Theorem genprndu 7484

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 7475	. . . . . . . . 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 2490	. . . . . . . . 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	bitrdi 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 475	. . . . . 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 7437	. . . . . . . . . . . . . . . 16 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
9		prnminu 7451	. . . . . . . . . . . . . . . 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 7437	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
12		prnminu 7451	. . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . 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 2639	. . . . . . . . . . . . 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 7472	. . . . . . . . . . . . . 14 |- ( ( c <Q a /\ d <Q b ) -> ( c G d ) <Q ( a G b ) )
21	20	reximi 2567	. . . . . . . . . . . . 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 2567	. . . . . . . . . . . 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 476	. . . . . . . . . 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 3993	. . . . . . . . . . . . . 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 2571	. . . . . . . . . . . 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 2571	. . . . . . . . . . 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 488	. . . . . . . . . 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 1890	. . . . . . 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 7477	. . . . . . . . 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 7444	. . . . . . . . . . . . 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 7444	. . . . . . . . . . . . 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 583	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) -> ( c e. Q. /\ d e. Q. ) )
43	2	caovcl 6007	. . . . . . . . . 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 3992	. . . . . . . . . . 11 |- ( q = ( c G d ) -> ( q <Q r <-> ( c G d ) <Q r ) )
46		eleq1 2233	. . . . . . . . . . 11 |- ( q = ( c G d ) -> ( q e. ( 2nd ` ( A F B ) ) <-> ( c G d ) e. ( 2nd ` ( A F B ) ) ) )
47	45, 46	anbi12d 470	. . . . . . . . . 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 2838	. . . . . . . 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 426	. . . . . . 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 2595	. . . . . 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 7482	. . . . . . . . . 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 1869	. . . . . . . . 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 3993	. . . . . . . . . . 11 |- ( x = r -> ( q <Q x <-> q <Q r ) )
59		eleq1 2233	. . . . . . . . . . 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 1910	. . . . . . . . 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 1504	. . . . . . . 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 485	. . . 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 2587	. . 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 2543	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 973   A.wal 1346   = wceq 1348   E.wex 1485   e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   <Q cltq 7247   P.cnp 7253

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 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  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-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-mi 7268  df-lti 7269  df-enq 7309  df-nqqs 7310  df-ltnqqs 7315  df-inp 7428

This theorem is referenced by:  addclpr  7499  mulclpr  7534