![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dplti | Structured version Visualization version GIF version |
Description: Comparing a decimal expansions with the next higher integer. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
Ref | Expression |
---|---|
dplti.a | ⊢ 𝐴 ∈ ℕ0 |
dplti.b | ⊢ 𝐵 ∈ ℝ+ |
dplti.c | ⊢ 𝐶 ∈ ℕ0 |
dplti.1 | ⊢ 𝐵 < ;10 |
dplti.2 | ⊢ (𝐴 + 1) = 𝐶 |
Ref | Expression |
---|---|
dplti | ⊢ (𝐴.𝐵) < 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dplti.a | . . . 4 ⊢ 𝐴 ∈ ℕ0 | |
2 | dplti.b | . . . . 5 ⊢ 𝐵 ∈ ℝ+ | |
3 | rpre 13008 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 𝐵 ∈ ℝ |
5 | 1, 4 | dpval2 32610 | . . 3 ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10)) |
6 | dplti.1 | . . . . 5 ⊢ 𝐵 < ;10 | |
7 | 10re 12720 | . . . . . . . 8 ⊢ ;10 ∈ ℝ | |
8 | 10pos 12718 | . . . . . . . 8 ⊢ 0 < ;10 | |
9 | 7, 8 | pm3.2i 470 | . . . . . . 7 ⊢ (;10 ∈ ℝ ∧ 0 < ;10) |
10 | elrp 13002 | . . . . . . 7 ⊢ (;10 ∈ ℝ+ ↔ (;10 ∈ ℝ ∧ 0 < ;10)) | |
11 | 9, 10 | mpbir 230 | . . . . . 6 ⊢ ;10 ∈ ℝ+ |
12 | divlt1lt 13069 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ ;10 ∈ ℝ+) → ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10)) | |
13 | 4, 11, 12 | mp2an 691 | . . . . 5 ⊢ ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10) |
14 | 6, 13 | mpbir 230 | . . . 4 ⊢ (𝐵 / ;10) < 1 |
15 | 0re 11240 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
16 | 15, 8 | gtneii 11350 | . . . . . 6 ⊢ ;10 ≠ 0 |
17 | 4, 7, 16 | redivcli 12005 | . . . . 5 ⊢ (𝐵 / ;10) ∈ ℝ |
18 | 1re 11238 | . . . . 5 ⊢ 1 ∈ ℝ | |
19 | nn0ssre 12500 | . . . . . 6 ⊢ ℕ0 ⊆ ℝ | |
20 | 19, 1 | sselii 3975 | . . . . 5 ⊢ 𝐴 ∈ ℝ |
21 | 17, 18, 20 | ltadd2i 11369 | . . . 4 ⊢ ((𝐵 / ;10) < 1 ↔ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1)) |
22 | 14, 21 | mpbi 229 | . . 3 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1) |
23 | 5, 22 | eqbrtri 5163 | . 2 ⊢ (𝐴.𝐵) < (𝐴 + 1) |
24 | dplti.2 | . 2 ⊢ (𝐴 + 1) = 𝐶 | |
25 | 23, 24 | breqtri 5167 | 1 ⊢ (𝐴.𝐵) < 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 class class class wbr 5142 (class class class)co 7414 ℝcr 11131 0cc0 11132 1c1 11133 + caddc 11135 < clt 11272 / cdiv 11895 ℕ0cn0 12496 ;cdc 12701 ℝ+crp 13000 .cdp 32605 |
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 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
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-rmo 3372 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-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-dec 12702 df-rp 13001 df-dp2 32589 df-dp 32606 |
This theorem is referenced by: hgt750lem 34277 |
Copyright terms: Public domain | W3C validator |