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| 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 13043 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
| 4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ 𝐵 ∈ ℝ |
| 5 | 1, 4 | dpval2 32875 | . . 3 ⊢ (𝐴.𝐵) = (𝐴 + (𝐵 / ;10)) |
| 6 | dplti.1 | . . . . 5 ⊢ 𝐵 < ;10 | |
| 7 | 10re 12752 | . . . . . . . 8 ⊢ ;10 ∈ ℝ | |
| 8 | 10pos 12750 | . . . . . . . 8 ⊢ 0 < ;10 | |
| 9 | 7, 8 | pm3.2i 470 | . . . . . . 7 ⊢ (;10 ∈ ℝ ∧ 0 < ;10) |
| 10 | elrp 13036 | . . . . . . 7 ⊢ (;10 ∈ ℝ+ ↔ (;10 ∈ ℝ ∧ 0 < ;10)) | |
| 11 | 9, 10 | mpbir 231 | . . . . . 6 ⊢ ;10 ∈ ℝ+ |
| 12 | divlt1lt 13104 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ ;10 ∈ ℝ+) → ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10)) | |
| 13 | 4, 11, 12 | mp2an 692 | . . . . 5 ⊢ ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10) |
| 14 | 6, 13 | mpbir 231 | . . . 4 ⊢ (𝐵 / ;10) < 1 |
| 15 | 0re 11263 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
| 16 | 15, 8 | gtneii 11373 | . . . . . 6 ⊢ ;10 ≠ 0 |
| 17 | 4, 7, 16 | redivcli 12034 | . . . . 5 ⊢ (𝐵 / ;10) ∈ ℝ |
| 18 | 1re 11261 | . . . . 5 ⊢ 1 ∈ ℝ | |
| 19 | nn0ssre 12530 | . . . . . 6 ⊢ ℕ0 ⊆ ℝ | |
| 20 | 19, 1 | sselii 3980 | . . . . 5 ⊢ 𝐴 ∈ ℝ |
| 21 | 17, 18, 20 | ltadd2i 11392 | . . . 4 ⊢ ((𝐵 / ;10) < 1 ↔ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1)) |
| 22 | 14, 21 | mpbi 230 | . . 3 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1) |
| 23 | 5, 22 | eqbrtri 5164 | . 2 ⊢ (𝐴.𝐵) < (𝐴 + 1) |
| 24 | dplti.2 | . 2 ⊢ (𝐴 + 1) = 𝐶 | |
| 25 | 23, 24 | breqtri 5168 | 1 ⊢ (𝐴.𝐵) < 𝐶 |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 < clt 11295 / cdiv 11920 ℕ0cn0 12526 ;cdc 12733 ℝ+crp 13034 .cdp 32870 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-dec 12734 df-rp 13035 df-dp2 32854 df-dp 32871 |
| This theorem is referenced by: hgt750lem 34666 |
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