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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 10796 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))) |
6 | 1, 5 | mpbid 234 | 1 ⊢ (𝜑 → (𝐶 + 𝐴) < (𝐶 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℝcr 10536 + caddc 10540 < clt 10675 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-addrcl 10598 ax-pre-lttri 10611 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 |
This theorem is referenced by: zltaddlt1le 12891 2tnp1ge0ge0 13200 ccatrn 13943 eirrlem 15557 prmreclem5 16256 iccntr 23429 icccmplem2 23431 ivthlem2 24053 uniioombllem3 24186 opnmbllem 24202 dvcnvre 24616 cosordlem 25115 efif1olem2 25127 atanlogaddlem 25491 pntibndlem2 26167 pntlemr 26178 dya2icoseg 31535 opnmbllem0 34943 fltnltalem 39294 binomcxplemdvbinom 40705 zltlesub 41571 supxrge 41626 ltadd12dd 41631 xrralrecnnle 41673 0ellimcdiv 41950 climleltrp 41977 ioodvbdlimc1lem2 42237 stoweidlem11 42316 stoweidlem14 42319 stoweidlem26 42331 stoweidlem44 42349 dirkertrigeqlem3 42405 dirkercncflem1 42408 dirkercncflem2 42409 fourierdlem4 42416 fourierdlem10 42422 fourierdlem28 42440 fourierdlem40 42452 fourierdlem50 42461 fourierdlem57 42468 fourierdlem59 42470 fourierdlem60 42471 fourierdlem61 42472 fourierdlem68 42479 fourierdlem74 42485 fourierdlem75 42486 fourierdlem76 42487 fourierdlem78 42489 fourierdlem79 42490 fourierdlem84 42495 fourierdlem93 42504 fourierdlem111 42522 fouriersw 42536 smfaddlem1 43059 smflimlem3 43069 |
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