Theorem genprndl 7634

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 7625	. . . . . . . . 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 2526	. . . . . . . . 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 196	. . . . . . . 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 296	. . . . . . 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 478	. . . . . 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 7588	. . . . . . . . . . . . . . . 16 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
9		prnmaxl 7601	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ a e. ( 1st ` A ) ) -> E. c e. ( 1st ` A ) a <Q c )
10	8, 9	sylan 283	. . . . . . . . . . . . . . 15 |- ( ( A e. P. /\ a e. ( 1st ` A ) ) -> E. c e. ( 1st ` A ) a <Q c )
11		prop 7588	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
12		prnmaxl 7601	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ b e. ( 1st ` B ) ) -> E. d e. ( 1st ` B ) b <Q d )
13	11, 12	sylan 283	. . . . . . . . . . . . . . 15 |- ( ( B e. P. /\ b e. ( 1st ` B ) ) -> E. d e. ( 1st ` B ) b <Q d )
14	10, 13	anim12i 338	. . . . . . . . . . . . . 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 588	. . . . . . . . . . . . 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 2676	. . . . . . . . . . . . 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 134	. . . . . . . . . . . 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 7623	. . . . . . . . . . . . . 14 |- ( ( a <Q c /\ b <Q d ) -> ( a G b ) <Q ( c G d ) )
21	20	reximi 2603	. . . . . . . . . . . . 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 2603	. . . . . . . . . . . 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 479	. . . . . . . . . 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 4047	. . . . . . . . . . . . . 14 |- ( q = ( a G b ) -> ( q <Q ( c G d ) <-> ( a G b ) <Q ( c G d ) ) )
26	25	biimprd 158	. . . . . . . . . . . . 13 |- ( q = ( a G b ) -> ( ( a G b ) <Q ( c G d ) -> q <Q ( c G d ) ) )
27	26	reximdv 2607	. . . . . . . . . . . 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 2607	. . . . . . . . . . 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 491	. . . . . . . . . 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 115	. . . . . . . 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 1921	. . . . . . 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 276	. . . . . 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 7627	. . . . . . . . 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 124	. . . . . . . 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 7594	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ c e. ( 1st ` A ) ) -> c e. Q. )
38	8, 37	sylan 283	. . . . . . . . . . . 12 |- ( ( A e. P. /\ c e. ( 1st ` A ) ) -> c e. Q. )
39		elprnql 7594	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ d e. ( 1st ` B ) ) -> d e. Q. )
40	11, 39	sylan 283	. . . . . . . . . . . 12 |- ( ( B e. P. /\ d e. ( 1st ` B ) ) -> d e. Q. )
41	38, 40	anim12i 338	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ c e. ( 1st ` A ) ) /\ ( B e. P. /\ d e. ( 1st ` B ) ) ) -> ( c e. Q. /\ d e. Q. ) )
42	41	an4s 588	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( c e. ( 1st ` A ) /\ d e. ( 1st ` B ) ) ) -> ( c e. Q. /\ d e. Q. ) )
43	2	caovcl 6101	. . . . . . . . . 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 4048	. . . . . . . . . . 11 |- ( r = ( c G d ) -> ( q <Q r <-> q <Q ( c G d ) ) )
46		eleq1 2268	. . . . . . . . . . 11 |- ( r = ( c G d ) -> ( r e. ( 1st ` ( A F B ) ) <-> ( c G d ) e. ( 1st ` ( A F B ) ) ) )
47	45, 46	anbi12d 473	. . . . . . . . . 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 277	. . . . . . . . 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 2881	. . . . . . . 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 428	. . . . . . 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 2631	. . . . . 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 276	. . . . 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 375	. . 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 7632	. . . . . . . . . 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 1899	. . . . . . . . 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 4047	. . . . . . . . . . 11 |- ( x = q -> ( x <Q r <-> q <Q r ) )
59		eleq1 2268	. . . . . . . . . . 11 |- ( x = q -> ( x e. ( 1st ` ( A F B ) ) <-> q e. ( 1st ` ( A F B ) ) ) )
60	58, 59	imbi12d 234	. . . . . . . . . 10 |- ( x = q -> ( ( x <Q r -> x e. ( 1st ` ( A F B ) ) ) <-> ( q <Q r -> q e. ( 1st ` ( A F B ) ) ) ) )
61	60	cbvalv 1941	. . . . . . . . 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	imbitrdi 161	. . . . . . . 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 1534	. . . . . . . 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 254	. . . . . 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 269	. . . . 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 488	. . . 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 2623	. . 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 129	. 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 2579	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 104   <-> wb 105   /\ w3a 981   A.wal 1371   = wceq 1373   E.wex 1515   e. wcel 2176   A.wral 2484   E.wrex 2485   {crab 2488   <.cop 3636   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   e. cmpo 5946   1stc1st 6224   2ndc2nd 6225   Q.cnq 7393   <Q cltq 7398   P.cnp 7404

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-eprel 4336  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-oadd 6506  df-omul 6507  df-er 6620  df-ec 6622  df-qs 6626  df-ni 7417  df-mi 7419  df-lti 7420  df-enq 7460  df-nqqs 7461  df-ltnqqs 7466  df-inp 7579

This theorem is referenced by:  addclpr  7650  mulclpr  7685