Theorem gt0add 8147

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 948	. . 3 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> 0 < ( A + B ) )
2		0red 7586	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> 0 e. RR )
3		simp1 946	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> A e. RR )
4		simp2 947	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> B e. RR )
5	3, 4	readdcld 7614	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( A + B ) e. RR )
6		axltwlin 7651	. . . 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 1181	. . 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 8103	. . 3 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( 0 < B <-> A < ( A + B ) ) )
10	9	orbi2d 742	. 2 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( ( 0 < A \/ 0 < B ) <-> ( 0 < A \/ A < ( A + B ) ) ) )
11	8, 10	mpbird 166	1 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( 0 < A \/ 0 < B ) )

Syntax hints:   -> wi 4   \/ wo 667   /\ w3a 927   e. wcel 1445   class class class wbr 3867  (class class class)co 5690   RRcr 7446   0cc0 7447   + caddc 7450   < clt 7619

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-addcom 7542  ax-addass 7544  ax-i2m1 7547  ax-0id 7550  ax-rnegex 7551  ax-pre-ltwlin 7555  ax-pre-ltadd 7558

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-xp 4473  df-iota 5014  df-fv 5057  df-ov 5693  df-pnf 7621  df-mnf 7622  df-ltxr 7624

This theorem is referenced by: (None)