Theorem genprndl 7141

Description: The lower 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. )
genprndl.ord	|- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )
genprndl.com	|- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) )
genprndl.lower	|- ( ( ( ( A e. P. /\ g e. ( 1st ` A ) ) /\ ( B e. P. /\ h e. ( 1st ` B ) ) ) /\ x e. Q. ) -> ( x <Q ( g G h ) -> x e. ( 1st ` ( A F B ) ) ) )

Assertion

Ref	Expression
genprndl	|- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. ( q e. ( 1st ` ( A F B ) ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( 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 genprndl

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	genpelvl 7132	. . . . . . . . 9 |- ( ( A e. P. /\ B e. P. ) -> ( q e. ( 1st ` ( A F B ) ) <-> E. a e. ( 1st ` A ) E. b e. ( 1st ` B ) q = ( a G b ) ) )
4		r2ex 2399	. . . . . . . . 9 |- ( E. a e. ( 1st ` A ) E. b e. ( 1st ` B ) q = ( a G b ) <-> E. a E. b ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) )
5	3, 4	syl6bb 195	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( q e. ( 1st ` ( A F B ) ) <-> E. a E. b ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) ) )
6	5	biimpa 291	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ q e. ( 1st ` ( A F B ) ) ) -> E. a E. b ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) )
7	6	adantrl 463	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ q e. ( 1st ` ( A F B ) ) ) ) -> E. a E. b ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) )
8		prop 7095	. . . . . . . . . . . . . . . 16 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
9		prnmaxl 7108	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ a e. ( 1st ` A ) ) -> E. c e. ( 1st ` A ) a <Q c )
10	8, 9	sylan 278	. . . . . . . . . . . . . . 15 |- ( ( A e. P. /\ a e. ( 1st ` A ) ) -> E. c e. ( 1st ` A ) a <Q c )
11		prop 7095	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
12		prnmaxl 7108	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ b e. ( 1st ` B ) ) -> E. d e. ( 1st ` B ) b <Q d )
13	11, 12	sylan 278	. . . . . . . . . . . . . . 15 |- ( ( B e. P. /\ b e. ( 1st ` B ) ) -> E. d e. ( 1st ` B ) b <Q d )
14	10, 13	anim12i 332	. . . . . . . . . . . . . 14 |- ( ( ( A e. P. /\ a e. ( 1st ` A ) ) /\ ( B e. P. /\ b e. ( 1st ` B ) ) ) -> ( E. c e. ( 1st ` A ) a <Q c /\ E. d e. ( 1st ` B ) b <Q d ) )
15	14	an4s 556	. . . . . . . . . . . . 13 |- ( ( ( A e. P. /\ B e. P. ) /\ ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) ) -> ( E. c e. ( 1st ` A ) a <Q c /\ E. d e. ( 1st ` B ) b <Q d ) )
16		reeanv 2537	. . . . . . . . . . . . 13 |- ( E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) ( a <Q c /\ b <Q d ) <-> ( E. c e. ( 1st ` A ) a <Q c /\ E. d e. ( 1st ` B ) b <Q d ) )
17	15, 16	sylibr 133	. . . . . . . . . . . 12 |- ( ( ( A e. P. /\ B e. P. ) /\ ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) ) -> E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) ( a <Q c /\ b <Q d ) )
18		genprndl.ord	. . . . . . . . . . . . . . 15 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )
19		genprndl.com	. . . . . . . . . . . . . . 15 |- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) )
20	18, 19	genplt2i 7130	. . . . . . . . . . . . . 14 |- ( ( a <Q c /\ b <Q d ) -> ( a G b ) <Q ( c G d ) )
21	20	reximi 2471	. . . . . . . . . . . . 13 |- ( E. d e. ( 1st ` B ) ( a <Q c /\ b <Q d ) -> E. d e. ( 1st ` B ) ( a G b ) <Q ( c G d ) )
22	21	reximi 2471	. . . . . . . . . . . 12 |- ( E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) ( a <Q c /\ b <Q d ) -> E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) ( a G b ) <Q ( c G d ) )
23	17, 22	syl 14	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) ) -> E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) ( a G b ) <Q ( c G d ) )
24	23	adantrr 464	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) ) -> E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) ( a G b ) <Q ( c G d ) )
25		breq1 3854	. . . . . . . . . . . . . 14 |- ( q = ( a G b ) -> ( q <Q ( c G d ) <-> ( a G b ) <Q ( c G d ) ) )
26	25	biimprd 157	. . . . . . . . . . . . 13 |- ( q = ( a G b ) -> ( ( a G b ) <Q ( c G d ) -> q <Q ( c G d ) ) )
27	26	reximdv 2475	. . . . . . . . . . . 12 |- ( q = ( a G b ) -> ( E. d e. ( 1st ` B ) ( a G b ) <Q ( c G d ) -> E. d e. ( 1st ` B ) q <Q ( c G d ) ) )
28	27	reximdv 2475	. . . . . . . . . . 11 |- ( q = ( a G b ) -> ( E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) ( a G b ) <Q ( c G d ) -> E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) q <Q ( c G d ) ) )
29	28	ad2antll 476	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) ) -> ( E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) ( a G b ) <Q ( c G d ) -> E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) q <Q ( c G d ) ) )
30	24, 29	mpd 13	. . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. ) /\ ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) ) -> E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) q <Q ( c G d ) )
31	30	ex 114	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) -> E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) q <Q ( c G d ) ) )
32	31	exlimdvv 1826	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( E. a E. b ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) -> E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) q <Q ( c G d ) ) )
33	32	adantr 271	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ q e. ( 1st ` ( A F B ) ) ) ) -> ( E. a E. b ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) -> E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) q <Q ( c G d ) ) )
34	7, 33	mpd 13	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ q e. ( 1st ` ( A F B ) ) ) ) -> E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) q <Q ( c G d ) )
35	1, 2	genpprecll 7134	. . . . . . . . 9 |- ( ( A e. P. /\ B e. P. ) -> ( ( c e. ( 1st ` A ) /\ d e. ( 1st ` B ) ) -> ( c G d ) e. ( 1st ` ( A F B ) ) ) )
36	35	imp 123	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 1st ` A ) /\ d e. ( 1st ` B ) ) ) -> ( c G d ) e. ( 1st ` ( A F B ) ) )
37		elprnql 7101	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ c e. ( 1st ` A ) ) -> c e. Q. )
38	8, 37	sylan 278	. . . . . . . . . . . 12 |- ( ( A e. P. /\ c e. ( 1st ` A ) ) -> c e. Q. )
39		elprnql 7101	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ d e. ( 1st ` B ) ) -> d e. Q. )
40	11, 39	sylan 278	. . . . . . . . . . . 12 |- ( ( B e. P. /\ d e. ( 1st ` B ) ) -> d e. Q. )
41	38, 40	anim12i 332	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ c e. ( 1st ` A ) ) /\ ( B e. P. /\ d e. ( 1st ` B ) ) ) -> ( c e. Q. /\ d e. Q. ) )
42	41	an4s 556	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 1st ` A ) /\ d e. ( 1st ` B ) ) ) -> ( c e. Q. /\ d e. Q. ) )
43	2	caovcl 5813	. . . . . . . . . 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. ( 1st ` A ) /\ d e. ( 1st ` B ) ) ) -> ( c G d ) e. Q. )
45		breq2 3855	. . . . . . . . . . 11 |- ( r = ( c G d ) -> ( q <Q r <-> q <Q ( c G d ) ) )
46		eleq1 2151	. . . . . . . . . . 11 |- ( r = ( c G d ) -> ( r e. ( 1st ` ( A F B ) ) <-> ( c G d ) e. ( 1st ` ( A F B ) ) ) )
47	45, 46	anbi12d 458	. . . . . . . . . 10 |- ( r = ( c G d ) -> ( ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) <-> ( q <Q ( c G d ) /\ ( c G d ) e. ( 1st ` ( A F B ) ) ) ) )
48	47	adantl 272	. . . . . . . . 9 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 1st ` A ) /\ d e. ( 1st ` B ) ) ) /\ r = ( c G d ) ) -> ( ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) <-> ( q <Q ( c G d ) /\ ( c G d ) e. ( 1st ` ( A F B ) ) ) ) )
49	44, 48	rspcedv 2727	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 1st ` A ) /\ d e. ( 1st ` B ) ) ) -> ( ( q <Q ( c G d ) /\ ( c G d ) e. ( 1st ` ( A F B ) ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) ) )
50	36, 49	mpan2d 420	. . . . . . 7 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 1st ` A ) /\ d e. ( 1st ` B ) ) ) -> ( q <Q ( c G d ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) ) )
51	50	rexlimdvva 2497	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) q <Q ( c G d ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) ) )
52	51	adantr 271	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ q e. ( 1st ` ( A F B ) ) ) ) -> ( E. c e. ( 1st ` A ) E. d e. ( 1st ` B ) q <Q ( c G d ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) ) )
53	34, 52	mpd 13	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ q e. ( 1st ` ( A F B ) ) ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) )
54	53	expr 368	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ q e. Q. ) -> ( q e. ( 1st ` ( A F B ) ) -> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) ) )
55		genprndl.lower	. . . . . . . . . . 11 |- ( ( ( ( A e. P. /\ g e. ( 1st ` A ) ) /\ ( B e. P. /\ h e. ( 1st ` B ) ) ) /\ x e. Q. ) -> ( x <Q ( g G h ) -> x e. ( 1st ` ( A F B ) ) ) )
56	1, 2, 55	genpcdl 7139	. . . . . . . . . 10 |- ( ( A e. P. /\ B e. P. ) -> ( r e. ( 1st ` ( A F B ) ) -> ( x <Q r -> x e. ( 1st ` ( A F B ) ) ) ) )
57	56	alrimdv 1805	. . . . . . . . 9 |- ( ( A e. P. /\ B e. P. ) -> ( r e. ( 1st ` ( A F B ) ) -> A. x ( x <Q r -> x e. ( 1st ` ( A F B ) ) ) ) )
58		breq1 3854	. . . . . . . . . . 11 |- ( x = q -> ( x <Q r <-> q <Q r ) )
59		eleq1 2151	. . . . . . . . . . 11 |- ( x = q -> ( x e. ( 1st ` ( A F B ) ) <-> q e. ( 1st ` ( A F B ) ) ) )
60	58, 59	imbi12d 233	. . . . . . . . . 10 |- ( x = q -> ( ( x <Q r -> x e. ( 1st ` ( A F B ) ) ) <-> ( q <Q r -> q e. ( 1st ` ( A F B ) ) ) ) )
61	60	cbvalv 1843	. . . . . . . . 9 |- ( A. x ( x <Q r -> x e. ( 1st ` ( A F B ) ) ) <-> A. q ( q <Q r -> q e. ( 1st ` ( A F B ) ) ) )
62	57, 61	syl6ib 160	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( r e. ( 1st ` ( A F B ) ) -> A. q ( q <Q r -> q e. ( 1st ` ( A F B ) ) ) ) )
63		sp 1447	. . . . . . . 8 |- ( A. q ( q <Q r -> q e. ( 1st ` ( A F B ) ) ) -> ( q <Q r -> q e. ( 1st ` ( A F B ) ) ) )
64	62, 63	syl6 33	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( r e. ( 1st ` ( A F B ) ) -> ( q <Q r -> q e. ( 1st ` ( A F B ) ) ) ) )
65	64	impd 252	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( r e. ( 1st ` ( A F B ) ) /\ q <Q r ) -> q e. ( 1st ` ( A F B ) ) ) )
66	65	ancomsd 266	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) -> q e. ( 1st ` ( A F B ) ) ) )
67	66	ad2antrr 473	. . . 4 |- ( ( ( ( A e. P. /\ B e. P. ) /\ q e. Q. ) /\ r e. Q. ) -> ( ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) -> q e. ( 1st ` ( A F B ) ) ) )
68	67	rexlimdva 2490	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ q e. Q. ) -> ( E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) -> q e. ( 1st ` ( A F B ) ) ) )
69	54, 68	impbid 128	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ q e. Q. ) -> ( q e. ( 1st ` ( A F B ) ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) ) )
70	69	ralrimiva 2447	1 |- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. ( q e. ( 1st ` ( A F B ) ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A F B ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 925   A.wal 1288   = wceq 1290   E.wex 1427   e. wcel 1439   A.wral 2360   E.wrex 2361   {crab 2364   <.cop 3453   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   |-> cmpt2 5668   1stc1st 5923   2ndc2nd 5924   Q.cnq 6900   <Q cltq 6905   P.cnp 6911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-mi 6926  df-lti 6927  df-enq 6967  df-nqqs 6968  df-ltnqqs 6973  df-inp 7086

This theorem is referenced by:  addclpr  7157  mulclpr  7192