Theorem genpdisj 7524

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 7513	. . . . . . . 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 2497	. . . . . . . 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 7514	. . . . . . . 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 2497	. . . . . . . 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 1934	. . . . . 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 7476	. . . . . . . . . . . . . . . 16 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
15		prltlu 7488	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ a e. ( 1st ` A ) /\ c e. ( 2nd ` A ) ) -> a <Q c )
16	15	3expib 1206	. . . . . . . . . . . . . . . 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 7476	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19		prltlu 7488	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ b e. ( 1st ` B ) /\ d e. ( 2nd ` B ) ) -> b <Q d )
20	19	3expib 1206	. . . . . . . . . . . . . . . 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 7511	. . . . . . . . . . . . . 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 2196	. . . . . . . . . . 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 7399	. . . . . . . . . . 11 |- <Q Or Q.
36		ltrelnq 7366	. . . . . . . . . . 11 |- <Q C_ ( Q. X. Q. )
37	35, 36	soirri 5025	. . . . . . . . . 10 |- -. ( a G b ) <Q ( a G b )
38		breq2 4009	. . . . . . . . . 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 674	. . . . . . . . 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 1373	. . . . . . 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 1897	. . . . 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 1897	. . . 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 1372	. 2 |- ( ( A e. P. /\ B e. P. ) -> -. ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) )
47	46	ralrimivw 2551	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 978   = wceq 1353   F. wfal 1358   E.wex 1492   e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   <.cop 3597   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   e. cmpo 5879   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281   <Q cltq 7286   P.cnp 7292

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-mi 7307  df-lti 7308  df-enq 7348  df-nqqs 7349  df-ltnqqs 7354  df-inp 7467

This theorem is referenced by:  addclpr  7538  mulclpr  7573