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| Mirrors > Home > MPE Home > Th. List > ltadd1dd | Structured version Visualization version GIF version | ||
| Description: Addition to both sides of 'less than'. Theorem I.18 of [Apostol] p. 20. (Contributed by Mario Carneiro, 30-May-2016.) |
| Ref | Expression |
|---|---|
| leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
| ltadd1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
| ltadd1dd.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
| Ref | Expression |
|---|---|
| ltadd1dd | ⊢ (𝜑 → (𝐴 + 𝐶) < (𝐵 + 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ltadd1dd.4 | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
| 2 | leidd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 3 | ltnegd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 4 | ltadd1d.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
| 5 | 2, 3, 4 | ltadd1d 11856 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶))) |
| 6 | 1, 5 | mpbid 232 | 1 ⊢ (𝜑 → (𝐴 + 𝐶) < (𝐵 + 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 + caddc 11158 < clt 11295 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 |
| This theorem is referenced by: nnne0 12300 fzoaddel 13756 elincfzoext 13762 fladdz 13865 fzsdom2 14467 sadcaddlem 16494 iserodd 16873 4sqlem12 16994 efif1olem1 26584 atanlogsublem 26958 subfacval3 35194 poimirlem15 37642 itg2addnclem3 37680 aks4d1p1p6 42074 aks4d1p1p5 42076 fltnlta 42673 3cubeslem1 42695 rmspecfund 42920 jm2.24nn 42971 ltadd12dd 45354 infleinflem2 45382 iooshift 45535 iblspltprt 45988 itgspltprt 45994 stirlinglem5 46093 dirkercncflem1 46118 fourierdlem19 46141 fourierdlem35 46157 fourierdlem41 46163 fourierdlem47 46168 fourierdlem48 46169 fourierdlem49 46170 fourierdlem51 46172 fourierdlem64 46185 fourierdlem79 46200 fourierdlem81 46202 fourierdlem92 46213 fourierdlem112 46233 sqwvfoura 46243 sqwvfourb 46244 fouriersw 46246 smflimlem4 46789 ormkglobd 46890 2pwp1prm 47576 |
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