )
3	1, 2	genpelvu 7345	. . . . . . . . 9 |- ( ( A e. P. /\ B e. P. ) -> ( r e. ( 2nd ` ( A F B ) ) <-> E. a e. ( 2nd ` A ) E. b e. ( 2nd ` B ) r = ( a G b ) ) )
4		r2ex 2458	. . . . . . . . 9 |- ( E. a e. ( 2nd ` A ) E. b e. ( 2nd ` B ) r = ( a G b ) <-> E. a E. b ( ( a e. ( 2nd ` A ) /\ b e. ( 2nd ` B ) ) /\ r = ( a G b ) ) )
5	3, 4	syl6bb 195	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( r e. ( 2nd ` ( A F B ) ) <-> E. a E. b ( ( a e. ( 2nd ` A ) /\ b e. ( 2nd ` B ) ) /\ r = ( a G b ) ) ) )
6	5	biimpa 294	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ r e. ( 2nd ` ( A F B ) ) ) -> E. a E. b ( ( a e. ( 2nd ` A ) /\ b e. ( 2nd ` B ) ) /\ r = ( a G b ) ) )
7	6	adantrl 470	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ ( r e. Q. /\ r e. ( 2nd ` ( A F B ) ) ) ) -> E. a E. b ( ( a e. ( 2nd ` A ) /\ b e. ( 2nd ` B ) ) /\ r = ( a G b ) ) )
8		prop 7307	. . . . . . . . . . . . . . . 16 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
9		prnminu 7321	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ a e. ( 2nd ` A ) ) -> E. c e. ( 2nd ` A ) c <Q a )
10	8, 9	sylan 281	. . . . . . . . . . . . . . 15 |- ( ( A e. P. /\ a e. ( 2nd ` A ) ) -> E. c e. ( 2nd ` A ) c <Q a )
11		prop 7307	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
12		prnminu 7321	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ b e. ( 2nd ` B ) ) -> E. d e. ( 2nd ` B ) d <Q b )
13	11, 12	sylan 281	. . . . . . . . . . . . . . 15 |- ( ( B e. P. /\ b e. ( 2nd ` B ) ) -> E. d e. ( 2nd ` B ) d <Q b )
14	10, 13	anim12i 336	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ a e. ( 2nd ` A ) ) /\ ( B e. P. /\ b e. ( 2nd ` B ) ) ) -> ( E. c e. ( 2nd ` A ) c <Q a /\ E. d e. ( 2nd ` B ) d <Q b ) )
15	14	an4s 578	. . . . . . . . . . . . 13 |- ( ( ( A e. P. /\ B e. P. ) /\ ( a e. ( 2nd ` A ) /\ b e. ( 2nd ` B ) ) ) -> ( E. c e. ( 2nd ` A ) c <Q a /\ E. d e. ( 2nd ` B ) d <Q b ) )
16		reeanv 2603	. . . . . . . . . . . . 13 |- ( E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c <Q a /\ d <Q b ) <-> ( E. c e. ( 2nd ` A ) c <Q a /\ E. d e. ( 2nd ` B ) d <Q b ) )
17	15, 16	sylibr 133	. . . . . . . . . . . 12 |- ( ( ( A e. P. /\ B e. P. ) /\ ( a e. ( 2nd ` A ) /\ b e. ( 2nd ` B ) ) ) -> E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c <Q a /\ d <Q b ) )
18		genprndu.ord	. . . . . . . . . . . . . . 15 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )
19		genprndu.com	. . . . . . . . . . . . . . 15 |- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) )
20	18, 19	genplt2i 7342	. . . . . . . . . . . . . 14 |- ( ( c <Q a /\ d <Q b ) -> ( c G d ) <Q ( a G b ) )
21	20	reximi 2532	. . . . . . . . . . . . 13 |- ( E. d e. ( 2nd ` B ) ( c <Q a /\ d <Q b ) -> E. d e. ( 2nd ` B ) ( c G d ) <Q ( a G b ) )
22	21	reximi 2532	. . . . . . . . . . . 12 |- ( E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c <Q a /\ d <Q b ) -> E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q ( a G b ) )
23	17, 22	syl 14	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( a e. ( 2nd ` A ) /\ b e. ( 2nd ` B ) ) ) -> E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q ( a G b ) )
24	23	adantrr 471	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( ( a e. ( 2nd ` A ) /\ b e. ( 2nd ` B ) ) /\ r = ( a G b ) ) ) -> E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q ( a G b ) )
25		breq2 3941	. . . . . . . . . . . . . 14 |- ( r = ( a G b ) -> ( ( c G d ) <Q r <-> ( c G d ) <Q ( a G b ) ) )
26	25	biimprd 157	. . . . . . . . . . . . 13 |- ( r = ( a G b ) -> ( ( c G d ) <Q ( a G b ) -> ( c G d ) <Q r ) )
27	26	reximdv 2536	. . . . . . . . . . . 12 |- ( r = ( a G b ) -> ( E. d e. ( 2nd ` B ) ( c G d ) <Q ( a G b ) -> E. d e. ( 2nd ` B ) ( c G d ) <Q r ) )
28	27	reximdv 2536	. . . . . . . . . . 11 |- ( r = ( a G b ) -> ( E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q ( a G b ) -> E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q r ) )
29	28	ad2antll 483	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( ( a e. ( 2nd ` A ) /\ b e. ( 2nd ` B ) ) /\ r = ( a G b ) ) ) -> ( E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q ( a G b ) -> E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q r ) )
30	24, 29	mpd 13	. . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. ) /\ ( ( a e. ( 2nd ` A ) /\ b e. ( 2nd ` B ) ) /\ r = ( a G b ) ) ) -> E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q r )
31	30	ex 114	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( a e. ( 2nd ` A ) /\ b e. ( 2nd ` B ) ) /\ r = ( a G b ) ) -> E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q r ) )
32	31	exlimdvv 1870	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( E. a E. b ( ( a e. ( 2nd ` A ) /\ b e. ( 2nd ` B ) ) /\ r = ( a G b ) ) -> E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q r ) )
33	32	adantr 274	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ ( r e. Q. /\ r e. ( 2nd ` ( A F B ) ) ) ) -> ( E. a E. b ( ( a e. ( 2nd ` A ) /\ b e. ( 2nd ` B ) ) /\ r = ( a G b ) ) -> E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q r ) )
34	7, 33	mpd 13	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( r e. Q. /\ r e. ( 2nd ` ( A F B ) ) ) ) -> E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q r )
35	1, 2	genppreclu 7347	. . . . . . . . 9 |- ( ( A e. P. /\ B e. P. ) -> ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) -> ( c G d ) e. ( 2nd ` ( A F B ) ) ) )
36	35	imp 123	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) -> ( c G d ) e. ( 2nd ` ( A F B ) ) )
37		elprnqu 7314	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ c e. ( 2nd ` A ) ) -> c e. Q. )
38	8, 37	sylan 281	. . . . . . . . . . . 12 |- ( ( A e. P. /\ c e. ( 2nd ` A ) ) -> c e. Q. )
39		elprnqu 7314	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ d e. ( 2nd ` B ) ) -> d e. Q. )
40	11, 39	sylan 281	. . . . . . . . . . . 12 |- ( ( B e. P. /\ d e. ( 2nd ` B ) ) -> d e. Q. )
41	38, 40	anim12i 336	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ c e. ( 2nd ` A ) ) /\ ( B e. P. /\ d e. ( 2nd ` B ) ) ) -> ( c e. Q. /\ d e. Q. ) )
42	41	an4s 578	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) -> ( c e. Q. /\ d e. Q. ) )
43	2	caovcl 5933	. . . . . . . . . 10 |- ( ( c e. Q. /\ d e. Q. ) -> ( c G d ) e. Q. )
44	42, 43	syl 14	. . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) -> ( c G d ) e. Q. )
45		breq1 3940	. . . . . . . . . . 11 |- ( q = ( c G d ) -> ( q <Q r <-> ( c G d ) <Q r ) )
46		eleq1 2203	. . . . . . . . . . 11 |- ( q = ( c G d ) -> ( q e. ( 2nd ` ( A F B ) ) <-> ( c G d ) e. ( 2nd ` ( A F B ) ) ) )
47	45, 46	anbi12d 465	. . . . . . . . . 10 |- ( q = ( c G d ) -> ( ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) <-> ( ( c G d ) <Q r /\ ( c G d ) e. ( 2nd ` ( A F B ) ) ) ) )
48	47	adantl 275	. . . . . . . . 9 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) /\ q = ( c G d ) ) -> ( ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) <-> ( ( c G d ) <Q r /\ ( c G d ) e. ( 2nd ` ( A F B ) ) ) ) )
49	44, 48	rspcedv 2797	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) -> ( ( ( c G d ) <Q r /\ ( c G d ) e. ( 2nd ` ( A F B ) ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) ) )
50	36, 49	mpan2d 425	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) -> ( ( c G d ) <Q r -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) ) )
51	50	rexlimdvva 2560	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q r -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) ) )
52	51	adantr 274	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( r e. Q. /\ r e. ( 2nd ` ( A F B ) ) ) ) -> ( E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) ( c G d ) <Q r -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) ) )
53	34, 52	mpd 13	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( r e. Q. /\ r e. ( 2nd ` ( A F B ) ) ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) )
54	53	expr 373	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ r e. Q. ) -> ( r e. ( 2nd ` ( A F B ) ) -> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) ) )
55		genprndu.upper	. . . . . . . . . . 11 |- ( ( ( ( A e. P. /\ g e. ( 2nd ` A ) ) /\ ( B e. P. /\ h e. ( 2nd ` B ) ) ) /\ x e. Q. ) -> ( ( g G h ) <Q x -> x e. ( 2nd ` ( A F B ) ) ) )
56	1, 2, 55	genpcuu 7352	. . . . . . . . . 10 |- ( ( A e. P. /\ B e. P. ) -> ( q e. ( 2nd ` ( A F B ) ) -> ( q <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) )
57	56	alrimdv 1849	. . . . . . . . 9 |- ( ( A e. P. /\ B e. P. ) -> ( q e. ( 2nd ` ( A F B ) ) -> A. x ( q <Q x -> x e. ( 2nd ` ( A F B ) ) ) ) )
58		breq2 3941	. . . . . . . . . . 11 |- ( x = r -> ( q <Q x <-> q <Q r ) )
59		eleq1 2203	. . . . . . . . . . 11 |- ( x = r -> ( x e. ( 2nd ` ( A F B ) ) <-> r e. ( 2nd ` ( A F B ) ) ) )
60	58, 59	imbi12d 233	. . . . . . . . . 10 |- ( x = r -> ( ( q <Q x -> x e. ( 2nd ` ( A F B ) ) ) <-> ( q <Q r -> r e. ( 2nd ` ( A F B ) ) ) ) )
61	60	cbvalv 1890	. . . . . . . . 9 |- ( A. x ( q <Q x -> x e. ( 2nd ` ( A F B ) ) ) <-> A. r ( q <Q r -> r e. ( 2nd ` ( A F B ) ) ) )
62	57, 61	syl6ib 160	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( q e. ( 2nd ` ( A F B ) ) -> A. r ( q <Q r -> r e. ( 2nd ` ( A F B ) ) ) ) )
63		sp 1489	. . . . . . . 8 |- ( A. r ( q <Q r -> r e. ( 2nd ` ( A F B ) ) ) -> ( q <Q r -> r e. ( 2nd ` ( A F B ) ) ) )
64	62, 63	syl6 33	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( q e. ( 2nd ` ( A F B ) ) -> ( q <Q r -> r e. ( 2nd ` ( A F B ) ) ) ) )
65	64	impd 252	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( q e. ( 2nd ` ( A F B ) ) /\ q <Q r ) -> r e. ( 2nd ` ( A F B ) ) ) )
66	65	ancomsd 267	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) -> r e. ( 2nd ` ( A F B ) ) ) )
67	66	ad2antrr 480	. . . 4 |- ( ( ( ( A e. P. /\ B e. P. ) /\ r e. Q. ) /\ q e. Q. ) -> ( ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) -> r e. ( 2nd ` ( A F B ) ) ) )
68	67	rexlimdva 2552	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ r e. Q. ) -> ( E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) -> r e. ( 2nd ` ( A F B ) ) ) )
69	54, 68	impbid 128	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ r e. Q. ) -> ( r e. ( 2nd ` ( A F B ) ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) ) )
70	69	ralrimiva 2508	1 |- ( ( A e. P. /\ B e. P. ) -> A. r e. Q. ( r e. ( 2nd ` ( A F B ) ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A F B ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   A.wal 1330   = wceq 1332   E.wex 1469   e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421   <.cop 3535   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   e. cmpo 5784   1stc1st 6044   2ndc2nd 6045   Q.cnq 7112   <Q cltq 7117   P.cnp 7123

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-mi 7138  df-lti 7139  df-enq 7179  df-nqqs 7180  df-ltnqqs 7185  df-inp 7298

This theorem is referenced by:  addclpr  7369  mulclpr  7404