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 30550 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
2 | dp2lt10.1 | . . . . . 6 ⊢ 𝐴 < ;10 | |
3 | 9p1e10 12103 | . . . . . 6 ⊢ (9 + 1) = ;10 | |
4 | 2, 3 | breqtrri 5095 | . . . . 5 ⊢ 𝐴 < (9 + 1) |
5 | dp2lt10.a | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
6 | 5 | nn0zi 12010 | . . . . . 6 ⊢ 𝐴 ∈ ℤ |
7 | 9nn0 11924 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
8 | 7 | nn0zi 12010 | . . . . . 6 ⊢ 9 ∈ ℤ |
9 | zleltp1 12036 | . . . . . 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 233 | . . . 4 ⊢ 𝐴 ≤ 9 |
12 | dp2lt10.2 | . . . . 5 ⊢ 𝐵 < ;10 | |
13 | rpssre 12399 | . . . . . . 7 ⊢ ℝ+ ⊆ ℝ | |
14 | dp2lt10.b | . . . . . . 7 ⊢ 𝐵 ∈ ℝ+ | |
15 | 13, 14 | sselii 3966 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
16 | 10re 12120 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
17 | 10pos 12118 | . . . . . . 7 ⊢ 0 < ;10 | |
18 | 16, 17 | elrpii 12395 | . . . . . 6 ⊢ ;10 ∈ ℝ+ |
19 | divlt1lt 12461 | . . . . . 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 233 | . . . 4 ⊢ (𝐵 / ;10) < 1 |
22 | 5 | nn0rei 11911 | . . . . . 6 ⊢ 𝐴 ∈ ℝ |
23 | 0re 10645 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
24 | 23, 17 | gtneii 10754 | . . . . . . 7 ⊢ ;10 ≠ 0 |
25 | 15, 16, 24 | redivcli 11409 | . . . . . 6 ⊢ (𝐵 / ;10) ∈ ℝ |
26 | 22, 25 | pm3.2i 473 | . . . . 5 ⊢ (𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) |
27 | 9re 11739 | . . . . . 6 ⊢ 9 ∈ ℝ | |
28 | 1re 10643 | . . . . . 6 ⊢ 1 ∈ ℝ | |
29 | 27, 28 | pm3.2i 473 | . . . . 5 ⊢ (9 ∈ ℝ ∧ 1 ∈ ℝ) |
30 | leltadd 11126 | . . . . 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 5093 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < ;10 |
34 | 1, 33 | eqbrtri 5089 | 1 ⊢ _𝐴𝐵 < ;10 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 < clt 10677 ≤ cle 10678 / cdiv 11299 9c9 11702 ℕ0cn0 11900 ℤcz 11984 ;cdc 12101 ℝ+crp 12392 _cdp2 30549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-rp 12393 df-dp2 30550 |
This theorem is referenced by: hgt750lem 31924 hgt750lem2 31925 |
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