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Mirrors > Home > ILE Home > Th. List > decle | GIF version |
Description: Comparing two decimal integers (equal higher places). (Contributed by AV, 17-Aug-2021.) (Revised by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
decle.1 | ⊢ 𝐴 ∈ ℕ0 |
decle.2 | ⊢ 𝐵 ∈ ℕ0 |
decle.3 | ⊢ 𝐶 ∈ ℕ0 |
decle.4 | ⊢ 𝐵 ≤ 𝐶 |
Ref | Expression |
---|---|
decle | ⊢ ;𝐴𝐵 ≤ ;𝐴𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decle.4 | . . 3 ⊢ 𝐵 ≤ 𝐶 | |
2 | decle.2 | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
3 | 2 | nn0rei 8956 | . . . 4 ⊢ 𝐵 ∈ ℝ |
4 | decle.3 | . . . . 5 ⊢ 𝐶 ∈ ℕ0 | |
5 | 4 | nn0rei 8956 | . . . 4 ⊢ 𝐶 ∈ ℝ |
6 | 10nn0 9167 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
7 | decle.1 | . . . . . 6 ⊢ 𝐴 ∈ ℕ0 | |
8 | 6, 7 | nn0mulcli 8983 | . . . . 5 ⊢ (;10 · 𝐴) ∈ ℕ0 |
9 | 8 | nn0rei 8956 | . . . 4 ⊢ (;10 · 𝐴) ∈ ℝ |
10 | 3, 5, 9 | leadd2i 8234 | . . 3 ⊢ (𝐵 ≤ 𝐶 ↔ ((;10 · 𝐴) + 𝐵) ≤ ((;10 · 𝐴) + 𝐶)) |
11 | 1, 10 | mpbi 144 | . 2 ⊢ ((;10 · 𝐴) + 𝐵) ≤ ((;10 · 𝐴) + 𝐶) |
12 | dfdec10 9153 | . 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
13 | dfdec10 9153 | . 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
14 | 11, 12, 13 | 3brtr4i 3928 | 1 ⊢ ;𝐴𝐵 ≤ ;𝐴𝐶 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 class class class wbr 3899 (class class class)co 5742 0cc0 7588 1c1 7589 + caddc 7591 · cmul 7593 ≤ cle 7769 ℕ0cn0 8945 ;cdc 9150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-inn 8689 df-2 8747 df-3 8748 df-4 8749 df-5 8750 df-6 8751 df-7 8752 df-8 8753 df-9 8754 df-n0 8946 df-dec 9151 |
This theorem is referenced by: (None) |
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