Theorem addlelt 9712

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 9609	. . . 4 |- ( A e. RR+ -> 0 < A )
2	1	3ad2ant3 1015	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> 0 < A )
3		rpre 9604	. . . . 5 |- ( A e. RR+ -> A e. RR )
4	3	3ad2ant3 1015	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> A e. RR )
5		simp1 992	. . . 4 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> M e. RR )
6	4, 5	ltaddposd 8435	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( 0 < A <-> M < ( M + A ) ) )
7	2, 6	mpbid 146	. 2 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> M < ( M + A ) )
8		simpl 108	. . . . 5 |- ( ( M e. RR /\ A e. RR+ ) -> M e. RR )
9	3	adantl 275	. . . . 5 |- ( ( M e. RR /\ A e. RR+ ) -> A e. RR )
10	8, 9	readdcld 7936	. . . 4 |- ( ( M e. RR /\ A e. RR+ ) -> ( M + A ) e. RR )
11	10	3adant2 1011	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( M + A ) e. RR )
12		simp2 993	. . 3 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> N e. RR )
13		ltletr 7996	. . 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 1233	. 2 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M < ( M + A ) /\ ( M + A ) <_ N ) -> M < N ) )
15	7, 14	mpand 427	1 |- ( ( M e. RR /\ N e. RR /\ A e. RR+ ) -> ( ( M + A ) <_ N -> M < N ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   e. wcel 2141   class class class wbr 3987  (class class class)co 5850   RRcr 7760   0cc0 7761   + caddc 7764   < clt 7941   <_ cle 7942   RR+crp 9597

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 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-addass 7863  ax-i2m1 7866  ax-0id 7869  ax-rnegex 7870  ax-pre-ltwlin 7874  ax-pre-ltadd 7877

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 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-xp 4615  df-cnv 4617  df-iota 5158  df-fv 5204  df-ov 5853  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-rp 9598

This theorem is referenced by:  zltaddlt1le  9951