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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 9190 | . . . 4 ⊢ 𝐵 ∈ ℝ |
4 | decle.3 | . . . . 5 ⊢ 𝐶 ∈ ℕ0 | |
5 | 4 | nn0rei 9190 | . . . 4 ⊢ 𝐶 ∈ ℝ |
6 | 10nn0 9404 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
7 | decle.1 | . . . . . 6 ⊢ 𝐴 ∈ ℕ0 | |
8 | 6, 7 | nn0mulcli 9217 | . . . . 5 ⊢ (;10 · 𝐴) ∈ ℕ0 |
9 | 8 | nn0rei 9190 | . . . 4 ⊢ (;10 · 𝐴) ∈ ℝ |
10 | 3, 5, 9 | leadd2i 8464 | . . 3 ⊢ (𝐵 ≤ 𝐶 ↔ ((;10 · 𝐴) + 𝐵) ≤ ((;10 · 𝐴) + 𝐶)) |
11 | 1, 10 | mpbi 145 | . 2 ⊢ ((;10 · 𝐴) + 𝐵) ≤ ((;10 · 𝐴) + 𝐶) |
12 | dfdec10 9390 | . 2 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
13 | dfdec10 9390 | . 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
14 | 11, 12, 13 | 3brtr4i 4035 | 1 ⊢ ;𝐴𝐵 ≤ ;𝐴𝐶 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 class class class wbr 4005 (class class class)co 5878 0cc0 7814 1c1 7815 + caddc 7817 · cmul 7819 ≤ cle 7996 ℕ0cn0 9179 ;cdc 9387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-mulcom 7915 ax-addass 7916 ax-mulass 7917 ax-distr 7918 ax-i2m1 7919 ax-1rid 7921 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-5 8984 df-6 8985 df-7 8986 df-8 8987 df-9 8988 df-n0 9180 df-dec 9388 |
This theorem is referenced by: (None) |
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