![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 32736 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
2 | dp2lt10.1 | . . . . . 6 ⊢ 𝐴 < ;10 | |
3 | 9p1e10 12725 | . . . . . 6 ⊢ (9 + 1) = ;10 | |
4 | 2, 3 | breqtrri 5172 | . . . . 5 ⊢ 𝐴 < (9 + 1) |
5 | dp2lt10.a | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
6 | 5 | nn0zi 12633 | . . . . . 6 ⊢ 𝐴 ∈ ℤ |
7 | 9nn0 12542 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
8 | 7 | nn0zi 12633 | . . . . . 6 ⊢ 9 ∈ ℤ |
9 | zleltp1 12659 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 9 ∈ ℤ) → (𝐴 ≤ 9 ↔ 𝐴 < (9 + 1))) | |
10 | 6, 8, 9 | mp2an 690 | . . . . 5 ⊢ (𝐴 ≤ 9 ↔ 𝐴 < (9 + 1)) |
11 | 4, 10 | mpbir 230 | . . . 4 ⊢ 𝐴 ≤ 9 |
12 | dp2lt10.2 | . . . . 5 ⊢ 𝐵 < ;10 | |
13 | rpssre 13029 | . . . . . . 7 ⊢ ℝ+ ⊆ ℝ | |
14 | dp2lt10.b | . . . . . . 7 ⊢ 𝐵 ∈ ℝ+ | |
15 | 13, 14 | sselii 3975 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
16 | 10re 12742 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
17 | 10pos 12740 | . . . . . . 7 ⊢ 0 < ;10 | |
18 | 16, 17 | elrpii 13025 | . . . . . 6 ⊢ ;10 ∈ ℝ+ |
19 | divlt1lt 13091 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ ;10 ∈ ℝ+) → ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10)) | |
20 | 15, 18, 19 | mp2an 690 | . . . . 5 ⊢ ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10) |
21 | 12, 20 | mpbir 230 | . . . 4 ⊢ (𝐵 / ;10) < 1 |
22 | 5 | nn0rei 12529 | . . . . . 6 ⊢ 𝐴 ∈ ℝ |
23 | 0re 11257 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
24 | 23, 17 | gtneii 11367 | . . . . . . 7 ⊢ ;10 ≠ 0 |
25 | 15, 16, 24 | redivcli 12026 | . . . . . 6 ⊢ (𝐵 / ;10) ∈ ℝ |
26 | 22, 25 | pm3.2i 469 | . . . . 5 ⊢ (𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) |
27 | 9re 12357 | . . . . . 6 ⊢ 9 ∈ ℝ | |
28 | 1re 11255 | . . . . . 6 ⊢ 1 ∈ ℝ | |
29 | 27, 28 | pm3.2i 469 | . . . . 5 ⊢ (9 ∈ ℝ ∧ 1 ∈ ℝ) |
30 | leltadd 11739 | . . . . 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 690 | . . . 4 ⊢ ((𝐴 ≤ 9 ∧ (𝐵 / ;10) < 1) → (𝐴 + (𝐵 / ;10)) < (9 + 1)) |
32 | 11, 21, 31 | mp2an 690 | . . 3 ⊢ (𝐴 + (𝐵 / ;10)) < (9 + 1) |
33 | 32, 3 | breqtri 5170 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < ;10 |
34 | 1, 33 | eqbrtri 5166 | 1 ⊢ _𝐴𝐵 < ;10 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2099 class class class wbr 5145 (class class class)co 7416 ℝcr 11148 0cc0 11149 1c1 11150 + caddc 11152 < clt 11289 ≤ cle 11290 / cdiv 11912 9c9 12320 ℕ0cn0 12518 ℤcz 12604 ;cdc 12723 ℝ+crp 13022 _cdp2 32735 |
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 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-z 12605 df-dec 12724 df-rp 13023 df-dp2 32736 |
This theorem is referenced by: hgt750lem 34510 hgt750lem2 34511 |
Copyright terms: Public domain | W3C validator |