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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 11235 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴 + 𝐶) < (𝐵 + 𝐶))) |
6 | 1, 5 | mpbid 234 | 1 ⊢ (𝜑 → (𝐴 + 𝐶) < (𝐵 + 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 + caddc 10542 < clt 10677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 |
This theorem is referenced by: nnne0 11674 fzoaddel 13093 elincfzoext 13098 fladdz 13198 fzsdom2 13792 sadcaddlem 15808 iserodd 16174 4sqlem12 16294 efif1olem1 25128 atanlogsublem 25495 subfacval3 32438 poimirlem15 34909 itg2addnclem3 34947 fltnlta 39282 3cubeslem1 39288 rmspecfund 39513 jm2.24nn 39563 ltadd12dd 41618 infleinflem2 41646 iooshift 41805 iblspltprt 42265 itgspltprt 42271 stirlinglem5 42370 dirkercncflem1 42395 fourierdlem19 42418 fourierdlem35 42434 fourierdlem41 42440 fourierdlem47 42445 fourierdlem48 42446 fourierdlem49 42447 fourierdlem51 42449 fourierdlem64 42462 fourierdlem79 42477 fourierdlem81 42479 fourierdlem92 42490 fourierdlem112 42510 sqwvfoura 42520 sqwvfourb 42521 fouriersw 42523 smflimlem4 43057 2pwp1prm 43758 |
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