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Mirrors > Home > MPE Home > Th. List > lesub1dd | Structured version Visualization version GIF version |
Description: Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016.) |
Ref | Expression |
---|---|
leidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltnegd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltadd1d.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
leadd1dd.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Ref | Expression |
---|---|
lesub1dd | ⊢ (𝜑 → (𝐴 − 𝐶) ≤ (𝐵 − 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leadd1dd.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 | lesub1d 11868 | . 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 2106 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 ≤ cle 11294 − cmin 11490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 |
This theorem is referenced by: eluzmn 12883 elfzmlbm 13675 modmulnn 13926 icodiamlt 15471 rlimrege0 15612 climsqz2 15675 rlimsqz2 15684 isercolllem1 15698 caucvgrlem 15706 climcndslem1 15882 bitsinv1lem 16475 hashdvds 16809 4sqlem6 16977 dvfsumlem2 26082 dvfsumlem2OLD 26083 dvfsumlem4 26085 dvfsum2 26090 isosctrlem1 26876 lgamgulmlem2 27088 basellem9 27147 ppiub 27263 chtub 27271 logfaclbnd 27281 bposlem1 27343 bposlem6 27348 selberg2lem 27609 pntpbnd2 27646 pntlemo 27666 ttgcontlem1 28914 axpaschlem 28970 axcontlem8 29001 cycpmco2lem7 33135 dnibndlem10 36470 unblimceq0 36490 unbdqndv2lem2 36493 poimirlem6 37613 poimirlem7 37614 itg2addnclem3 37660 iccbnd 37827 lcmineqlem23 42033 sticksstones12a 42139 sticksstones12 42140 bcled 42160 bcle2d 42161 metakunt30 42216 jm2.24nn 42948 fzmaxdif 42970 areaquad 43205 monoords 45248 iccshift 45471 climinf 45562 sumnnodd 45586 dvnmul 45899 itgiccshift 45936 itgperiod 45937 itgsbtaddcnst 45938 stoweidlem13 45969 stoweidlem26 45982 stoweidlem34 45990 fourierdlem19 46082 fourierdlem42 46105 fourierdlem74 46136 fourierdlem75 46137 fourierdlem79 46141 fourierdlem81 46143 fourierdlem82 46144 fourierdlem103 46165 fourierdlem104 46166 fouriersw 46187 hoidmvlelem1 46551 bgoldbtbndlem2 47731 |
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