Theorem genprndl 7487

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 7478	. . . . . . . . 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 2491	. . . . . . . . 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	bitrdi 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 294	. . . . . . 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 476	. . . . . 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 7441	. . . . . . . . . . . . . . . 16 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
9		prnmaxl 7454	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ a e. ( 1st ` A ) ) -> E. c e. ( 1st ` A ) a <Q c )
10	8, 9	sylan 281	. . . . . . . . . . . . . . 15 |- ( ( A e. P. /\ a e. ( 1st ` A ) ) -> E. c e. ( 1st ` A ) a <Q c )
11		prop 7441	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
12		prnmaxl 7454	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ b e. ( 1st ` B ) ) -> E. d e. ( 1st ` B ) b <Q d )
13	11, 12	sylan 281	. . . . . . . . . . . . . . 15 |- ( ( B e. P. /\ b e. ( 1st ` B ) ) -> E. d e. ( 1st ` B ) b <Q d )
14	10, 13	anim12i 336	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 2640	. . . . . . . . . . . . 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 7476	. . . . . . . . . . . . . 14 |- ( ( a <Q c /\ b <Q d ) -> ( a G b ) <Q ( c G d ) )
21	20	reximi 2568	. . . . . . . . . . . . 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 2568	. . . . . . . . . . . 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 477	. . . . . . . . . 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 3993	. . . . . . . . . . . . . 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 2572	. . . . . . . . . . . 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 2572	. . . . . . . . . . 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 489	. . . . . . . . . 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 1891	. . . . . . 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 274	. . . . . 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 7480	. . . . . . . . 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 7447	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ c e. ( 1st ` A ) ) -> c e. Q. )
38	8, 37	sylan 281	. . . . . . . . . . . 12 |- ( ( A e. P. /\ c e. ( 1st ` A ) ) -> c e. Q. )
39		elprnql 7447	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ d e. ( 1st ` B ) ) -> d e. Q. )
40	11, 39	sylan 281	. . . . . . . . . . . 12 |- ( ( B e. P. /\ d e. ( 1st ` B ) ) -> d e. Q. )
41	38, 40	anim12i 336	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ c e. ( 1st ` A ) ) /\ ( B e. P. /\ d e. ( 1st ` B ) ) ) -> ( c e. Q. /\ d e. Q. ) )
42	41	an4s 584	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 1st ` A ) /\ d e. ( 1st ` B ) ) ) -> ( c e. Q. /\ d e. Q. ) )
43	2	caovcl 6011	. . . . . . . . . 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 3994	. . . . . . . . . . 11 |- ( r = ( c G d ) -> ( q <Q r <-> q <Q ( c G d ) ) )
46		eleq1 2234	. . . . . . . . . . 11 |- ( r = ( c G d ) -> ( r e. ( 1st ` ( A F B ) ) <-> ( c G d ) e. ( 1st ` ( A F B ) ) ) )
47	45, 46	anbi12d 471	. . . . . . . . . 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 275	. . . . . . . . 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 2839	. . . . . . . 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 426	. . . . . . 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 2596	. . . . . 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 274	. . . . 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 373	. . 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 7485	. . . . . . . . . 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 1870	. . . . . . . . 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 3993	. . . . . . . . . . 11 |- ( x = q -> ( x <Q r <-> q <Q r ) )
59		eleq1 2234	. . . . . . . . . . 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 1911	. . . . . . . . 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 1505	. . . . . . . 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 267	. . . . 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 486	. . . 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 2588	. . 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 2544	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 974   A.wal 1347   = wceq 1349   E.wex 1486   e. wcel 2142   A.wral 2449   E.wrex 2450   {crab 2453   <.cop 3587   class class class wbr 3990   ` cfv 5200  (class class class)co 5857   e. cmpo 5859   1stc1st 6121   2ndc2nd 6122   Q.cnq 7246   <Q cltq 7251   P.cnp 7257

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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-nul 4116  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-iinf 4573

This theorem depends on definitions:  df-bi 116  df-dc 831  df-3or 975  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-ral 2454  df-rex 2455  df-reu 2456  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-nul 3416  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-br 3991  df-opab 4052  df-mpt 4053  df-tr 4089  df-eprel 4275  df-id 4279  df-po 4282  df-iso 4283  df-iord 4352  df-on 4354  df-suc 4357  df-iom 4576  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-ov 5860  df-oprab 5861  df-mpo 5862  df-1st 6123  df-2nd 6124  df-recs 6288  df-irdg 6353  df-oadd 6403  df-omul 6404  df-er 6517  df-ec 6519  df-qs 6523  df-ni 7270  df-mi 7272  df-lti 7273  df-enq 7313  df-nqqs 7314  df-ltnqqs 7319  df-inp 7432

This theorem is referenced by:  addclpr  7503  mulclpr  7538