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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 12985 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
5 | 10re 12697 | . . . . . 6 ⊢ ;10 ∈ ℝ | |
6 | 10pos 12695 | . . . . . 6 ⊢ 0 < ;10 | |
7 | 4, 5, 5, 6 | ltdiv1ii 12144 | . . . . 5 ⊢ (𝐵 < ;10 ↔ (𝐵 / ;10) < (;10 / ;10)) |
8 | 1, 7 | mpbi 229 | . . . 4 ⊢ (𝐵 / ;10) < (;10 / ;10) |
9 | 5 | recni 11229 | . . . . 5 ⊢ ;10 ∈ ℂ |
10 | 10nn 12694 | . . . . . 6 ⊢ ;10 ∈ ℕ | |
11 | 10 | nnne0i 12253 | . . . . 5 ⊢ ;10 ≠ 0 |
12 | 9, 11 | dividi 11948 | . . . 4 ⊢ (;10 / ;10) = 1 |
13 | 8, 12 | breqtri 5166 | . . 3 ⊢ (𝐵 / ;10) < 1 |
14 | 4, 5, 11 | redivcli 11982 | . . . 4 ⊢ (𝐵 / ;10) ∈ ℝ |
15 | 1re 11215 | . . . 4 ⊢ 1 ∈ ℝ | |
16 | dp2lt.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
17 | 16 | nn0rei 12484 | . . . 4 ⊢ 𝐴 ∈ ℝ |
18 | 14, 15, 17 | ltadd2i 11346 | . . 3 ⊢ ((𝐵 / ;10) < 1 ↔ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1)) |
19 | 13, 18 | mpbi 229 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1) |
20 | df-dp2 32540 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
21 | dp2ltsuc.2 | . . 3 ⊢ (𝐴 + 1) = 𝐶 | |
22 | 21 | eqcomi 2735 | . 2 ⊢ 𝐶 = (𝐴 + 1) |
23 | 19, 20, 22 | 3brtr4i 5171 | 1 ⊢ _𝐴𝐵 < 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 class class class wbr 5141 (class class class)co 7404 ℝcr 11108 0cc0 11109 1c1 11110 + caddc 11112 < clt 11249 / cdiv 11872 ℕ0cn0 12473 ;cdc 12678 ℝ+crp 12977 _cdp2 32539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-dec 12679 df-rp 12978 df-dp2 32540 |
This theorem is referenced by: hgt750lem2 34192 |
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