| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dp2ltc | Structured version Visualization version GIF version | ||
| Description: Comparing two decimal expansions (unequal higher places). (Contributed by Thierry Arnoux, 16-Dec-2021.) |
| Ref | Expression |
|---|---|
| dp2lt.a | ⊢ 𝐴 ∈ ℕ0 |
| dp2lt.b | ⊢ 𝐵 ∈ ℝ+ |
| dp2ltc.c | ⊢ 𝐶 ∈ ℕ0 |
| dp2ltc.d | ⊢ 𝐷 ∈ ℝ+ |
| dp2ltc.s | ⊢ 𝐵 < ;10 |
| dp2ltc.l | ⊢ 𝐴 < 𝐶 |
| Ref | Expression |
|---|---|
| dp2ltc | ⊢ _𝐴𝐵 < _𝐶𝐷 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dp2ltc.s | . . . . . 6 ⊢ 𝐵 < ;10 | |
| 2 | rpssre 13042 | . . . . . . . 8 ⊢ ℝ+ ⊆ ℝ | |
| 3 | dp2lt.b | . . . . . . . 8 ⊢ 𝐵 ∈ ℝ+ | |
| 4 | 2, 3 | sselii 3980 | . . . . . . 7 ⊢ 𝐵 ∈ ℝ |
| 5 | 10re 12752 | . . . . . . . 8 ⊢ ;10 ∈ ℝ | |
| 6 | 10pos 12750 | . . . . . . . 8 ⊢ 0 < ;10 | |
| 7 | elrp 13036 | . . . . . . . 8 ⊢ (;10 ∈ ℝ+ ↔ (;10 ∈ ℝ ∧ 0 < ;10)) | |
| 8 | 5, 6, 7 | mpbir2an 711 | . . . . . . 7 ⊢ ;10 ∈ ℝ+ |
| 9 | divlt1lt 13104 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ ;10 ∈ ℝ+) → ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10)) | |
| 10 | 4, 8, 9 | mp2an 692 | . . . . . 6 ⊢ ((𝐵 / ;10) < 1 ↔ 𝐵 < ;10) |
| 11 | 1, 10 | mpbir 231 | . . . . 5 ⊢ (𝐵 / ;10) < 1 |
| 12 | 5, 6 | gt0ne0ii 11799 | . . . . . . 7 ⊢ ;10 ≠ 0 |
| 13 | 4, 5, 12 | redivcli 12034 | . . . . . 6 ⊢ (𝐵 / ;10) ∈ ℝ |
| 14 | 1re 11261 | . . . . . 6 ⊢ 1 ∈ ℝ | |
| 15 | dp2lt.a | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
| 16 | 15 | nn0rei 12537 | . . . . . 6 ⊢ 𝐴 ∈ ℝ |
| 17 | ltadd2 11365 | . . . . . 6 ⊢ (((𝐵 / ;10) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝐵 / ;10) < 1 ↔ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1))) | |
| 18 | 13, 14, 16, 17 | mp3an 1463 | . . . . 5 ⊢ ((𝐵 / ;10) < 1 ↔ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1)) |
| 19 | 11, 18 | mpbi 230 | . . . 4 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐴 + 1) |
| 20 | dp2ltc.l | . . . . 5 ⊢ 𝐴 < 𝐶 | |
| 21 | 15 | nn0zi 12642 | . . . . . 6 ⊢ 𝐴 ∈ ℤ |
| 22 | dp2ltc.c | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 | |
| 23 | 22 | nn0zi 12642 | . . . . . 6 ⊢ 𝐶 ∈ ℤ |
| 24 | zltp1le 12667 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶)) | |
| 25 | 21, 23, 24 | mp2an 692 | . . . . 5 ⊢ (𝐴 < 𝐶 ↔ (𝐴 + 1) ≤ 𝐶) |
| 26 | 20, 25 | mpbi 230 | . . . 4 ⊢ (𝐴 + 1) ≤ 𝐶 |
| 27 | 16, 13 | readdcli 11276 | . . . . 5 ⊢ (𝐴 + (𝐵 / ;10)) ∈ ℝ |
| 28 | 16, 14 | readdcli 11276 | . . . . 5 ⊢ (𝐴 + 1) ∈ ℝ |
| 29 | 22 | nn0rei 12537 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
| 30 | 27, 28, 29 | ltletri 11389 | . . . 4 ⊢ (((𝐴 + (𝐵 / ;10)) < (𝐴 + 1) ∧ (𝐴 + 1) ≤ 𝐶) → (𝐴 + (𝐵 / ;10)) < 𝐶) |
| 31 | 19, 26, 30 | mp2an 692 | . . 3 ⊢ (𝐴 + (𝐵 / ;10)) < 𝐶 |
| 32 | dp2ltc.d | . . . . . 6 ⊢ 𝐷 ∈ ℝ+ | |
| 33 | 32, 8 | pm3.2i 470 | . . . . 5 ⊢ (𝐷 ∈ ℝ+ ∧ ;10 ∈ ℝ+) |
| 34 | rpdivcl 13060 | . . . . 5 ⊢ ((𝐷 ∈ ℝ+ ∧ ;10 ∈ ℝ+) → (𝐷 / ;10) ∈ ℝ+) | |
| 35 | 33, 34 | ax-mp 5 | . . . 4 ⊢ (𝐷 / ;10) ∈ ℝ+ |
| 36 | ltaddrp 13072 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ (𝐷 / ;10) ∈ ℝ+) → 𝐶 < (𝐶 + (𝐷 / ;10))) | |
| 37 | 29, 35, 36 | mp2an 692 | . . 3 ⊢ 𝐶 < (𝐶 + (𝐷 / ;10)) |
| 38 | 2, 32 | sselii 3980 | . . . . . 6 ⊢ 𝐷 ∈ ℝ |
| 39 | 38, 5, 12 | redivcli 12034 | . . . . 5 ⊢ (𝐷 / ;10) ∈ ℝ |
| 40 | 29, 39 | readdcli 11276 | . . . 4 ⊢ (𝐶 + (𝐷 / ;10)) ∈ ℝ |
| 41 | 27, 29, 40 | lttri 11387 | . . 3 ⊢ (((𝐴 + (𝐵 / ;10)) < 𝐶 ∧ 𝐶 < (𝐶 + (𝐷 / ;10))) → (𝐴 + (𝐵 / ;10)) < (𝐶 + (𝐷 / ;10))) |
| 42 | 31, 37, 41 | mp2an 692 | . 2 ⊢ (𝐴 + (𝐵 / ;10)) < (𝐶 + (𝐷 / ;10)) |
| 43 | df-dp2 32854 | . 2 ⊢ _𝐴𝐵 = (𝐴 + (𝐵 / ;10)) | |
| 44 | df-dp2 32854 | . 2 ⊢ _𝐶𝐷 = (𝐶 + (𝐷 / ;10)) | |
| 45 | 42, 43, 44 | 3brtr4i 5173 | 1 ⊢ _𝐴𝐵 < _𝐶𝐷 |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ 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 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: dpltc 32889 |
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