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Mirrors > Home > MPE Home > Th. List > lem1d | Structured version Visualization version GIF version |
Description: A number minus 1 is less than or equal to itself. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
ltp1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
lem1d | ⊢ (𝜑 → (𝐴 − 1) ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltp1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | lem1 11818 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ≤ 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 − 1) ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℝcr 10870 1c1 10872 ≤ cle 11010 − cmin 11205 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 |
This theorem is referenced by: fzossrbm1 13416 seqcoll 14178 efgsp1 19343 efgredlemd 19350 efgredlem 19353 2lgslem1c 26541 rplogsumlem1 26632 logdivbnd 26704 wwlksm1edg 28246 clwlkclwwlklem2 28364 clwlkclwwlk 28366 clwwisshclwwslem 28378 clwwlkf 28411 wwlksubclwwlk 28422 fzspl 31111 pfxlsw2ccat 31224 wrdt2ind 31225 psgnfzto1stlem 31367 submateqlem1 31757 elfzm12 33633 knoppndvlem14 34705 poimirlem6 35783 poimirlem7 35784 poimirlem13 35790 aks4d1p1p2 40078 sticksstones10 40111 sticksstones12a 40113 sticksstones12 40114 oddfl 42816 fmul01lt1lem2 43126 stoweidlem11 43552 wallispilem3 43608 etransclem23 43798 iccpartipre 44873 flnn0div2ge 45879 upwordnul 46515 |
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