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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dp2ltsuc | Structured version Visualization version GIF version |
Description: Comparing a decimal fraction with the next integer. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
Ref | Expression |
---|---|
dp2lt.a | ⊢ 𝐴 ∈ ℕ0 |
dp2lt.b | ⊢ 𝐵 ∈ ℝ+ |
dp2ltsuc.1 | ⊢ 𝐵 < ;10 |
dp2ltsuc.2 | ⊢ (𝐴 + 1) = 𝐶 |
Ref | Expression |
---|---|
dp2ltsuc | ⊢ _𝐴𝐵 < 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dp2ltsuc.1 | . . . . 5 ⊢ 𝐵 < ;10 | |
2 | dp2lt.b | . . . . . . 7 ⊢ 𝐵 ∈ ℝ+ | |
3 | rpre 12120 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
5 | 10re 11840 | . . . . . 6 ⊢ ;10 ∈ ℝ | |
6 | 10pos 11838 | . . . . . 6 ⊢ 0 < ;10 | |
7 | 4, 5, 5, 6 | ltdiv1ii 11283 | . . . . 5 ⊢ (𝐵 < ;10 ↔ (𝐵 / ;10) < (;10 / ;10)) |
8 | 1, 7 | mpbi 222 | . . . 4 ⊢ (𝐵 / ;10) < (;10 / ;10) |
9 | 5 | recni 10371 | . . . . 5 ⊢ ;10 ∈ ℂ |
10 | 10nn 11837 | . . . . . 6 ⊢ ;10 ∈ ℕ | |
11 | 10 | nnne0i 11391 | . . . . 5 ⊢ ;10 ≠ 0 |
12 | 9, 11 | dividi 11084 | . . . 4 ⊢ (;10 / ;10) = 1 |
13 | 8, 12 | breqtri 4898 | . . 3 ⊢ (𝐵 / ;10) < 1 |
14 | 4, 5, 11 | redivcli 11118 | . . . 4 ⊢ (𝐵 / ;10) ∈ ℝ |
15 | 1re 10356 | . . . 4 ⊢ 1 ∈ ℝ | |
16 | dp2lt.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
17 | 16 | nn0rei 11630 | . . . 4 ⊢ 𝐴 ∈ ℝ |
18 | 14, 15, 17 | ltadd2i 10487 | . . 3 ⊢ ((𝐵 / ;10) < 1 ↔ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1)) |
19 | 13, 18 | mpbi 222 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1) |
20 | df-dp2 30114 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
21 | dp2ltsuc.2 | . . 3 ⊢ (𝐴 + 1) = 𝐶 | |
22 | 21 | eqcomi 2834 | . 2 ⊢ 𝐶 = (𝐴 + 1) |
23 | 19, 20, 22 | 3brtr4i 4903 | 1 ⊢ _𝐴𝐵 < 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 class class class wbr 4873 (class class class)co 6905 ℝcr 10251 0cc0 10252 1c1 10253 + caddc 10255 < clt 10391 / cdiv 11009 ℕ0cn0 11618 ;cdc 11821 ℝ+crp 12112 _cdp2 30113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-dec 11822 df-rp 12113 df-dp2 30114 |
This theorem is referenced by: hgt750lem2 31268 |
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