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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dpgti | Structured version Visualization version GIF version |
Description: Comparing a decimal expansions with the next lower integer. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
Ref | Expression |
---|---|
dpgti.a | ⊢ 𝐴 ∈ ℕ0 |
dpgti.b | ⊢ 𝐵 ∈ ℝ+ |
Ref | Expression |
---|---|
dpgti | ⊢ 𝐴 < (𝐴.𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dpgti.a | . . . 4 ⊢ 𝐴 ∈ ℕ0 | |
2 | 1 | nn0rei 11718 | . . 3 ⊢ 𝐴 ∈ ℝ |
3 | dpgti.b | . . . 4 ⊢ 𝐵 ∈ ℝ+ | |
4 | 10re 11929 | . . . . . 6 ⊢ ;10 ∈ ℝ | |
5 | 10pos 11927 | . . . . . 6 ⊢ 0 < ;10 | |
6 | 4, 5 | pm3.2i 463 | . . . . 5 ⊢ (;10 ∈ ℝ ∧ 0 < ;10) |
7 | elrp 12205 | . . . . 5 ⊢ (;10 ∈ ℝ+ ↔ (;10 ∈ ℝ ∧ 0 < ;10)) | |
8 | 6, 7 | mpbir 223 | . . . 4 ⊢ ;10 ∈ ℝ+ |
9 | rpdivcl 12230 | . . . 4 ⊢ ((𝐵 ∈ ℝ+ ∧ ;10 ∈ ℝ+) → (𝐵 / ;10) ∈ ℝ+) | |
10 | 3, 8, 9 | mp2an 680 | . . 3 ⊢ (𝐵 / ;10) ∈ ℝ+ |
11 | ltaddrp 12242 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ+) → 𝐴 < (𝐴 + (𝐵 / ;10))) | |
12 | 2, 10, 11 | mp2an 680 | . 2 ⊢ 𝐴 < (𝐴 + (𝐵 / ;10)) |
13 | rpre 12211 | . . . 4 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
14 | 3, 13 | ax-mp 5 | . . 3 ⊢ 𝐵 ∈ ℝ |
15 | 1, 14 | dpval2 30340 | . 2 ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10)) |
16 | 12, 15 | breqtrri 4953 | 1 ⊢ 𝐴 < (𝐴.𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 ∈ wcel 2051 class class class wbr 4926 (class class class)co 6975 ℝcr 10333 0cc0 10334 1c1 10335 + caddc 10337 < clt 10473 / cdiv 11097 ℕ0cn0 11706 ;cdc 11910 ℝ+crp 12203 .cdp 30335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-5 11505 df-6 11506 df-7 11507 df-8 11508 df-9 11509 df-n0 11707 df-dec 11911 df-rp 12204 df-dp2 30319 df-dp 30336 |
This theorem is referenced by: hgt750lem 31603 |
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