Theorem genpdisj 7590

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 7579	. . . . . . . 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 2517	. . . . . . . 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	bitrdi 196	. . . . . . 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 7580	. . . . . . . 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 2517	. . . . . . . 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	bitrdi 196	. . . . . . 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 473	. . . . . 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 1953	. . . . . 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	bitr4di 198	. . . . 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 296	. . . 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 586	. . . . . . . . . . . . 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 7542	. . . . . . . . . . . . . . . 16 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
15		prltlu 7554	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ a e. ( 1st ` A ) /\ c e. ( 2nd ` A ) ) -> a <Q c )
16	15	3expib 1208	. . . . . . . . . . . . . . . 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 7542	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19		prltlu 7554	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ b e. ( 1st ` B ) /\ d e. ( 2nd ` B ) ) -> b <Q d )
20	19	3expib 1208	. . . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . 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 7577	. . . . . . . . . . . . . 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	biimtrrid 153	. . . . . . . . . . . 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 124	. . . . . . . . . . 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 477	. . . . . . . . . 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 485	. . . . . . . . 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 487	. . . . . . . 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 2215	. . . . . . . . . . 11 |- ( ( q = ( a G b ) /\ q = ( c G d ) ) -> ( a G b ) = ( c G d ) )
33	32	ad2ant2l 508	. . . . . . . . . 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 277	. . . . . . . . 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 7465	. . . . . . . . . . 11 |- <Q Or Q.
36		ltrelnq 7432	. . . . . . . . . . 11 |- <Q C_ ( Q. X. Q. )
37	35, 36	soirri 5064	. . . . . . . . . 10 |- -. ( a G b ) <Q ( a G b )
38		breq2 4037	. . . . . . . . . 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 675	. . . . . . . . 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 1384	. . . . . . 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 115	. . . . . 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 1912	. . . . 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 1912	. . . 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 1383	. 2 |- ( ( A e. P. /\ B e. P. ) -> -. ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) )
47	46	ralrimivw 2571	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 104   <-> wb 105   /\ w3a 980   = wceq 1364   F. wfal 1369   E.wex 1506   e. wcel 2167   A.wral 2475   E.wrex 2476   {crab 2479   <.cop 3625   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   e. cmpo 5924   1stc1st 6196   2ndc2nd 6197   Q.cnq 7347   <Q cltq 7352   P.cnp 7358

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-mi 7373  df-lti 7374  df-enq 7414  df-nqqs 7415  df-ltnqqs 7420  df-inp 7533

This theorem is referenced by:  addclpr  7604  mulclpr  7639