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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 11101 | . . . . 5 ⊢ <R ⊆ (R × R) | |
2 | 1 | brel 5747 | . . . 4 ⊢ (0R <R 𝐴 → (0R ∈ R ∧ 𝐴 ∈ R)) |
3 | ltasr 11133 | . . . . 5 ⊢ (𝐴 ∈ R → (0R <R 𝐵 ↔ (𝐴 +R 0R) <R (𝐴 +R 𝐵))) | |
4 | 0idsr 11130 | . . . . . 6 ⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) | |
5 | 4 | breq1d 5162 | . . . . 5 ⊢ (𝐴 ∈ R → ((𝐴 +R 0R) <R (𝐴 +R 𝐵) ↔ 𝐴 <R (𝐴 +R 𝐵))) |
6 | 3, 5 | bitrd 278 | . . . 4 ⊢ (𝐴 ∈ R → (0R <R 𝐵 ↔ 𝐴 <R (𝐴 +R 𝐵))) |
7 | 2, 6 | simpl2im 502 | . . 3 ⊢ (0R <R 𝐴 → (0R <R 𝐵 ↔ 𝐴 <R (𝐴 +R 𝐵))) |
8 | 7 | biimpa 475 | . 2 ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 𝐴 <R (𝐴 +R 𝐵)) |
9 | ltsosr 11127 | . . 3 ⊢ <R Or R | |
10 | 9, 1 | sotri 6138 | . 2 ⊢ ((0R <R 𝐴 ∧ 𝐴 <R (𝐴 +R 𝐵)) → 0R <R (𝐴 +R 𝐵)) |
11 | 8, 10 | syldan 589 | 1 ⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 +R 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 class class class wbr 5152 (class class class)co 7426 Rcnr 10898 0Rc0r 10899 +R cplr 10902 <R cltr 10904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-oadd 8499 df-omul 8500 df-er 8733 df-ec 8735 df-qs 8739 df-ni 10905 df-pli 10906 df-mi 10907 df-lti 10908 df-plpq 10941 df-mpq 10942 df-ltpq 10943 df-enq 10944 df-nq 10945 df-erq 10946 df-plq 10947 df-mq 10948 df-1nq 10949 df-rq 10950 df-ltnq 10951 df-np 11014 df-1p 11015 df-plp 11016 df-ltp 11018 df-enr 11088 df-nr 11089 df-plr 11090 df-ltr 11092 df-0r 11093 |
This theorem is referenced by: (None) |
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