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Mirrors > Home > ILE Home > Th. List > ltm1 | 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 7355 | . . 3 ⊢ 0 < 1 | |
2 | 0re 7233 | . . . 4 ⊢ 0 ∈ ℝ | |
3 | 1re 7232 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | ltsub2 7682 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < 1 ↔ (𝐴 − 1) < (𝐴 − 0))) | |
5 | 2, 3, 4 | mp3an12 1259 | . . 3 ⊢ (𝐴 ∈ ℝ → (0 < 1 ↔ (𝐴 − 1) < (𝐴 − 0))) |
6 | 1, 5 | mpbii 146 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < (𝐴 − 0)) |
7 | recn 7220 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
8 | 7 | subid1d 7527 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 − 0) = 𝐴) |
9 | 6, 8 | breqtrd 3829 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∈ wcel 1434 class class class wbr 3805 (class class class)co 5563 ℝcr 7094 0cc0 7095 1c1 7096 < clt 7267 − cmin 7398 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-cnex 7181 ax-resscn 7182 ax-1cn 7183 ax-1re 7184 ax-icn 7185 ax-addcl 7186 ax-addrcl 7187 ax-mulcl 7188 ax-addcom 7190 ax-addass 7192 ax-distr 7194 ax-i2m1 7195 ax-0lt1 7196 ax-0id 7198 ax-rnegex 7199 ax-cnre 7201 ax-pre-ltadd 7206 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2825 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-br 3806 df-opab 3860 df-id 4076 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-iota 4917 df-fun 4954 df-fv 4960 df-riota 5519 df-ov 5566 df-oprab 5567 df-mpt2 5568 df-pnf 7269 df-mnf 7270 df-ltxr 7272 df-sub 7400 df-neg 7401 |
This theorem is referenced by: lem1 8044 ltm1d 8129 qbtwnxr 9396 bcpasc 9842 |
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