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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 10290 | . . . . 5 ⊢ <R ⊆ (R × R) | |
2 | 1 | brel 5468 | . . . 4 ⊢ (0R <R 𝐴 → (0R ∈ R ∧ 𝐴 ∈ R)) |
3 | ltasr 10322 | . . . . 5 ⊢ (𝐴 ∈ R → (0R <R 𝐵 ↔ (𝐴 +R 0R) <R (𝐴 +R 𝐵))) | |
4 | 0idsr 10319 | . . . . . 6 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) | |
5 | 4 | breq1d 4940 | . . . . 5 ⊢ (𝐴 ∈ R → ((𝐴 +R 0R) <R (𝐴 +R 𝐵) ↔ 𝐴 <R (𝐴 +R 𝐵))) |
6 | 3, 5 | bitrd 271 | . . . 4 ⊢ (𝐴 ∈ R → (0R <R 𝐵 ↔ 𝐴 <R (𝐴 +R 𝐵))) |
7 | 2, 6 | simpl2im 496 | . . 3 ⊢ (0R <R 𝐴 → (0R <R 𝐵 ↔ 𝐴 <R (𝐴 +R 𝐵))) |
8 | 7 | biimpa 469 | . 2 ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 𝐴 <R (𝐴 +R 𝐵)) |
9 | ltsosr 10316 | . . 3 ⊢ <R Or R | |
10 | 9, 1 | sotri 5829 | . 2 ⊢ ((0R <R 𝐴 ∧ 𝐴 <R (𝐴 +R 𝐵)) → 0R <R (𝐴 +R 𝐵)) |
11 | 8, 10 | syldan 582 | 1 ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 +R 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∈ wcel 2050 class class class wbr 4930 (class class class)co 6978 Rcnr 10087 0Rc0r 10088 +R cplr 10091 <R cltr 10093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-inf2 8900 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-omul 7912 df-er 8091 df-ec 8093 df-qs 8097 df-ni 10094 df-pli 10095 df-mi 10096 df-lti 10097 df-plpq 10130 df-mpq 10131 df-ltpq 10132 df-enq 10133 df-nq 10134 df-erq 10135 df-plq 10136 df-mq 10137 df-1nq 10138 df-rq 10139 df-ltnq 10140 df-np 10203 df-1p 10204 df-plp 10205 df-ltp 10207 df-enr 10277 df-nr 10278 df-plr 10279 df-ltr 10281 df-0r 10282 |
This theorem is referenced by: (None) |
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