Theorem addextpr 7562

Description: Strong extensionality of addition (ordering version). This is similar to addext 8508 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 7478	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
2	1	adantr 274	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( A +P. B ) e. P. )
3		addclpr 7478	. . . 4 |- ( ( C e. P. /\ D e. P. ) -> ( C +P. D ) e. P. )
4	3	adantl 275	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( C +P. D ) e. P. )
5		simprl 521	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> C e. P. )
6		simplr 520	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> B e. P. )
7		addclpr 7478	. . . 4 |- ( ( C e. P. /\ B e. P. ) -> ( C +P. B ) e. P. )
8	5, 6, 7	syl2anc 409	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( C +P. B ) e. P. )
9		ltsopr 7537	. . . 4 |- <P Or P.
10		sowlin 4298	. . . 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 421	. . 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 1228	. 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 519	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> A e. P. )
14		ltaprg 7560	. . . . 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 1228	. . . 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 7519	. . . . . . 7 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
17	16	adantl 275	. . . . . 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 5998	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( A +P. B ) = ( B +P. A ) )
19	17, 5, 6	caovcomd 5998	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( C +P. B ) = ( B +P. C ) )
20	18, 19	breq12d 3995	. . . 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 190	. . 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 522	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> D e. P. )
23		ltaprg 7560	. . . 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 1228	. . 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 783	. 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 168	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 103   <-> wb 104   \/ wo 698   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982   Or wor 4273  (class class class)co 5842   P.cnp 7232   +P. cpp 7234   <P cltp 7236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409  df-iltp 7411

This theorem is referenced by:  mulextsr1lem  7721