Theorem lt2add 8548

Description: Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
lt2add	|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A < C /\ B < D ) -> ( A + B ) < ( C + D ) ) )

Proof of Theorem lt2add

Step	Hyp	Ref	Expression
1		simpll 527	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> A e. RR )
2		simprl 529	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. RR )
3		simplr 528	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> B e. RR )
4		ltadd1 8532	. . . 4 |- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
5	1, 2, 3, 4	syl3anc 1250	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
6		simprr 531	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. RR )
7	3, 6, 2	ltadd2d 8524	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( B < D <-> ( C + B ) < ( C + D ) ) )
8	5, 7	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 ) ) ) )
9	1, 3	readdcld 8132	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A + B ) e. RR )
10	2, 3	readdcld 8132	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + B ) e. RR )
11	2, 6	readdcld 8132	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + D ) e. RR )
12		lttr 8176	. . 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 ) ) )
13	9, 10, 11, 12	syl3anc 1250	. 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 ) ) )
14	8, 13	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 2177   class class class wbr 4054  (class class class)co 5962   RRcr 7954   + caddc 7958   < clt 8137

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-i2m1 8060  ax-0id 8063  ax-rnegex 8064  ax-pre-lttrn 8069  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-xp 4694  df-iota 5246  df-fv 5293  df-ov 5965  df-pnf 8139  df-mnf 8140  df-ltxr 8142

This theorem is referenced by:  addgt0  8551  lt2addi  8613  lt2halves  9303