Theorem gt0add 8492

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 994	. . 3 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> 0 < ( A + B ) )
2		0red 7921	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> 0 e. RR )
3		simp1 992	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> A e. RR )
4		simp2 993	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> B e. RR )
5	3, 4	readdcld 7949	. . . 4 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( A + B ) e. RR )
6		axltwlin 7987	. . . 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 1233	. . 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 8448	. . 3 |- ( ( A e. RR /\ B e. RR /\ 0 < ( A + B ) ) -> ( 0 < B <-> A < ( A + B ) ) )
10	9	orbi2d 785	. 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 703   /\ w3a 973   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   + caddc 7777   < clt 7954

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 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-pre-ltwlin 7887  ax-pre-ltadd 7890

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 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-iota 5160  df-fv 5206  df-ov 5856  df-pnf 7956  df-mnf 7957  df-ltxr 7959

This theorem is referenced by: (None)