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Mirrors > Home > MPE Home > Th. List > ltaddrp | Structured version Visualization version GIF version |
Description: Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007.) |
Ref | Expression |
---|---|
ltaddrp | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrp 12916 | . 2 ⊢ (𝐵 ∈ ℝ+ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) | |
2 | ltaddpos 11644 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐵 ↔ 𝐴 < (𝐴 + 𝐵))) | |
3 | 2 | biimpd 228 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 𝐵 → 𝐴 < (𝐴 + 𝐵))) |
4 | 3 | expcom 414 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐵 ∈ ℝ → (0 < 𝐵 → 𝐴 < (𝐴 + 𝐵)))) |
5 | 4 | imp32 419 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 𝐴 < (𝐴 + 𝐵)) |
6 | 1, 5 | sylan2b 594 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → 𝐴 < (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 class class class wbr 5105 (class class class)co 7356 ℝcr 11049 0cc0 11050 + caddc 11053 < clt 11188 ℝ+crp 12914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-ltxr 11193 df-rp 12915 |
This theorem is referenced by: ltaddrpd 12989 lswccatn0lsw 14478 efgt1 15997 efgsfo 19519 efgredlemd 19524 efgredlem 19527 iccntr 24182 reconnlem2 24188 opnreen 24192 minveclem3b 24790 logimul 25967 emcllem2 26344 emcllem4 26346 emcllem6 26348 perfectlem2 26576 bclbnd 26626 pntibndlem1 26935 pntlemd 26940 pntlemc 26941 pntlemr 26948 pntlemp 26956 smcnlem 29637 dp2ltc 31738 dpgti 31757 ballotlem2 33079 poimir 36102 stoweidlem59 44272 perfectALTVlem2 45886 |
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