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Mirrors > Home > MPE Home > Th. List > ltm1 | Structured version Visualization version GIF version |
Description: A number minus 1 is less than itself. (Contributed by NM, 9-Apr-2006.) |
Ref | Expression |
---|---|
ltm1 | ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0lt1 10843 | . . 3 ⊢ 0 < 1 | |
2 | 0re 10331 | . . . 4 ⊢ 0 ∈ ℝ | |
3 | 1re 10329 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | ltsub2 10818 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 1 ↔ (𝐴 − 1) < (𝐴 − 0))) | |
5 | 2, 3, 4 | mp3an12 1576 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 < 1 ↔ (𝐴 − 1) < (𝐴 − 0))) |
6 | 1, 5 | mpbii 225 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < (𝐴 − 0)) |
7 | recn 10315 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
8 | 7 | subid1d 10674 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 − 0) = 𝐴) |
9 | 6, 8 | breqtrd 4870 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2157 class class class wbr 4844 (class class class)co 6879 ℝcr 10224 0cc0 10225 1c1 10226 < clt 10364 − cmin 10557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-po 5234 df-so 5235 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-er 7983 df-en 8197 df-dom 8198 df-sdom 8199 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 |
This theorem is referenced by: lem1 11157 ltm1d 11249 nnunb 11575 qbtwnxr 12279 xrinfmsslem 12386 xrub 12390 reltre 12418 bcpasc 13360 arisum2 14930 vdwap0 16012 icoopnst 23065 abelthlem6 24530 ballotlemfrceq 31106 poimirlem13 33910 poimirlem14 33911 poimirlem31 33928 poimirlem32 33929 stoweidlem34 40989 smfresal 41736 altgsumbcALT 42925 |
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