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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 8892 | . . . 4 ⊢ 𝐵 ∈ ℝ |
4 | decle.3 | . . . . 5 ⊢ 𝐶 ∈ ℕ0 | |
5 | 4 | nn0rei 8892 | . . . 4 ⊢ 𝐶 ∈ ℝ |
6 | 10nn0 9103 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
7 | decle.1 | . . . . . 6 ⊢ 𝐴 ∈ ℕ0 | |
8 | 6, 7 | nn0mulcli 8919 | . . . . 5 ⊢ (;10 · 𝐴) ∈ ℕ0 |
9 | 8 | nn0rei 8892 | . . . 4 ⊢ (;10 · 𝐴) ∈ ℝ |
10 | 3, 5, 9 | leadd2i 8185 | . . 3 ⊢ (𝐵 ≤ 𝐶 ↔ ((;10 · 𝐴) + 𝐵) ≤ ((;10 · 𝐴) + 𝐶)) |
11 | 1, 10 | mpbi 144 | . 2 ⊢ ((;10 · 𝐴) + 𝐵) ≤ ((;10 · 𝐴) + 𝐶) |
12 | dfdec10 9089 | . 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
13 | dfdec10 9089 | . 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
14 | 11, 12, 13 | 3brtr4i 3923 | 1 ⊢ ;𝐴𝐵 ≤ ;𝐴𝐶 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1463 class class class wbr 3895 (class class class)co 5728 0cc0 7547 1c1 7548 + caddc 7550 · cmul 7552 ≤ cle 7725 ℕ0cn0 8881 ;cdc 9086 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-addcom 7645 ax-mulcom 7646 ax-addass 7647 ax-mulass 7648 ax-distr 7649 ax-i2m1 7650 ax-1rid 7652 ax-0id 7653 ax-rnegex 7654 ax-cnre 7656 ax-pre-ltadd 7661 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-opab 3950 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-iota 5046 df-fun 5083 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-pnf 7726 df-mnf 7727 df-xr 7728 df-ltxr 7729 df-le 7730 df-sub 7858 df-inn 8631 df-2 8689 df-3 8690 df-4 8691 df-5 8692 df-6 8693 df-7 8694 df-8 8695 df-9 8696 df-n0 8882 df-dec 9087 |
This theorem is referenced by: (None) |
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