Theorem addextpr 7681

Description: Strong extensionality of addition (ordering version). This is similar to addext 8629 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 7597	. . . 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 7597	. . . 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 529	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> C e. P. )
6		simplr 528	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> B e. P. )
7		addclpr 7597	. . . 4 |- ( ( C e. P. /\ B e. P. ) -> ( C +P. B ) e. P. )
8	5, 6, 7	syl2anc 411	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( C +P. B ) e. P. )
9		ltsopr 7656	. . . 4 |- <P Or P.
10		sowlin 4351	. . . 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 424	. . 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 1249	. 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 527	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> A e. P. )
14		ltaprg 7679	. . . . 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 1249	. . . 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 7638	. . . . . . 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 6075	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( A +P. B ) = ( B +P. A ) )
19	17, 5, 6	caovcomd 6075	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> ( C +P. B ) = ( B +P. C ) )
20	18, 19	breq12d 4042	. . . 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 531	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( C e. P. /\ D e. P. ) ) -> D e. P. )
23		ltaprg 7679	. . . 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 1249	. . 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 794	. 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 709   /\ w3a 980   = wceq 1364   e. wcel 2164   class class class wbr 4029   Or wor 4326  (class class class)co 5918   P.cnp 7351   +P. cpp 7353   <P cltp 7355

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-iltp 7530

This theorem is referenced by:  mulextsr1lem  7840