Theorem ltleadd 8625

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 8608	. . . . . 6 |- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
2	1	3com23 1235	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
3	2	3expa 1229	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
4	3	adantrr 479	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
5		leadd2 8610	. . . . . 6 |- ( ( B e. RR /\ D e. RR /\ C e. RR ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
6	5	3com23 1235	. . . . 5 |- ( ( B e. RR /\ C e. RR /\ D e. RR ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
7	6	3expb 1230	. . . 4 |- ( ( B e. RR /\ ( C e. RR /\ D e. RR ) ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
8	7	adantll 476	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
9	4, 8	anbi12d 473	. 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 8157	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
11	10	adantr 276	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A + B ) e. RR )
12		readdcl 8157	. . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR )
13	12	ancoms 268	. . . 4 |- ( ( B e. RR /\ C e. RR ) -> ( C + B ) e. RR )
14	13	ad2ant2lr 510	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + B ) e. RR )
15		readdcl 8157	. . . 4 |- ( ( C e. RR /\ D e. RR ) -> ( C + D ) e. RR )
16	15	adantl 277	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + D ) e. RR )
17		ltletr 8268	. . 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 1273	. 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 150	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 104   <-> wb 105   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   RRcr 8030   + caddc 8034   < clt 8213   <_ cle 8214

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-pre-ltwlin 8144  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-iota 5286  df-fv 5334  df-ov 6020  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219

This theorem is referenced by:  leltadd  8626  addgtge0  8629  ltleaddd  8744