Theorem ltleadd 8365

Description: Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)

Assertion

Ref	Expression
ltleadd	|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A < C /\ B <_ D ) -> ( A + B ) < ( C + D ) ) )

Proof of Theorem ltleadd

Step	Hyp	Ref	Expression
1		ltadd1 8348	. . . . . 6 |- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
2	1	3com23 1204	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
3	2	3expa 1198	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
4	3	adantrr 476	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
5		leadd2 8350	. . . . . 6 |- ( ( B e. RR /\ D e. RR /\ C e. RR ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
6	5	3com23 1204	. . . . 5 |- ( ( B e. RR /\ C e. RR /\ D e. RR ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
7	6	3expb 1199	. . . 4 |- ( ( B e. RR /\ ( C e. RR /\ D e. RR ) ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
8	7	adantll 473	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
9	4, 8	anbi12d 470	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A < C /\ B <_ D ) <-> ( ( A + B ) < ( C + B ) /\ ( C + B ) <_ ( C + D ) ) ) )
10		readdcl 7900	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
11	10	adantr 274	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A + B ) e. RR )
12		readdcl 7900	. . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR )
13	12	ancoms 266	. . . 4 |- ( ( B e. RR /\ C e. RR ) -> ( C + B ) e. RR )
14	13	ad2ant2lr 507	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + B ) e. RR )
15		readdcl 7900	. . . 4 |- ( ( C e. RR /\ D e. RR ) -> ( C + D ) e. RR )
16	15	adantl 275	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + D ) e. RR )
17		ltletr 8009	. . 3 |- ( ( ( A + B ) e. RR /\ ( C + B ) e. RR /\ ( C + D ) e. RR ) -> ( ( ( A + B ) < ( C + B ) /\ ( C + B ) <_ ( C + D ) ) -> ( A + B ) < ( C + D ) ) )
18	11, 14, 16, 17	syl3anc 1233	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( ( A + B ) < ( C + B ) /\ ( C + B ) <_ ( C + D ) ) -> ( A + B ) < ( C + D ) ) )
19	9, 18	sylbid 149	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A < C /\ B <_ D ) -> ( A + B ) < ( C + D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   + caddc 7777   < clt 7954   <_ cle 7955

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-pre-ltwlin 7887  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-iota 5160  df-fv 5206  df-ov 5856  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960

This theorem is referenced by:  leltadd  8366  addgtge0  8369  ltleaddd  8484