Theorem genpdisj 7355

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 7344	. . . . . . . 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 2458	. . . . . . . 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 195	. . . . . . 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 7345	. . . . . . . 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 2458	. . . . . . . 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 195	. . . . . . 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 465	. . . . . 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 1907	. . . . . 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 197	. . . . 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 294	. . . 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 576	. . . . . . . . . . . . 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 7307	. . . . . . . . . . . . . . . 16 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
15		prltlu 7319	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ a e. ( 1st ` A ) /\ c e. ( 2nd ` A ) ) -> a <Q c )
16	15	3expib 1185	. . . . . . . . . . . . . . . 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 7307	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19		prltlu 7319	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ b e. ( 1st ` B ) /\ d e. ( 2nd ` B ) ) -> b <Q d )
20	19	3expib 1185	. . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . 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 7342	. . . . . . . . . . . . . 14 |- ( ( a <Q c /\ b <Q d ) -> ( a G b ) <Q ( c G d ) )
26	22, 25	syl6 33	. . . . . . . . . . . . 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 152	. . . . . . . . . . . 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 123	. . . . . . . . . . 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 469	. . . . . . . . . 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 477	. . . . . . . . 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 479	. . . . . . . 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 2159	. . . . . . . . . . 11 |- ( ( q = ( a G b ) /\ q = ( c G d ) ) -> ( a G b ) = ( c G d ) )
33	32	ad2ant2l 500	. . . . . . . . . 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 275	. . . . . . . . 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 7230	. . . . . . . . . . 11 |- <Q Or Q.
36		ltrelnq 7197	. . . . . . . . . . 11 |- <Q C_ ( Q. X. Q. )
37	35, 36	soirri 4941	. . . . . . . . . 10 |- -. ( a G b ) <Q ( a G b )
38		breq2 3941	. . . . . . . . . 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 664	. . . . . . . . 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 1352	. . . . . . 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 114	. . . . . 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 1870	. . . . 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 1870	. . . 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 1351	. 2 |- ( ( A e. P. /\ B e. P. ) -> -. ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) )
47	46	ralrimivw 2509	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 103   <-> wb 104   /\ w3a 963   = wceq 1332   F. wfal 1337   E.wex 1469   e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421   <.cop 3535   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   e. cmpo 5784   1stc1st 6044   2ndc2nd 6045   Q.cnq 7112   <Q cltq 7117   P.cnp 7123

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-mi 7138  df-lti 7139  df-enq 7179  df-nqqs 7180  df-ltnqqs 7185  df-inp 7298

This theorem is referenced by:  addclpr  7369  mulclpr  7404