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Mirrors > Home > ILE Home > Th. List > 3dec | GIF version |
Description: A "decimal constructor" which is used to build up "decimal integers" or "numeric terms" in base 10 with 3 "digits". (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
Ref | Expression |
---|---|
3dec.a | ⊢ 𝐴 ∈ ℕ0 |
3dec.b | ⊢ 𝐵 ∈ ℕ0 |
Ref | Expression |
---|---|
3dec | ⊢ ;;𝐴𝐵𝐶 = ((((;10↑2) · 𝐴) + (;10 · 𝐵)) + 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdec10 8878 | . 2 ⊢ ;;𝐴𝐵𝐶 = ((;10 · ;𝐴𝐵) + 𝐶) | |
2 | dfdec10 8878 | . . . . . 6 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
3 | 2 | oveq2i 5663 | . . . . 5 ⊢ (;10 · ;𝐴𝐵) = (;10 · ((;10 · 𝐴) + 𝐵)) |
4 | 1nn 8431 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
5 | 4 | decnncl2 8898 | . . . . . . 7 ⊢ ;10 ∈ ℕ |
6 | 5 | nncni 8430 | . . . . . 6 ⊢ ;10 ∈ ℂ |
7 | 3dec.a | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
8 | 7 | nn0cni 8683 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
9 | 6, 8 | mulcli 7491 | . . . . . 6 ⊢ (;10 · 𝐴) ∈ ℂ |
10 | 3dec.b | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
11 | 10 | nn0cni 8683 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
12 | 6, 9, 11 | adddii 7496 | . . . . 5 ⊢ (;10 · ((;10 · 𝐴) + 𝐵)) = ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) |
13 | 3, 12 | eqtri 2108 | . . . 4 ⊢ (;10 · ;𝐴𝐵) = ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) |
14 | 6, 6, 8 | mulassi 7495 | . . . . . . 7 ⊢ ((;10 · ;10) · 𝐴) = (;10 · (;10 · 𝐴)) |
15 | 14 | eqcomi 2092 | . . . . . 6 ⊢ (;10 · (;10 · 𝐴)) = ((;10 · ;10) · 𝐴) |
16 | 6 | sqvali 10030 | . . . . . . . 8 ⊢ (;10↑2) = (;10 · ;10) |
17 | 16 | eqcomi 2092 | . . . . . . 7 ⊢ (;10 · ;10) = (;10↑2) |
18 | 17 | oveq1i 5662 | . . . . . 6 ⊢ ((;10 · ;10) · 𝐴) = ((;10↑2) · 𝐴) |
19 | 15, 18 | eqtri 2108 | . . . . 5 ⊢ (;10 · (;10 · 𝐴)) = ((;10↑2) · 𝐴) |
20 | 19 | oveq1i 5662 | . . . 4 ⊢ ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) = (((;10↑2) · 𝐴) + (;10 · 𝐵)) |
21 | 13, 20 | eqtri 2108 | . . 3 ⊢ (;10 · ;𝐴𝐵) = (((;10↑2) · 𝐴) + (;10 · 𝐵)) |
22 | 21 | oveq1i 5662 | . 2 ⊢ ((;10 · ;𝐴𝐵) + 𝐶) = ((((;10↑2) · 𝐴) + (;10 · 𝐵)) + 𝐶) |
23 | 1, 22 | eqtri 2108 | 1 ⊢ ;;𝐴𝐵𝐶 = ((((;10↑2) · 𝐴) + (;10 · 𝐵)) + 𝐶) |
Colors of variables: wff set class |
Syntax hints: = wceq 1289 ∈ wcel 1438 (class class class)co 5652 0cc0 7348 1c1 7349 + caddc 7351 · cmul 7353 2c2 8471 ℕ0cn0 8671 ;cdc 8875 ↑cexp 9950 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 ax-cnex 7434 ax-resscn 7435 ax-1cn 7436 ax-1re 7437 ax-icn 7438 ax-addcl 7439 ax-addrcl 7440 ax-mulcl 7441 ax-mulrcl 7442 ax-addcom 7443 ax-mulcom 7444 ax-addass 7445 ax-mulass 7446 ax-distr 7447 ax-i2m1 7448 ax-0lt1 7449 ax-1rid 7450 ax-0id 7451 ax-rnegex 7452 ax-precex 7453 ax-cnre 7454 ax-pre-ltirr 7455 ax-pre-ltwlin 7456 ax-pre-lttrn 7457 ax-pre-apti 7458 ax-pre-ltadd 7459 ax-pre-mulgt0 7460 ax-pre-mulext 7461 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-if 3394 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-id 4120 df-po 4123 df-iso 4124 df-iord 4193 df-on 4195 df-ilim 4196 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-frec 6156 df-pnf 7522 df-mnf 7523 df-xr 7524 df-ltxr 7525 df-le 7526 df-sub 7653 df-neg 7654 df-reap 8050 df-ap 8057 df-div 8138 df-inn 8421 df-2 8479 df-3 8480 df-4 8481 df-5 8482 df-6 8483 df-7 8484 df-8 8485 df-9 8486 df-n0 8672 df-z 8749 df-dec 8876 df-uz 9018 df-iseq 9849 df-seq3 9850 df-exp 9951 |
This theorem is referenced by: 3dvds2dec 11140 |
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