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| Mirrors > Home > MPE Home > Th. List > numltc | Structured version Visualization version GIF version | ||
| Description: Comparing two decimal integers (unequal higher places). (Contributed by Mario Carneiro, 18-Feb-2014.) |
| Ref | Expression |
|---|---|
| numlt.1 | ⊢ 𝑇 ∈ ℕ |
| numlt.2 | ⊢ 𝐴 ∈ ℕ0 |
| numlt.3 | ⊢ 𝐵 ∈ ℕ0 |
| numltc.3 | ⊢ 𝐶 ∈ ℕ0 |
| numltc.4 | ⊢ 𝐷 ∈ ℕ0 |
| numltc.5 | ⊢ 𝐶 < 𝑇 |
| numltc.6 | ⊢ 𝐴 < 𝐵 |
| Ref | Expression |
|---|---|
| numltc | ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | numlt.1 | . . . . 5 ⊢ 𝑇 ∈ ℕ | |
| 2 | numlt.2 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
| 3 | numltc.3 | . . . . 5 ⊢ 𝐶 ∈ ℕ0 | |
| 4 | numltc.5 | . . . . 5 ⊢ 𝐶 < 𝑇 | |
| 5 | 1, 2, 3, 1, 4 | numlt 12758 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐴) + 𝑇) |
| 6 | 1 | nnrei 12275 | . . . . . . 7 ⊢ 𝑇 ∈ ℝ |
| 7 | 6 | recni 11275 | . . . . . 6 ⊢ 𝑇 ∈ ℂ |
| 8 | 2 | nn0rei 12537 | . . . . . . 7 ⊢ 𝐴 ∈ ℝ |
| 9 | 8 | recni 11275 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
| 10 | ax-1cn 11213 | . . . . . 6 ⊢ 1 ∈ ℂ | |
| 11 | 7, 9, 10 | adddii 11273 | . . . . 5 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + (𝑇 · 1)) |
| 12 | 7 | mulridi 11265 | . . . . . 6 ⊢ (𝑇 · 1) = 𝑇 |
| 13 | 12 | oveq2i 7442 | . . . . 5 ⊢ ((𝑇 · 𝐴) + (𝑇 · 1)) = ((𝑇 · 𝐴) + 𝑇) |
| 14 | 11, 13 | eqtri 2765 | . . . 4 ⊢ (𝑇 · (𝐴 + 1)) = ((𝑇 · 𝐴) + 𝑇) |
| 15 | 5, 14 | breqtrri 5170 | . . 3 ⊢ ((𝑇 · 𝐴) + 𝐶) < (𝑇 · (𝐴 + 1)) |
| 16 | numltc.6 | . . . . 5 ⊢ 𝐴 < 𝐵 | |
| 17 | numlt.3 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
| 18 | nn0ltp1le 12676 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵)) | |
| 19 | 2, 17, 18 | mp2an 692 | . . . . 5 ⊢ (𝐴 < 𝐵 ↔ (𝐴 + 1) ≤ 𝐵) |
| 20 | 16, 19 | mpbi 230 | . . . 4 ⊢ (𝐴 + 1) ≤ 𝐵 |
| 21 | 1 | nngt0i 12305 | . . . . 5 ⊢ 0 < 𝑇 |
| 22 | peano2re 11434 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | |
| 23 | 8, 22 | ax-mp 5 | . . . . . 6 ⊢ (𝐴 + 1) ∈ ℝ |
| 24 | 17 | nn0rei 12537 | . . . . . 6 ⊢ 𝐵 ∈ ℝ |
| 25 | 23, 24, 6 | lemul2i 12191 | . . . . 5 ⊢ (0 < 𝑇 → ((𝐴 + 1) ≤ 𝐵 ↔ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵))) |
| 26 | 21, 25 | ax-mp 5 | . . . 4 ⊢ ((𝐴 + 1) ≤ 𝐵 ↔ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵)) |
| 27 | 20, 26 | mpbi 230 | . . 3 ⊢ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵) |
| 28 | 6, 8 | remulcli 11277 | . . . . 5 ⊢ (𝑇 · 𝐴) ∈ ℝ |
| 29 | 3 | nn0rei 12537 | . . . . 5 ⊢ 𝐶 ∈ ℝ |
| 30 | 28, 29 | readdcli 11276 | . . . 4 ⊢ ((𝑇 · 𝐴) + 𝐶) ∈ ℝ |
| 31 | 6, 23 | remulcli 11277 | . . . 4 ⊢ (𝑇 · (𝐴 + 1)) ∈ ℝ |
| 32 | 6, 24 | remulcli 11277 | . . . 4 ⊢ (𝑇 · 𝐵) ∈ ℝ |
| 33 | 30, 31, 32 | ltletri 11389 | . . 3 ⊢ ((((𝑇 · 𝐴) + 𝐶) < (𝑇 · (𝐴 + 1)) ∧ (𝑇 · (𝐴 + 1)) ≤ (𝑇 · 𝐵)) → ((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵)) |
| 34 | 15, 27, 33 | mp2an 692 | . 2 ⊢ ((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵) |
| 35 | numltc.4 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
| 36 | 32, 35 | nn0addge1i 12574 | . 2 ⊢ (𝑇 · 𝐵) ≤ ((𝑇 · 𝐵) + 𝐷) |
| 37 | 35 | nn0rei 12537 | . . . 4 ⊢ 𝐷 ∈ ℝ |
| 38 | 32, 37 | readdcli 11276 | . . 3 ⊢ ((𝑇 · 𝐵) + 𝐷) ∈ ℝ |
| 39 | 30, 32, 38 | ltletri 11389 | . 2 ⊢ ((((𝑇 · 𝐴) + 𝐶) < (𝑇 · 𝐵) ∧ (𝑇 · 𝐵) ≤ ((𝑇 · 𝐵) + 𝐷)) → ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷)) |
| 40 | 34, 36, 39 | mp2an 692 | 1 ⊢ ((𝑇 · 𝐴) + 𝐶) < ((𝑇 · 𝐵) + 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 ℕcn 12266 ℕ0cn0 12526 |
| 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-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-nn 12267 df-n0 12527 df-z 12614 |
| This theorem is referenced by: decltc 12762 numlti 12770 |
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