Theorem gt0add 8847

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 1026	. . 3 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> 0 < ( A + B ) )
2		0red 8275	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> 0 e. RR )
3		simp1 1024	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> A e. RR )
4		simp2 1025	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> B e. RR )
5	3, 4	readdcld 8303	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( A + B ) e. RR )
6		axltwlin 8341	. . . 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 1274	. . 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 8803	. . 3 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( 0 < B <-> A < ( A + B ) ) )
10	9	orbi2d 798	. 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 716   /\ w3a 1005   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   RRcr 8126   0cc0 8127   + caddc 8130   < clt 8308

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236  ax-pre-ltwlin 8240  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-xp 4755  df-iota 5312  df-fv 5360  df-ov 6053  df-pnf 8310  df-mnf 8311  df-ltxr 8313

This theorem is referenced by: (None)