![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > addgt0sr | Structured version Visualization version GIF version |
Description: The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
addgt0sr | ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 +R 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelsr 10091 | . . . . . 6 ⊢ <R ⊆ (R × R) | |
2 | 1 | brel 5306 | . . . . 5 ⊢ (0R <R 𝐴 → (0R ∈ R ∧ 𝐴 ∈ R)) |
3 | 2 | simprd 483 | . . . 4 ⊢ (0R <R 𝐴 → 𝐴 ∈ R) |
4 | ltasr 10123 | . . . . 5 ⊢ (𝐴 ∈ R → (0R <R 𝐵 ↔ (𝐴 +R 0R) <R (𝐴 +R 𝐵))) | |
5 | 0idsr 10120 | . . . . . 6 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) | |
6 | 5 | breq1d 4796 | . . . . 5 ⊢ (𝐴 ∈ R → ((𝐴 +R 0R) <R (𝐴 +R 𝐵) ↔ 𝐴 <R (𝐴 +R 𝐵))) |
7 | 4, 6 | bitrd 268 | . . . 4 ⊢ (𝐴 ∈ R → (0R <R 𝐵 ↔ 𝐴 <R (𝐴 +R 𝐵))) |
8 | 3, 7 | syl 17 | . . 3 ⊢ (0R <R 𝐴 → (0R <R 𝐵 ↔ 𝐴 <R (𝐴 +R 𝐵))) |
9 | 8 | biimpa 462 | . 2 ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 𝐴 <R (𝐴 +R 𝐵)) |
10 | ltsosr 10117 | . . 3 ⊢ <R Or R | |
11 | 10, 1 | sotri 5662 | . 2 ⊢ ((0R <R 𝐴 ∧ 𝐴 <R (𝐴 +R 𝐵)) → 0R <R (𝐴 +R 𝐵)) |
12 | 9, 11 | syldan 579 | 1 ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 +R 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∈ wcel 2145 class class class wbr 4786 (class class class)co 6792 Rcnr 9889 0Rc0r 9890 +R cplr 9893 <R cltr 9895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-omul 7718 df-er 7896 df-ec 7898 df-qs 7902 df-ni 9896 df-pli 9897 df-mi 9898 df-lti 9899 df-plpq 9932 df-mpq 9933 df-ltpq 9934 df-enq 9935 df-nq 9936 df-erq 9937 df-plq 9938 df-mq 9939 df-1nq 9940 df-rq 9941 df-ltnq 9942 df-np 10005 df-1p 10006 df-plp 10007 df-ltp 10009 df-enr 10079 df-nr 10080 df-plr 10081 df-ltr 10083 df-0r 10084 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |