Theorem genprndu 6712

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 6703	. . . . . . . . 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 2386	. . . . . . . . 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 194	. . . . . . . 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 290	. . . . . . 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 461	. . . . . 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 6665	. . . . . . . . . . . . . . . 16 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
9		prnminu 6679	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ a e. ( 2nd ` A ) ) -> E. c e. ( 2nd ` A ) c <Q a )
10	8, 9	sylan 277	. . . . . . . . . . . . . . 15 |- ( ( A e. P. /\ a e. ( 2nd ` A ) ) -> E. c e. ( 2nd ` A ) c <Q a )
11		prop 6665	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
12		prnminu 6679	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ b e. ( 2nd ` B ) ) -> E. d e. ( 2nd ` B ) d <Q b )
13	11, 12	sylan 277	. . . . . . . . . . . . . . 15 |- ( ( B e. P. /\ b e. ( 2nd ` B ) ) -> E. d e. ( 2nd ` B ) d <Q b )
14	10, 13	anim12i 331	. . . . . . . . . . . . . 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 552	. . . . . . . . . . . . 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 2523	. . . . . . . . . . . . 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 132	. . . . . . . . . . . 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 6700	. . . . . . . . . . . . . 14 |- ( ( c <Q a /\ d <Q b ) -> ( c G d ) <Q ( a G b ) )
21	20	reximi 2458	. . . . . . . . . . . . 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 2458	. . . . . . . . . . . 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 462	. . . . . . . . . 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 3789	. . . . . . . . . . . . . 14 |- ( r = ( a G b ) -> ( ( c G d ) <Q r <-> ( c G d ) <Q ( a G b ) ) )
26	25	biimprd 156	. . . . . . . . . . . . 13 |- ( r = ( a G b ) -> ( ( c G d ) <Q ( a G b ) -> ( c G d ) <Q r ) )
27	26	reximdv 2462	. . . . . . . . . . . 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 2462	. . . . . . . . . . 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 474	. . . . . . . . . 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 113	. . . . . . . 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 1818	. . . . . . 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 270	. . . . . 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 6705	. . . . . . . . 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 122	. . . . . . . 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 6672	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ c e. ( 2nd ` A ) ) -> c e. Q. )
38	8, 37	sylan 277	. . . . . . . . . . . 12 |- ( ( A e. P. /\ c e. ( 2nd ` A ) ) -> c e. Q. )
39		elprnqu 6672	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ d e. ( 2nd ` B ) ) -> d e. Q. )
40	11, 39	sylan 277	. . . . . . . . . . . 12 |- ( ( B e. P. /\ d e. ( 2nd ` B ) ) -> d e. Q. )
41	38, 40	anim12i 331	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ c e. ( 2nd ` A ) ) /\ ( B e. P. /\ d e. ( 2nd ` B ) ) ) -> ( c e. Q. /\ d e. Q. ) )
42	41	an4s 552	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) -> ( c e. Q. /\ d e. Q. ) )
43	2	caovcl 5675	. . . . . . . . . 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 3788	. . . . . . . . . . 11 |- ( q = ( c G d ) -> ( q <Q r <-> ( c G d ) <Q r ) )
46		eleq1 2141	. . . . . . . . . . 11 |- ( q = ( c G d ) -> ( q e. ( 2nd ` ( A F B ) ) <-> ( c G d ) e. ( 2nd ` ( A F B ) ) ) )
47	45, 46	anbi12d 456	. . . . . . . . . 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 271	. . . . . . . . 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 2705	. . . . . . . 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 418	. . . . . . 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 2484	. . . . . 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 270	. . . . 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 367	. . 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 6710	. . . . . . . . . 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 1797	. . . . . . . . 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 3789	. . . . . . . . . . 11 |- ( x = r -> ( q <Q x <-> q <Q r ) )
59		eleq1 2141	. . . . . . . . . . 11 |- ( x = r -> ( x e. ( 2nd ` ( A F B ) ) <-> r e. ( 2nd ` ( A F B ) ) ) )
60	58, 59	imbi12d 232	. . . . . . . . . 10 |- ( x = r -> ( ( q <Q x -> x e. ( 2nd ` ( A F B ) ) ) <-> ( q <Q r -> r e. ( 2nd ` ( A F B ) ) ) ) )
61	60	cbvalv 1835	. . . . . . . . 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 159	. . . . . . . 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 1441	. . . . . . . 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 251	. . . . . 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 265	. . . . 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 471	. . . 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 2477	. . 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 127	. 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 2434	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 102   <-> wb 103   /\ w3a 919   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   A.wral 2348   E.wrex 2349   {crab 2352   <.cop 3401   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   |-> cmpt2 5534   1stc1st 5785   2ndc2nd 5786   Q.cnq 6470   <Q cltq 6475   P.cnp 6481

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-mi 6496  df-lti 6497  df-enq 6537  df-nqqs 6538  df-ltnqqs 6543  df-inp 6656

This theorem is referenced by:  addclpr  6727  mulclpr  6762