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 29578 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
2 | dp2lt10.1 | . . . . . 6 ⊢ 𝐴 < ;10 | |
3 | 9p1e10 11496 | . . . . . 6 ⊢ (9 + 1) = ;10 | |
4 | 2, 3 | breqtrri 4680 | . . . . 5 ⊢ 𝐴 < (9 + 1) |
5 | dp2lt10.a | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
6 | 5 | nn0zi 11402 | . . . . . 6 ⊢ 𝐴 ∈ ℤ |
7 | 9nn0 11316 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
8 | 7 | nn0zi 11402 | . . . . . 6 ⊢ 9 ∈ ℤ |
9 | zleltp1 11428 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 9 ∈ ℤ) → (𝐴 ≤ 9 ↔ 𝐴 < (9 + 1))) | |
10 | 6, 8, 9 | mp2an 708 | . . . . 5 ⊢ (𝐴 ≤ 9 ↔ 𝐴 < (9 + 1)) |
11 | 4, 10 | mpbir 221 | . . . 4 ⊢ 𝐴 ≤ 9 |
12 | dp2lt10.2 | . . . . 5 ⊢ 𝐵 < ;10 | |
13 | rpssre 11843 | . . . . . . 7 ⊢ ℝ+ ⊆ ℝ | |
14 | dp2lt10.b | . . . . . . 7 ⊢ 𝐵 ∈ ℝ+ | |
15 | 13, 14 | sselii 3600 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
16 | 10re 11517 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
17 | 10pos 11515 | . . . . . . 7 ⊢ 0 < ;10 | |
18 | 16, 17 | elrpii 11835 | . . . . . 6 ⊢ ;10 ∈ ℝ+ |
19 | divlt1lt 11899 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ ;10 ∈ ℝ+) → ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10)) | |
20 | 15, 18, 19 | mp2an 708 | . . . . 5 ⊢ ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10) |
21 | 12, 20 | mpbir 221 | . . . 4 ⊢ (𝐵 / ;10) < 1 |
22 | 5 | nn0rei 11303 | . . . . . 6 ⊢ 𝐴 ∈ ℝ |
23 | 0re 10040 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
24 | 23, 17 | gtneii 10149 | . . . . . . 7 ⊢ ;10 ≠ 0 |
25 | 15, 16, 24 | redivcli 10792 | . . . . . 6 ⊢ (𝐵 / ;10) ∈ ℝ |
26 | 22, 25 | pm3.2i 471 | . . . . 5 ⊢ (𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) |
27 | 9re 11107 | . . . . . 6 ⊢ 9 ∈ ℝ | |
28 | 1re 10039 | . . . . . 6 ⊢ 1 ∈ ℝ | |
29 | 27, 28 | pm3.2i 471 | . . . . 5 ⊢ (9 ∈ ℝ ∧ 1 ∈ ℝ) |
30 | leltadd 10512 | . . . . 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 708 | . . . 4 ⊢ ((𝐴 ≤ 9 ∧ (𝐵 / ;10) < 1) → (𝐴 + (𝐵 / ;10)) < (9 + 1)) |
32 | 11, 21, 31 | mp2an 708 | . . 3 ⊢ (𝐴 + (𝐵 / ;10)) < (9 + 1) |
33 | 32, 3 | breqtri 4678 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < ;10 |
34 | 1, 33 | eqbrtri 4674 | 1 ⊢ _𝐴𝐵 < ;10 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 / cdiv 10684 9c9 11077 ℕ0cn0 11292 ℤcz 11377 ;cdc 11493 ℝ+crp 11832 _cdp2 29577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-rp 11833 df-dp2 29578 |
This theorem is referenced by: hgt750lem 30729 hgt750lem2 30730 |
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