Theorem addlelt 9770

Description: If the sum of a real number and a positive real number is less than or equal to a third real number, the first real number is less than the third real number. (Contributed by AV, 1-Jul-2021.)

Assertion

Ref	Expression
addlelt	|- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M + A ) <_ N -> M < N ) )

Proof of Theorem addlelt

Step	Hyp	Ref	Expression
1		rpgt0 9667	. . . 4 |- ( A e. RR+ -> 0 < A )
2	1	3ad2ant3 1020	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> 0 < A )
3		rpre 9662	. . . . 5 |- ( A e. RR+ -> A e. RR )
4	3	3ad2ant3 1020	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> A e. RR )
5		simp1 997	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> M e. RR )
6	4, 5	ltaddposd 8488	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( 0 < A <-> M < ( M + A ) ) )
7	2, 6	mpbid 147	. 2 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> M < ( M + A ) )
8		simpl 109	. . . . 5 |- ( ( M e. RR /\ A e. RR+ ) -> M e. RR )
9	3	adantl 277	. . . . 5 |- ( ( M e. RR /\ A e. RR+ ) -> A e. RR )
10	8, 9	readdcld 7989	. . . 4 |- ( ( M e. RR /\ A e. RR+ ) -> ( M + A ) e. RR )
11	10	3adant2 1016	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( M + A ) e. RR )
12		simp2 998	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> N e. RR )
13		ltletr 8049	. . 3 |- ( ( M e. RR /\ ( M + A ) e. RR /\ N e. RR ) -> ( ( M < ( M + A ) /\ ( M + A ) <_ N ) -> M < N ) )
14	5, 11, 12, 13	syl3anc 1238	. 2 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M < ( M + A ) /\ ( M + A ) <_ N ) -> M < N ) )
15	7, 14	mpand 429	1 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M + A ) <_ N -> M < N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   e. wcel 2148   class class class wbr 4005  (class class class)co 5877   RRcr 7812   0cc0 7813   + caddc 7816   < clt 7994   <_ cle 7995   RR+crp 9655

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0id 7921  ax-rnegex 7922  ax-pre-ltwlin 7926  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-iota 5180  df-fv 5226  df-ov 5880  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-rp 9656

This theorem is referenced by:  zltaddlt1le  10009