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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 10193 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐶 + 𝐴) < (𝐶 + 𝐵))) |
6 | 1, 5 | mpbid 222 | 1 ⊢ (𝜑 → (𝐶 + 𝐴) < (𝐶 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 + caddc 9939 < clt 10074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-addrcl 9997 ax-pre-lttri 10010 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 |
This theorem is referenced by: zltaddlt1le 12324 2tnp1ge0ge0 12630 ccatrn 13372 eirrlem 14932 prmreclem5 15624 iccntr 22624 icccmplem2 22626 ivthlem2 23221 uniioombllem3 23353 opnmbllem 23369 dvcnvre 23782 cosordlem 24277 efif1olem2 24289 atanlogaddlem 24640 pntibndlem2 25280 pntlemr 25291 dya2icoseg 30339 opnmbllem0 33445 binomcxplemdvbinom 38552 zltlesub 39497 supxrge 39554 ltadd12dd 39559 xrralrecnnle 39602 0ellimcdiv 39881 climleltrp 39908 ioodvbdlimc1lem2 40147 stoweidlem11 40228 stoweidlem14 40231 stoweidlem26 40243 stoweidlem44 40261 dirkertrigeqlem3 40317 dirkercncflem1 40320 dirkercncflem2 40321 fourierdlem4 40328 fourierdlem10 40334 fourierdlem28 40352 fourierdlem40 40364 fourierdlem50 40373 fourierdlem57 40380 fourierdlem59 40382 fourierdlem60 40383 fourierdlem61 40384 fourierdlem68 40391 fourierdlem74 40397 fourierdlem75 40398 fourierdlem76 40399 fourierdlem78 40401 fourierdlem79 40402 fourierdlem84 40407 fourierdlem93 40416 fourierdlem111 40434 fouriersw 40448 smfaddlem1 40971 smflimlem3 40981 |
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