Theorem gt0add 8592

Description: A positive sum must have a positive addend. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.)

Assertion

Ref	Expression
gt0add	|- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( 0 < A \/ 0 < B ) )

Proof of Theorem gt0add

Step	Hyp	Ref	Expression
1		simp3 1001	. . 3 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> 0 < ( A + B ) )
2		0red 8020	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> 0 e. RR )
3		simp1 999	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> A e. RR )
4		simp2 1000	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> B e. RR )
5	3, 4	readdcld 8049	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( A + B ) e. RR )
6		axltwlin 8087	. . . 4 |- ( ( 0 e. RR /\ ( A + B ) e. RR /\ A e. RR ) -> ( 0 < ( A + B ) -> ( 0 < A \/ A < ( A + B ) ) ) )
7	2, 5, 3, 6	syl3anc 1249	. . 3 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( 0 < ( A + B ) -> ( 0 < A \/ A < ( A + B ) ) ) )
8	1, 7	mpd 13	. 2 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( 0 < A \/ A < ( A + B ) ) )
9	4, 3	ltaddposd 8548	. . 3 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( 0 < B <-> A < ( A + B ) ) )
10	9	orbi2d 791	. 2 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( ( 0 < A \/ 0 < B ) <-> ( 0 < A \/ A < ( A + B ) ) ) )
11	8, 10	mpbird 167	1 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( 0 < A \/ 0 < B ) )

Syntax hints:   -> wi 4   \/ wo 709   /\ w3a 980   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   RRcr 7871   0cc0 7872   + caddc 7875   < clt 8054

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-pre-ltwlin 7985  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-iota 5215  df-fv 5262  df-ov 5921  df-pnf 8056  df-mnf 8057  df-ltxr 8059

This theorem is referenced by: (None)