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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 11853 | . 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 2105 class class class wbr 5147 (class class class)co 7430 ℝcr 11151 + caddc 11155 < clt 11292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-ltxr 11297 |
This theorem is referenced by: nnne0 12297 fzoaddel 13752 elincfzoext 13758 fladdz 13861 fzsdom2 14463 sadcaddlem 16490 iserodd 16868 4sqlem12 16989 efif1olem1 26598 atanlogsublem 26972 subfacval3 35173 poimirlem15 37621 itg2addnclem3 37659 aks4d1p1p6 42054 aks4d1p1p5 42056 fltnlta 42649 3cubeslem1 42671 rmspecfund 42896 jm2.24nn 42947 ltadd12dd 45292 infleinflem2 45320 iooshift 45474 iblspltprt 45928 itgspltprt 45934 stirlinglem5 46033 dirkercncflem1 46058 fourierdlem19 46081 fourierdlem35 46097 fourierdlem41 46103 fourierdlem47 46108 fourierdlem48 46109 fourierdlem49 46110 fourierdlem51 46112 fourierdlem64 46125 fourierdlem79 46140 fourierdlem81 46142 fourierdlem92 46153 fourierdlem112 46173 sqwvfoura 46183 sqwvfourb 46184 fouriersw 46186 smflimlem4 46729 2pwp1prm 47513 |
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