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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 12082 | . 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 2099 class class class wbr 5143 (class class class)co 7415 ℝcr 11132 1c1 11134 ≤ cle 11274 − cmin 11469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-po 5585 df-so 5586 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 |
This theorem is referenced by: fzossrbm1 13688 seqcoll 14452 efgsp1 19686 efgredlemd 19693 efgredlem 19696 2lgslem1c 27320 rplogsumlem1 27411 logdivbnd 27483 wwlksm1edg 29686 clwlkclwwlklem2 29804 clwlkclwwlk 29806 clwwisshclwwslem 29818 clwwlkf 29851 wwlksubclwwlk 29862 fzspl 32553 pfxlsw2ccat 32668 wrdt2ind 32669 psgnfzto1stlem 32816 submateqlem1 33403 elfzm12 35274 knoppndvlem14 35995 poimirlem6 37094 poimirlem7 37095 poimirlem13 37101 aks4d1p1p2 41536 sticksstones10 41622 sticksstones12a 41624 sticksstones12 41625 bcle2d 41646 aks6d1c7lem1 41647 oddfl 44650 fmul01lt1lem2 44964 stoweidlem11 45390 wallispilem3 45446 etransclem23 45636 upwordnul 46257 iccpartipre 46752 flnn0div2ge 47597 |
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