![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 13009 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
5 | 10re 12721 | . . . . . 6 ⊢ ;10 ∈ ℝ | |
6 | 10pos 12719 | . . . . . 6 ⊢ 0 < ;10 | |
7 | 4, 5, 5, 6 | ltdiv1ii 12168 | . . . . 5 ⊢ (𝐵 < ;10 ↔ (𝐵 / ;10) < (;10 / ;10)) |
8 | 1, 7 | mpbi 229 | . . . 4 ⊢ (𝐵 / ;10) < (;10 / ;10) |
9 | 5 | recni 11253 | . . . . 5 ⊢ ;10 ∈ ℂ |
10 | 10nn 12718 | . . . . . 6 ⊢ ;10 ∈ ℕ | |
11 | 10 | nnne0i 12277 | . . . . 5 ⊢ ;10 ≠ 0 |
12 | 9, 11 | dividi 11972 | . . . 4 ⊢ (;10 / ;10) = 1 |
13 | 8, 12 | breqtri 5168 | . . 3 ⊢ (𝐵 / ;10) < 1 |
14 | 4, 5, 11 | redivcli 12006 | . . . 4 ⊢ (𝐵 / ;10) ∈ ℝ |
15 | 1re 11239 | . . . 4 ⊢ 1 ∈ ℝ | |
16 | dp2lt.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
17 | 16 | nn0rei 12508 | . . . 4 ⊢ 𝐴 ∈ ℝ |
18 | 14, 15, 17 | ltadd2i 11370 | . . 3 ⊢ ((𝐵 / ;10) < 1 ↔ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1)) |
19 | 13, 18 | mpbi 229 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1) |
20 | df-dp2 32590 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
21 | dp2ltsuc.2 | . . 3 ⊢ (𝐴 + 1) = 𝐶 | |
22 | 21 | eqcomi 2737 | . 2 ⊢ 𝐶 = (𝐴 + 1) |
23 | 19, 20, 22 | 3brtr4i 5173 | 1 ⊢ _𝐴𝐵 < 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 class class class wbr 5143 (class class class)co 7415 ℝcr 11132 0cc0 11133 1c1 11134 + caddc 11136 < clt 11273 / cdiv 11896 ℕ0cn0 12497 ;cdc 12702 ℝ+crp 13001 _cdp2 32589 |
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-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 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 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-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-om 7866 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 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 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-dec 12703 df-rp 13002 df-dp2 32590 |
This theorem is referenced by: hgt750lem2 34279 |
Copyright terms: Public domain | W3C validator |