Theorem genpdisj 7671

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 7660	. . . . . . . 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 2528	. . . . . . . 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 7661	. . . . . . . 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 2528	. . . . . . . 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 1963	. . . . . 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 7623	. . . . . . . . . . . . . . . 16 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
15		prltlu 7635	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ a e. ( 1st ` A ) /\ c e. ( 2nd ` A ) ) -> a <Q c )
16	15	3expib 1209	. . . . . . . . . . . . . . . 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 7623	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
19		prltlu 7635	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ b e. ( 1st ` B ) /\ d e. ( 2nd ` B ) ) -> b <Q d )
20	19	3expib 1209	. . . . . . . . . . . . . . . 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 7658	. . . . . . . . . . . . . 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 2226	. . . . . . . . . . 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 7546	. . . . . . . . . . 11 |- <Q Or Q.
36		ltrelnq 7513	. . . . . . . . . . 11 |- <Q C_ ( Q. X. Q. )
37	35, 36	soirri 5096	. . . . . . . . . 10 |- -. ( a G b ) <Q ( a G b )
38		breq2 4063	. . . . . . . . . 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 676	. . . . . . . . 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 1393	. . . . . . 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 1922	. . . . 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 1922	. . . 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 1392	. 2 |- ( ( A e. P. /\ B e. P. ) -> -. ( q e. ( 1st ` ( A F B ) ) /\ q e. ( 2nd ` ( A F B ) ) ) )
47	46	ralrimivw 2582	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 981   = wceq 1373   F. wfal 1378   E.wex 1516   e. wcel 2178   A.wral 2486   E.wrex 2487   {crab 2490   <.cop 3646   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   1stc1st 6247   2ndc2nd 6248   Q.cnq 7428   <Q cltq 7433   P.cnp 7439

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-mi 7454  df-lti 7455  df-enq 7495  df-nqqs 7496  df-ltnqqs 7501  df-inp 7614

This theorem is referenced by:  addclpr  7685  mulclpr  7720