Theorem genpdisj 6506

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 Gz)) } , {x e. Q. | E.y e. Q. E.z e. Q. (y e. ( 2nd ` w) /\ z e. ( 2nd ` v) /\ x = (y Gz)) } >.)
genpelvl.2	|- ((y e. Q. /\ z e. Q.) -> (y Gz) e. Q.)
genpdisj.ord	|- ((x e. Q. /\ y e. Q. /\ z e. Q.) -> (x <Q y <-> (z Gx) <Q (z Gy)))
genpdisj.com	|- ((x e. Q. /\ y e. Q.) -> (x Gy) = (y Gx))

Assertion

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

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 Gz)) } , {x e. Q. | E.y e. Q. E.z e. Q. (y e. ( 2nd ` w) /\ z e. ( 2nd ` v) /\ x = (y Gz)) } >.)
2		genpelvl.2	. . . . . . . . 9 |- ((y e. Q. /\ z e. Q.) -> (y Gz) e. Q.)
3	1, 2	genpelvl 6495	. . . . . . . 8 |- ((A e. P. /\ B e. P.) -> ( q e. ( 1st ` (A FB)) <-> E. a e. ( 1st ` A)E. b e. ( 1st ` B) q = ( a G b)))
4		r2ex 2338	. . . . . . . 8 |- (E. a e. ( 1st ` A)E. b e. ( 1st ` B) q = ( a G b) <-> E. aE. 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 FB)) <-> E. aE. b(( a e. ( 1st ` A) /\ b e. ( 1st ` B)) /\ q = ( a G b))))
6	1, 2	genpelvu 6496	. . . . . . . 8 |- ((A e. P. /\ B e. P.) -> ( q e. ( 2nd ` (A FB)) <-> E. c e. ( 2nd ` A)E. d e. ( 2nd ` B) q = ( c G d)))
7		r2ex 2338	. . . . . . . 8 |- (E. c e. ( 2nd ` A)E. d e. ( 2nd ` B) q = ( c G d) <-> E. cE. 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 FB)) <-> E. cE. 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 FB)) /\ q e. ( 2nd ` (A FB))) <-> (E. aE. b(( a e. ( 1st ` A) /\ b e. ( 1st ` B)) /\ q = ( a G b)) /\ E. cE. d(( c e. ( 2nd ` A) /\ d e. ( 2nd ` B)) /\ q = ( c G d)))))
10		ee4anv 1806	. . . . . 6 |- (E. aE. bE. cE. 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. aE. b(( a e. ( 1st ` A) /\ b e. ( 1st ` B)) /\ q = ( a G b)) /\ E. cE. 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 FB)) /\ q e. ( 2nd ` (A FB))) <-> E. aE. bE. cE. 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 FB)) /\ q e. ( 2nd ` (A FB)))) -> E. aE. bE. cE. 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 6458	. . . . . . . . . . . . . . . 16 |- (A e. P. -> <.( 1st ` A) , ( 2nd ` A) >. e. P.)
15		prltlu 6470	. . . . . . . . . . . . . . . . 17 |- (( <.( 1st ` A) , ( 2nd ` A) >. e. P. /\ a e. ( 1st ` A) /\ c e. ( 2nd ` A)) -> a <Q c)
16	15	3expib 1106	. . . . . . . . . . . . . . . 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 6458	. . . . . . . . . . . . . . . 16 |- (B e. P. -> <.( 1st ` B) , ( 2nd ` B) >. e. P.)
19		prltlu 6470	. . . . . . . . . . . . . . . . 17 |- (( <.( 1st ` B) , ( 2nd ` B) >. e. P. /\ b e. ( 1st ` B) /\ d e. ( 2nd ` B)) -> b <Q d)
20	19	3expib 1106	. . . . . . . . . . . . . . . 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 Gx) <Q (z Gy)))
24		genpdisj.com	. . . . . . . . . . . . . . 15 |- ((x e. Q. /\ y e. Q.) -> (x Gy) = (y Gx))
25	23, 24	genplt2i 6493	. . . . . . . . . . . . . 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 FB)) /\ q e. ( 2nd ` (A FB)))) /\ (( 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 FB)) /\ q e. ( 2nd ` (A FB)))) /\ ((( 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 FB)) /\ q e. ( 2nd ` (A FB)))) /\ ((( 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 2055	. . . . . . . . . . 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 FB)) /\ q e. ( 2nd ` (A FB)))) /\ ((( 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 6382	. . . . . . . . . . 11 |- <Q Or Q.
36		ltrelnq 6349	. . . . . . . . . . 11 |- <Q C_ ( Q. X. Q.)
37	35, 36	soirri 4662	. . . . . . . . . 10 |- -. ( a G b) <Q ( a G b)
38		breq2 3759	. . . . . . . . . 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 598	. . . . . . . . 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 FB)) /\ q e. ( 2nd ` (A FB)))) /\ ((( 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 1263	. . . . . . 7 |- ((((A e. P. /\ B e. P.) /\ ( q e. ( 1st ` (A FB)) /\ q e. ( 2nd ` (A FB)))) /\ ((( 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 FB)) /\ q e. ( 2nd ` (A FB)))) -> (((( 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 1774	. . . . 5 |- (((A e. P. /\ B e. P.) /\ ( q e. ( 1st ` (A FB)) /\ q e. ( 2nd ` (A FB)))) -> (E. cE. 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 1774	. . . 4 |- (((A e. P. /\ B e. P.) /\ ( q e. ( 1st ` (A FB)) /\ q e. ( 2nd ` (A FB)))) -> (E. aE. bE. cE. 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 FB)) /\ q e. ( 2nd ` (A FB)))) -> F. )
46	45	inegd 1262	. 2 |- ((A e. P. /\ B e. P.) -> -. ( q e. ( 1st ` (A FB)) /\ q e. ( 2nd ` (A FB))))
47	46	ralrimivw 2387	1 |- ((A e. P. /\ B e. P.) -> A. q e. Q. -. ( q e. ( 1st ` (A FB)) /\ q e. ( 2nd ` (A FB))))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 884   = wceq 1242   F. wfal 1247  E.wex 1378   e. wcel 1390  A.wral 2300  E.wrex 2301   {crab 2304   <.cop 3370   class class class wbr 3755   ` cfv 4845  (class class class)co 5455   |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264   <Q cltq 6269   P.cnp 6275

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-mi 6290  df-lti 6291  df-enq 6331  df-nqqs 6332  df-ltnqqs 6337  df-inp 6449

This theorem is referenced by:  addclpr  6520  mulclpr  6553