Theorem gt0add 8901

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 1030	. . 3 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> 0 < ( A + B ) )
2		0red 8327	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> 0 e. RR )
3		simp1 1028	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> A e. RR )
4		simp2 1029	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> B e. RR )
5	3, 4	readdcld 8355	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( A + B ) e. RR )
6		axltwlin 8393	. . . 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 1278	. . 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 8857	. . 3 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( 0 < B <-> A < ( A + B ) ) )
10	9	orbi2d 802	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   \/ wo 720   /\ w3a 1009   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   RRcr 8178   0cc0 8179   + caddc 8182   < clt 8360

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-pre-ltwlin 8292  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-iota 5337  df-fv 5385  df-ov 6088  df-pnf 8362  df-mnf 8363  df-ltxr 8365

This theorem is used by: (None)