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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 32854 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
| 2 | dp2lt10.1 | . . . . . 6 ⊢ 𝐴 < ;10 | |
| 3 | 9p1e10 12735 | . . . . . 6 ⊢ (9 + 1) = ;10 | |
| 4 | 2, 3 | breqtrri 5170 | . . . . 5 ⊢ 𝐴 < (9 + 1) |
| 5 | dp2lt10.a | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
| 6 | 5 | nn0zi 12642 | . . . . . 6 ⊢ 𝐴 ∈ ℤ |
| 7 | 9nn0 12550 | . . . . . . 7 ⊢ 9 ∈ ℕ0 | |
| 8 | 7 | nn0zi 12642 | . . . . . 6 ⊢ 9 ∈ ℤ |
| 9 | zleltp1 12668 | . . . . . 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 13042 | . . . . . . 7 ⊢ ℝ+ ⊆ ℝ | |
| 14 | dp2lt10.b | . . . . . . 7 ⊢ 𝐵 ∈ ℝ+ | |
| 15 | 13, 14 | sselii 3980 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
| 16 | 10re 12752 | . . . . . . 7 ⊢ ;10 ∈ ℝ | |
| 17 | 10pos 12750 | . . . . . . 7 ⊢ 0 < ;10 | |
| 18 | 16, 17 | elrpii 13037 | . . . . . 6 ⊢ ;10 ∈ ℝ+ |
| 19 | divlt1lt 13104 | . . . . . 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 12537 | . . . . . 6 ⊢ 𝐴 ∈ ℝ |
| 23 | 0re 11263 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
| 24 | 23, 17 | gtneii 11373 | . . . . . . 7 ⊢ ;10 ≠ 0 |
| 25 | 15, 16, 24 | redivcli 12034 | . . . . . 6 ⊢ (𝐵 / ;10) ∈ ℝ |
| 26 | 22, 25 | pm3.2i 470 | . . . . 5 ⊢ (𝐴 ∈ ℝ ∧ (𝐵 / ;10) ∈ ℝ) |
| 27 | 9re 12365 | . . . . . 6 ⊢ 9 ∈ ℝ | |
| 28 | 1re 11261 | . . . . . 6 ⊢ 1 ∈ ℝ | |
| 29 | 27, 28 | pm3.2i 470 | . . . . 5 ⊢ (9 ∈ ℝ ∧ 1 ∈ ℝ) |
| 30 | leltadd 11747 | . . . . 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 5168 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < ;10 |
| 34 | 1, 33 | eqbrtri 5164 | 1 ⊢ _𝐴𝐵 < ;10 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 < clt 11295 ≤ cle 11296 / cdiv 11920 9c9 12328 ℕ0cn0 12526 ℤcz 12613 ;cdc 12733 ℝ+crp 13034 _cdp2 32853 |
| 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-z 12614 df-dec 12734 df-rp 13035 df-dp2 32854 |
| This theorem is referenced by: hgt750lem 34666 hgt750lem2 34667 |
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