Theorem genprndl 7581

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 7572	. . . . . . . . 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 2514	. . . . . . . . 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 7535	. . . . . . . . . . . . . . . 16 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
9		prnmaxl 7548	. . . . . . . . . . . . . . . 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 7535	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
12		prnmaxl 7548	. . . . . . . . . . . . . . . 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 2664	. . . . . . . . . . . . 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 7570	. . . . . . . . . . . . . 14 |- ( ( a <Q c /\ b <Q d ) -> ( a G b ) <Q ( c G d ) )
21	20	reximi 2591	. . . . . . . . . . . . 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 2591	. . . . . . . . . . . 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 4032	. . . . . . . . . . . . . 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 2595	. . . . . . . . . . . 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 2595	. . . . . . . . . . 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 1909	. . . . . . 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 7574	. . . . . . . . 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 7541	. . . . . . . . . . . . 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 7541	. . . . . . . . . . . . 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 6073	. . . . . . . . . 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 4033	. . . . . . . . . . 11 |- ( r = ( c G d ) -> ( q <Q r <-> q <Q ( c G d ) ) )
46		eleq1 2256	. . . . . . . . . . 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 2868	. . . . . . . 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 2619	. . . . . 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 7579	. . . . . . . . . 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 1887	. . . . . . . . 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 4032	. . . . . . . . . . 11 |- ( x = q -> ( x <Q r <-> q <Q r ) )
59		eleq1 2256	. . . . . . . . . . 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 1929	. . . . . . . . 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 1522	. . . . . . . 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 2611	. . 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 2567	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 980   A.wal 1362   = wceq 1364   E.wex 1503   e. wcel 2164   A.wral 2472   E.wrex 2473   {crab 2476   <.cop 3621   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   e. cmpo 5920   1stc1st 6191   2ndc2nd 6192   Q.cnq 7340   <Q cltq 7345   P.cnp 7351

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-mi 7366  df-lti 7367  df-enq 7407  df-nqqs 7408  df-ltnqqs 7413  df-inp 7526

This theorem is referenced by:  addclpr  7597  mulclpr  7632