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Mirrors > Home > MPE Home > Th. List > ltadd2dd | Structured version Visualization version GIF version |
Description: Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
letrd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
ltletrd.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
Ref | Expression |
---|---|
ltadd2dd | ⊢ (𝜑 → (𝐶 + 𝐴) < (𝐶 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltletrd.4 | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
2 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | letrd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
5 | 2, 3, 4 | ltadd2d 10784 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))) |
6 | 1, 5 | mpbid 233 | 1 ⊢ (𝜑 → (𝐶 + 𝐴) < (𝐶 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℝcr 10524 + caddc 10528 < clt 10663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-addrcl 10586 ax-pre-lttri 10599 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 |
This theorem is referenced by: zltaddlt1le 12878 2tnp1ge0ge0 13187 ccatrn 13931 eirrlem 15545 prmreclem5 16244 iccntr 23356 icccmplem2 23358 ivthlem2 23980 uniioombllem3 24113 opnmbllem 24129 dvcnvre 24543 cosordlem 25042 efif1olem2 25054 atanlogaddlem 25418 pntibndlem2 26094 pntlemr 26105 dya2icoseg 31434 opnmbllem0 34809 fltnltalem 39152 binomcxplemdvbinom 40562 zltlesub 41427 supxrge 41482 ltadd12dd 41487 xrralrecnnle 41529 0ellimcdiv 41806 climleltrp 41833 ioodvbdlimc1lem2 42093 stoweidlem11 42173 stoweidlem14 42176 stoweidlem26 42188 stoweidlem44 42206 dirkertrigeqlem3 42262 dirkercncflem1 42265 dirkercncflem2 42266 fourierdlem4 42273 fourierdlem10 42279 fourierdlem28 42297 fourierdlem40 42309 fourierdlem50 42318 fourierdlem57 42325 fourierdlem59 42327 fourierdlem60 42328 fourierdlem61 42329 fourierdlem68 42336 fourierdlem74 42342 fourierdlem75 42343 fourierdlem76 42344 fourierdlem78 42346 fourierdlem79 42347 fourierdlem84 42352 fourierdlem93 42361 fourierdlem111 42379 fouriersw 42393 smfaddlem1 42916 smflimlem3 42926 |
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