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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 11554 | . 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 2113 class class class wbr 5027 (class class class)co 7164 ℝcr 10607 1c1 10609 ≤ cle 10747 − cmin 10941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 |
This theorem is referenced by: fzossrbm1 13150 seqcoll 13909 efgsp1 18974 efgredlemd 18981 efgredlem 18984 2lgslem1c 26121 rplogsumlem1 26212 logdivbnd 26284 wwlksm1edg 27811 clwlkclwwlklem2 27929 clwlkclwwlk 27931 clwwisshclwwslem 27943 clwwlkf 27976 wwlksubclwwlk 27987 fzspl 30678 pfxlsw2ccat 30791 wrdt2ind 30792 psgnfzto1stlem 30936 submateqlem1 31321 elfzm12 33196 knoppndvlem14 34335 poimirlem6 35395 poimirlem7 35396 poimirlem13 35402 aks4d1p1p2 39686 oddfl 42337 fmul01lt1lem2 42652 stoweidlem11 43078 wallispilem3 43134 etransclem23 43324 iccpartipre 44391 flnn0div2ge 45397 |
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