Theorem lt2addnq 7060

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 7056	. . . . . 6 |- ( ( A e. Q. /\ B e. Q. /\ C e. Q. ) -> ( A <Q B <-> ( C +Q A ) <Q ( C +Q B ) ) )
2	1	3expa 1146	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ C e. Q. ) -> ( A <Q B <-> ( C +Q A ) <Q ( C +Q B ) ) )
3	2	adantrr 464	. . . 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 7037	. . . . . . 7 |- ( ( C e. Q. /\ A e. Q. ) -> ( C +Q A ) = ( A +Q C ) )
5	4	ancoms 265	. . . . . 6 |- ( ( A e. Q. /\ C e. Q. ) -> ( C +Q A ) = ( A +Q C ) )
6	5	ad2ant2r 494	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( C +Q A ) = ( A +Q C ) )
7		addcomnqg 7037	. . . . . . 7 |- ( ( C e. Q. /\ B e. Q. ) -> ( C +Q B ) = ( B +Q C ) )
8	7	ancoms 265	. . . . . 6 |- ( ( B e. Q. /\ C e. Q. ) -> ( C +Q B ) = ( B +Q C ) )
9	8	ad2ant2lr 495	. . . . 5 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( C e. Q. /\ D e. Q. ) ) -> ( C +Q B ) = ( B +Q C ) )
10	6, 9	breq12d 3880	. . . 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 187	. . 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 7056	. . . . . 6 |- ( ( C e. Q. /\ D e. Q. /\ B e. Q. ) -> ( C <Q D <-> ( B +Q C ) <Q ( B +Q D ) ) )
13	12	3expa 1146	. . . . 5 |- ( ( ( C e. Q. /\ D e. Q. ) /\ B e. Q. ) -> ( C <Q D <-> ( B +Q C ) <Q ( B +Q D ) ) )
14	13	ancoms 265	. . . 4 |- ( ( B e. Q. /\ ( C e. Q. /\ D e. Q. ) ) -> ( C <Q D <-> ( B +Q C ) <Q ( B +Q D ) ) )
15	14	adantll 461	. . 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 458	. 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 7054	. . 3 |- <Q Or Q.
18		ltrelnq 7021	. . 3 |- <Q C_ ( Q. X. Q. )
19	17, 18	sotri 4860	. 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	syl6bi 162	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 103   <-> wb 104   = wceq 1296   e. wcel 1445   class class class wbr 3867  (class class class)co 5690   Q.cnq 6936   +Q cplq 6938   <Q cltq 6941

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-ltnqqs 7009

This theorem is referenced by:  addlocprlemeqgt  7188  addnqprlemrl  7213  addnqprlemru  7214  cauappcvgprlemladdfl  7311  caucvgprlemloc  7331  caucvgprprlemloccalc  7340