Theorem lt2addnq 7719

Description: Ordering property of addition for positive fractions. (Contributed by Jim Kingdon, 7-Dec-2019.)

Assertion

Ref	Expression
lt2addnq	|- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( ( A <Q B /\ C <Q D ) -> ( A +Q C ) <Q ( B +Q D ) ) )

Proof of Theorem lt2addnq

Step	Hyp	Ref	Expression
1		ltanqg 7715	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A <Q B <-> ( C +Q A ) <Q ( C +Q B ) ) )
2	1	3expa 1230	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ C e. Q. ) -> ( A <Q B <-> ( C +Q A ) <Q ( C +Q B ) ) )
3	2	adantrr 479	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( A <Q B <-> ( C +Q A ) <Q ( C +Q B ) ) )
4		addcomnqg 7696	. . . . . . 7 |- ( ( C e. Q. /\ A e. Q. ) -> ( C +Q A ) = ( A +Q C ) )
5	4	ancoms 268	. . . . . 6 |- ( ( A e. Q. /\ C e. Q. ) -> ( C +Q A ) = ( A +Q C ) )
6	5	ad2ant2r 509	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( C +Q A ) = ( A +Q C ) )
7		addcomnqg 7696	. . . . . . 7 |- ( ( C e. Q. /\ B e. Q. ) -> ( C +Q B ) = ( B +Q C ) )
8	7	ancoms 268	. . . . . 6 |- ( ( B e. Q. /\ C e. Q. ) -> ( C +Q B ) = ( B +Q C ) )
9	8	ad2ant2lr 510	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( C +Q B ) = ( B +Q C ) )
10	6, 9	breq12d 4122	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( ( C +Q A ) <Q ( C +Q B ) <-> ( A +Q C ) <Q ( B +Q C ) ) )
11	3, 10	bitrd 188	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( A <Q B <-> ( A +Q C ) <Q ( B +Q C ) ) )
12		ltanqg 7715	. . . . . 6 |- ( ( C e. Q. /\ D e. Q. /\ B e. Q. ) -> ( C <Q D <-> ( B +Q C ) <Q ( B +Q D ) ) )
13	12	3expa 1230	. . . . 5 |- ( ( ( C e. Q. /\ D e. Q. ) /\ B e. Q. ) -> ( C <Q D <-> ( B +Q C ) <Q ( B +Q D ) ) )
14	13	ancoms 268	. . . 4 |- ( ( B e. Q. /\ ( C e. Q. /\ D e. Q. ) ) -> ( C <Q D <-> ( B +Q C ) <Q ( B +Q D ) ) )
15	14	adantll 476	. . 3 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( C <Q D <-> ( B +Q C ) <Q ( B +Q D ) ) )
16	11, 15	anbi12d 473	. 2 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( ( A <Q B /\ C <Q D ) <-> ( ( A +Q C ) <Q ( B +Q C ) /\ ( B +Q C ) <Q ( B +Q D ) ) ) )
17		ltsonq 7713	. . 3 |- <Q Or Q.
18		ltrelnq 7680	. . 3 |- <Q C_ ( Q. X. Q. )
19	17, 18	sotri 5158	. 2 |- ( ( ( A +Q C ) <Q ( B +Q C ) /\ ( B +Q C ) <Q ( B +Q D ) ) -> ( A +Q C ) <Q ( B +Q D ) )
20	16, 19	biimtrdi 163	1 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( ( A <Q B /\ C <Q D ) -> ( A +Q C ) <Q ( B +Q D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   Q.cnq 7595   +Q cplq 7597   <Q cltq 7600

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-ltnqqs 7668

This theorem is referenced by:  addlocprlemeqgt  7847  addnqprlemrl  7872  addnqprlemru  7873  cauappcvgprlemladdfl  7970  caucvgprlemloc  7990  caucvgprprlemloccalc  7999