Theorem addlelt 9860

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 9757	. . . 4 |- ( A e. RR+ -> 0 < A )
2	1	3ad2ant3 1022	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> 0 < A )
3		rpre 9752	. . . . 5 |- ( A e. RR+ -> A e. RR )
4	3	3ad2ant3 1022	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> A e. RR )
5		simp1 999	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> M e. RR )
6	4, 5	ltaddposd 8573	. . 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 8073	. . . 4 |- ( ( M e. RR /\ A e. RR+ ) -> ( M + A ) e. RR )
11	10	3adant2 1018	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( M + A ) e. RR )
12		simp2 1000	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> N e. RR )
13		ltletr 8133	. . 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 1249	. 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 980   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   RRcr 7895   0cc0 7896   + caddc 7899   < clt 8078   <_ cle 8079   RR+crp 9745

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0id 8004  ax-rnegex 8005  ax-pre-ltwlin 8009  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-iota 5220  df-fv 5267  df-ov 5928  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-rp 9746

This theorem is referenced by:  zltaddlt1le  10099