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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dp2lt10 | Structured version Visualization version GIF version |
Description: Decimal fraction builds real numbers less than 10. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
Ref | Expression |
---|---|
dp2lt10.a | ⊢ 𝐴 ∈ ℕ0 |
dp2lt10.b | ⊢ 𝐵 ∈ ℝ+ |
dp2lt10.1 | ⊢ 𝐴 < ;10 |
dp2lt10.2 | ⊢ 𝐵 < ;10 |
Ref | Expression |
---|---|
dp2lt10 | ⊢ _𝐴𝐵 < ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dp2 32838 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
2 | dp2lt10.1 | . . . . . 6 ⊢ 𝐴 < ;10 | |
3 | 9p1e10 12732 | . . . . . 6 ⊢ (9 + 1) = ;10 | |
4 | 2, 3 | breqtrri 5174 | . . . . 5 ⊢ 𝐴 < (9 + 1) |
5 | dp2lt10.a | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
6 | 5 | nn0zi 12639 | . . . . . 6 ⊢ 𝐴 ∈ ℤ |
7 | 9nn0 12547 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
8 | 7 | nn0zi 12639 | . . . . . 6 ⊢ 9 ∈ ℤ |
9 | zleltp1 12665 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 9 ∈ ℤ) → (𝐴 ≤ 9 ↔ 𝐴 < (9 + 1))) | |
10 | 6, 8, 9 | mp2an 692 | . . . . 5 ⊢ (𝐴 ≤ 9 ↔ 𝐴 < (9 + 1)) |
11 | 4, 10 | mpbir 231 | . . . 4 ⊢ 𝐴 ≤ 9 |
12 | dp2lt10.2 | . . . . 5 ⊢ 𝐵 < ;10 | |
13 | rpssre 13039 | . . . . . . 7 ⊢ ℝ+ ⊆ ℝ | |
14 | dp2lt10.b | . . . . . . 7 ⊢ 𝐵 ∈ ℝ+ | |
15 | 13, 14 | sselii 3991 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
16 | 10re 12749 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
17 | 10pos 12747 | . . . . . . 7 ⊢ 0 < ;10 | |
18 | 16, 17 | elrpii 13034 | . . . . . 6 ⊢ ;10 ∈ ℝ+ |
19 | divlt1lt 13101 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ ;10 ∈ ℝ+) → ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10)) | |
20 | 15, 18, 19 | mp2an 692 | . . . . 5 ⊢ ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10) |
21 | 12, 20 | mpbir 231 | . . . 4 ⊢ (𝐵 / ;10) < 1 |
22 | 5 | nn0rei 12534 | . . . . . 6 ⊢ 𝐴 ∈ ℝ |
23 | 0re 11260 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
24 | 23, 17 | gtneii 11370 | . . . . . . 7 ⊢ ;10 ≠ 0 |
25 | 15, 16, 24 | redivcli 12031 | . . . . . 6 ⊢ (𝐵 / ;10) ∈ ℝ |
26 | 22, 25 | pm3.2i 470 | . . . . 5 ⊢ (𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) |
27 | 9re 12362 | . . . . . 6 ⊢ 9 ∈ ℝ | |
28 | 1re 11258 | . . . . . 6 ⊢ 1 ∈ ℝ | |
29 | 27, 28 | pm3.2i 470 | . . . . 5 ⊢ (9 ∈ ℝ ∧ 1 ∈ ℝ) |
30 | leltadd 11744 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) ∧ (9 ∈ ℝ ∧ 1 ∈ ℝ)) → ((𝐴 ≤ 9 ∧ (𝐵 / ;10) < 1) → (𝐴 + (𝐵 / ;10)) < (9 + 1))) | |
31 | 26, 29, 30 | mp2an 692 | . . . 4 ⊢ ((𝐴 ≤ 9 ∧ (𝐵 / ;10) < 1) → (𝐴 + (𝐵 / ;10)) < (9 + 1)) |
32 | 11, 21, 31 | mp2an 692 | . . 3 ⊢ (𝐴 + (𝐵 / ;10)) < (9 + 1) |
33 | 32, 3 | breqtri 5172 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < ;10 |
34 | 1, 33 | eqbrtri 5168 | 1 ⊢ _𝐴𝐵 < ;10 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2105 class class class wbr 5147 (class class class)co 7430 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 < clt 11292 ≤ cle 11293 / cdiv 11917 9c9 12325 ℕ0cn0 12523 ℤcz 12610 ;cdc 12730 ℝ+crp 13031 _cdp2 32837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-rp 13032 df-dp2 32838 |
This theorem is referenced by: hgt750lem 34644 hgt750lem2 34645 |
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