Theorem addextpr 7978

Description: Strong extensionality of addition (ordering version). This is similar to addext 8928 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.)

Assertion

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

Proof of Theorem addextpr

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addclpr 7894	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
2	1	adantr 276	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( A +P. B ) e. P. )
3		addclpr 7894	. . . 4 |- ( ( C e. P. /\ D e. P. ) -> ( C +P. D ) e. P. )
4	3	adantl 277	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( C +P. D ) e. P. )
5		simprl 535	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> C e. P. )
6		simplr 533	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> B e. P. )
7		addclpr 7894	. . . 4 |- ( ( C e. P. /\ B e. P. ) -> ( C +P. B ) e. P. )
8	5, 6, 7	syl2anc 415	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( C +P. B ) e. P. )
9		ltsopr 7953	. . . 4 |- <P Or P.
10		sowlin 4460	. . . 4 |- ( ( <P Or P. /\ ( ( A +P. B ) e. P. /\ ( C +P. D ) e. P. /\ ( C +P. B ) e. P. ) ) -> ( ( A +P. B ) <P ( C +P. D ) -> ( ( A +P. B ) <P ( C +P. B ) \/ ( C +P. B ) <P ( C +P. D ) ) ) )
11	9, 10	mpan 428	. . 3 |- ( ( ( A +P. B ) e. P. /\ ( C +P. D ) e. P. /\ ( C +P. B ) e. P. ) -> ( ( A +P. B ) <P ( C +P. D ) -> ( ( A +P. B ) <P ( C +P. B ) \/ ( C +P. B ) <P ( C +P. D ) ) ) )
12	2, 4, 8, 11	syl3anc 1278	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( ( A +P. B ) <P ( C +P. D ) -> ( ( A +P. B ) <P ( C +P. B ) \/ ( C +P. B ) <P ( C +P. D ) ) ) )
13		simpll 531	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> A e. P. )
14		ltaprg 7976	. . . . 5 |- ( ( A e. P. /\ C e. P. /\ B e. P. ) -> ( A <P C <-> ( B +P. A ) <P ( B +P. C ) ) )
15	13, 5, 6, 14	syl3anc 1278	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( A <P C <-> ( B +P. A ) <P ( B +P. C ) ) )
16		addcomprg 7935	. . . . . . 7 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
17	16	adantl 277	. . . . . 6 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
18	17, 13, 6	caovcomd 6236	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( A +P. B ) = ( B +P. A ) )
19	17, 5, 6	caovcomd 6236	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( C +P. B ) = ( B +P. C ) )
20	18, 19	breq12d 4138	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( ( A +P. B ) <P ( C +P. B ) <-> ( B +P. A ) <P ( B +P. C ) ) )
21	15, 20	bitr4d 191	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( A <P C <-> ( A +P. B ) <P ( C +P. B ) ) )
22		simprr 537	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> D e. P. )
23		ltaprg 7976	. . . 4 |- ( ( B e. P. /\ D e. P. /\ C e. P. ) -> ( B <P D <-> ( C +P. B ) <P ( C +P. D ) ) )
24	6, 22, 5, 23	syl3anc 1278	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( B <P D <-> ( C +P. B ) <P ( C +P. D ) ) )
25	21, 24	orbi12d 805	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( ( A <P C \/ B <P D ) <-> ( ( A +P. B ) <P ( C +P. B ) \/ ( C +P. B ) <P ( C +P. D ) ) ) )
26	12, 25	sylibrd 169	1 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( ( A +P. B ) <P ( C +P. D ) -> ( A <P C \/ B <P D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125   Or wor 4435  (class class class)co 6075   P.cnp 7648   +P. cpp 7650   <P cltp 7652

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-iplp 7825  df-iltp 7827

This theorem is referenced by:  mulextsr1lem  8137