Theorem genprndu 7802

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 7793	. . . . . . . . 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 2553	. . . . . . . . 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 196	. . . . . . . 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 296	. . . . . . 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 478	. . . . . 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 7755	. . . . . . . . . . . . . . . 16 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
9		prnminu 7769	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ a e. ( 2nd ` A ) ) -> E. c e. ( 2nd ` A ) c <Q a )
10	8, 9	sylan 283	. . . . . . . . . . . . . . 15 |- ( ( A e. P. /\ a e. ( 2nd ` A ) ) -> E. c e. ( 2nd ` A ) c <Q a )
11		prop 7755	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
12		prnminu 7769	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ b e. ( 2nd ` B ) ) -> E. d e. ( 2nd ` B ) d <Q b )
13	11, 12	sylan 283	. . . . . . . . . . . . . . 15 |- ( ( B e. P. /\ b e. ( 2nd ` B ) ) -> E. d e. ( 2nd ` B ) d <Q b )
14	10, 13	anim12i 338	. . . . . . . . . . . . . 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 592	. . . . . . . . . . . . 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 2704	. . . . . . . . . . . . 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 134	. . . . . . . . . . . 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 7790	. . . . . . . . . . . . . 14 |- ( ( c <Q a /\ d <Q b ) -> ( c G d ) <Q ( a G b ) )
21	20	reximi 2630	. . . . . . . . . . . . 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 2630	. . . . . . . . . . . 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 479	. . . . . . . . . 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 4097	. . . . . . . . . . . . . 14 |- ( r = ( a G b ) -> ( ( c G d ) <Q r <-> ( c G d ) <Q ( a G b ) ) )
26	25	biimprd 158	. . . . . . . . . . . . 13 |- ( r = ( a G b ) -> ( ( c G d ) <Q ( a G b ) -> ( c G d ) <Q r ) )
27	26	reximdv 2634	. . . . . . . . . . . 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 2634	. . . . . . . . . . 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 491	. . . . . . . . . 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 115	. . . . . . . 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 1946	. . . . . . 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 276	. . . . . 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 7795	. . . . . . . . 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 124	. . . . . . . 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 7762	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ c e. ( 2nd ` A ) ) -> c e. Q. )
38	8, 37	sylan 283	. . . . . . . . . . . 12 |- ( ( A e. P. /\ c e. ( 2nd ` A ) ) -> c e. Q. )
39		elprnqu 7762	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ d e. ( 2nd ` B ) ) -> d e. Q. )
40	11, 39	sylan 283	. . . . . . . . . . . 12 |- ( ( B e. P. /\ d e. ( 2nd ` B ) ) -> d e. Q. )
41	38, 40	anim12i 338	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ c e. ( 2nd ` A ) ) /\ ( B e. P. /\ d e. ( 2nd ` B ) ) ) -> ( c e. Q. /\ d e. Q. ) )
42	41	an4s 592	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) -> ( c e. Q. /\ d e. Q. ) )
43	2	caovcl 6187	. . . . . . . . . 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 4096	. . . . . . . . . . 11 |- ( q = ( c G d ) -> ( q <Q r <-> ( c G d ) <Q r ) )
46		eleq1 2294	. . . . . . . . . . 11 |- ( q = ( c G d ) -> ( q e. ( 2nd ` ( A F B ) ) <-> ( c G d ) e. ( 2nd ` ( A F B ) ) ) )
47	45, 46	anbi12d 473	. . . . . . . . . 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 277	. . . . . . . . 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 2915	. . . . . . . 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 428	. . . . . . 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 2659	. . . . . 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 276	. . . . 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 375	. . 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 7800	. . . . . . . . . 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 1924	. . . . . . . . 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 4097	. . . . . . . . . . 11 |- ( x = r -> ( q <Q x <-> q <Q r ) )
59		eleq1 2294	. . . . . . . . . . 11 |- ( x = r -> ( x e. ( 2nd ` ( A F B ) ) <-> r e. ( 2nd ` ( A F B ) ) ) )
60	58, 59	imbi12d 234	. . . . . . . . . 10 |- ( x = r -> ( ( q <Q x -> x e. ( 2nd ` ( A F B ) ) ) <-> ( q <Q r -> r e. ( 2nd ` ( A F B ) ) ) ) )
61	60	cbvalv 1966	. . . . . . . . 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	imbitrdi 161	. . . . . . . 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 1560	. . . . . . . 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 254	. . . . . 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 269	. . . . 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 488	. . . 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 2651	. . 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 129	. 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 2606	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 104   <-> wb 105   /\ w3a 1005   A.wal 1396   = wceq 1398   E.wex 1541   e. wcel 2202   A.wral 2511   E.wrex 2512   {crab 2515   <.cop 3676   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   1stc1st 6310   2ndc2nd 6311   Q.cnq 7560   <Q cltq 7565   P.cnp 7571

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-mi 7586  df-lti 7587  df-enq 7627  df-nqqs 7628  df-ltnqqs 7633  df-inp 7746

This theorem is referenced by:  addclpr  7817  mulclpr  7852