Theorem genpdisj 6621

Description: The lower and upper cuts produced by addition or multiplication on positive reals are disjoint. (Contributed by Jim Kingdon, 15-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. )
genpdisj.ord	|- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )
genpdisj.com	|- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) )

Assertion

Ref	Expression
genpdisj	|- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. -. ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) )

Distinct variable groups:   x, y, z, w, v, q, A   x, B, y, z, w, v, q   x, G, y, z, w, v, q   F, q

Allowed substitution hints:   F( x, y, z, w, v)

Proof of Theorem genpdisj

Dummy variables a b c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		genpelvl.1	. . . . . . . . 9 |- 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	. . . . . . . . 9 |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )
3	1, 2	genpelvl 6610	. . . . . . . 8 |- ( ( 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 2344	. . . . . . . 8 |- ( 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 185	. . . . . . 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 ) ) ) )
6	1, 2	genpelvu 6611	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( q e. ( 2nd ` ( A F B ) ) <-> E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) q = ( c G d ) ) )
7		r2ex 2344	. . . . . . . 8 |- ( E. c e. ( 2nd ` A ) E. d e. ( 2nd ` B ) q = ( c G d ) <-> E. c E. d ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) )
8	6, 7	syl6bb 185	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( q e. ( 2nd ` ( A F B ) ) <-> E. c E. d ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) )
9	5, 8	anbi12d 442	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) <-> ( E. a E. b ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ E. c E. d ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) ) )
10		ee4anv 1809	. . . . . 6 |- ( E. a E. b E. c E. d ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) <-> ( E. a E. b ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ E. c E. d ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) )
11	9, 10	syl6bbr 187	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) <-> E. a E. b E. c E. d ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) ) )
12	11	biimpa 280	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) ) -> E. a E. b E. c E. d ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) )
13		an4 520	. . . . . . . . . . . . 13 |- ( ( ( a e. ( 1st ` A ) /\ c e. ( 2nd ` A ) ) /\ ( b e. ( 1st ` B ) /\ d e. ( 2nd ` B ) ) ) <-> ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) )
14		prop 6573	. . . . . . . . . . . . . . . 16 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
15		prltlu 6585	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ a e. ( 1st ` A ) /\ c e. ( 2nd ` A ) ) -> a <Q c )
16	15	3expib 1107	. . . . . . . . . . . . . . . 16 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> ( ( a e. ( 1st ` A ) /\ c e. ( 2nd ` A ) ) -> a <Q c ) )
17	14, 16	syl 14	. . . . . . . . . . . . . . 15 |- ( A e. P. -> ( ( a e. ( 1st ` A ) /\ c e. ( 2nd ` A ) ) -> a <Q c ) )
18		prop 6573	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19		prltlu 6585	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ b e. ( 1st ` B ) /\ d e. ( 2nd ` B ) ) -> b <Q d )
20	19	3expib 1107	. . . . . . . . . . . . . . . 16 |- ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. -> ( ( b e. ( 1st ` B ) /\ d e. ( 2nd ` B ) ) -> b <Q d ) )
21	18, 20	syl 14	. . . . . . . . . . . . . . 15 |- ( B e. P. -> ( ( b e. ( 1st ` B ) /\ d e. ( 2nd ` B ) ) -> b <Q d ) )
22	17, 21	im2anan9 530	. . . . . . . . . . . . . 14 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( a e. ( 1st ` A ) /\ c e. ( 2nd ` A ) ) /\ ( b e. ( 1st ` B ) /\ d e. ( 2nd ` B ) ) ) -> ( a <Q c /\ b <Q d ) ) )
23		genpdisj.ord	. . . . . . . . . . . . . . 15 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z G x ) <Q ( z G y ) ) )
24		genpdisj.com	. . . . . . . . . . . . . . 15 |- ( ( x e. Q. /\ y e. Q. ) -> ( x G y ) = ( y G x ) )
25	23, 24	genplt2i 6608	. . . . . . . . . . . . . 14 |- ( ( a <Q c /\ b <Q d ) -> ( a G b ) <Q ( c G d ) )
26	22, 25	syl6 29	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( a e. ( 1st ` A ) /\ c e. ( 2nd ` A ) ) /\ ( b e. ( 1st ` B ) /\ d e. ( 2nd ` B ) ) ) -> ( a G b ) <Q ( c G d ) ) )
27	13, 26	syl5bir 142	. . . . . . . . . . . 12 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) -> ( a G b ) <Q ( c G d ) ) )
28	27	imp 115	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) ) -> ( a G b ) <Q ( c G d ) )
29	28	adantlr 446	. . . . . . . . . 10 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) ) /\ ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) ) -> ( a G b ) <Q ( c G d ) )
30	29	adantrlr 454	. . . . . . . . 9 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) ) /\ ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) ) ) -> ( a G b ) <Q ( c G d ) )
31	30	adantrrr 456	. . . . . . . 8 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) ) /\ ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) ) -> ( a G b ) <Q ( c G d ) )
32		eqtr2 2058	. . . . . . . . . . 11 |- ( ( q = ( a G b ) /\ q = ( c G d ) ) -> ( a G b ) = ( c G d ) )
33	32	ad2ant2l 477	. . . . . . . . . 10 |- ( ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) -> ( a G b ) = ( c G d ) )
34	33	adantl 262	. . . . . . . . 9 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) ) /\ ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) ) -> ( a G b ) = ( c G d ) )
35		ltsonq 6496	. . . . . . . . . . 11 |- <Q Or Q.
36		ltrelnq 6463	. . . . . . . . . . 11 |- <Q C_ ( Q. X. Q. )
37	35, 36	soirri 4719	. . . . . . . . . 10 |- -. ( a G b ) <Q ( a G b )
38		breq2 3768	. . . . . . . . . 10 |- ( ( a G b ) = ( c G d ) -> ( ( a G b ) <Q ( a G b ) <-> ( a G b ) <Q ( c G d ) ) )
39	37, 38	mtbii 599	. . . . . . . . 9 |- ( ( a G b ) = ( c G d ) -> -. ( a G b ) <Q ( c G d ) )
40	34, 39	syl 14	. . . . . . . 8 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) ) /\ ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) ) -> -. ( a G b ) <Q ( c G d ) )
41	31, 40	pm2.21fal 1264	. . . . . . 7 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) ) /\ ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) ) -> F. )
42	41	ex 108	. . . . . 6 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) ) -> ( ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) -> F. ) )
43	42	exlimdvv 1777	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) ) -> ( E. c E. d ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) -> F. ) )
44	43	exlimdvv 1777	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) ) -> ( E. a E. b E. c E. d ( ( ( a e. ( 1st ` A ) /\ b e. ( 1st ` B ) ) /\ q = ( a G b ) ) /\ ( ( c e. ( 2nd ` A ) /\ d e. ( 2nd ` B ) ) /\ q = ( c G d ) ) ) -> F. ) )
45	12, 44	mpd 13	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) ) -> F. )
46	45	inegd 1263	. 2 |- ( ( A e. P. /\ B e. P. ) -> -. ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) )
47	46	ralrimivw 2393	1 |- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. -. ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   F. wfal 1248   E.wex 1381   e. wcel 1393   A.wral 2306   E.wrex 2307   {crab 2310   <.cop 3378   class class class wbr 3764   ` cfv 4902  (class class class)co 5512   |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378   <Q cltq 6383   P.cnp 6389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-lti 6405  df-enq 6445  df-nqqs 6446  df-ltnqqs 6451  df-inp 6564

This theorem is referenced by:  addclpr  6635  mulclpr  6670