![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 3dec | Structured version Visualization version 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 12089 | . 2 ⊢ ;;𝐴𝐵𝐶 = ((;10 · ;𝐴𝐵) + 𝐶) | |
2 | dfdec10 12089 | . . . . . 6 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
3 | 2 | oveq2i 7146 | . . . . 5 ⊢ (;10 · ;𝐴𝐵) = (;10 · ((;10 · 𝐴) + 𝐵)) |
4 | 10nn 12102 | . . . . . . 7 ⊢ ;10 ∈ ℕ | |
5 | 4 | nncni 11635 | . . . . . 6 ⊢ ;10 ∈ ℂ |
6 | 3dec.a | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
7 | 6 | nn0cni 11897 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
8 | 5, 7 | mulcli 10637 | . . . . . 6 ⊢ (;10 · 𝐴) ∈ ℂ |
9 | 3dec.b | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
10 | 9 | nn0cni 11897 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
11 | 5, 8, 10 | adddii 10642 | . . . . 5 ⊢ (;10 · ((;10 · 𝐴) + 𝐵)) = ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) |
12 | 3, 11 | eqtri 2821 | . . . 4 ⊢ (;10 · ;𝐴𝐵) = ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) |
13 | 5, 5, 7 | mulassi 10641 | . . . . . . 7 ⊢ ((;10 · ;10) · 𝐴) = (;10 · (;10 · 𝐴)) |
14 | 13 | eqcomi 2807 | . . . . . 6 ⊢ (;10 · (;10 · 𝐴)) = ((;10 · ;10) · 𝐴) |
15 | 5 | sqvali 13539 | . . . . . . . 8 ⊢ (;10↑2) = (;10 · ;10) |
16 | 15 | eqcomi 2807 | . . . . . . 7 ⊢ (;10 · ;10) = (;10↑2) |
17 | 16 | oveq1i 7145 | . . . . . 6 ⊢ ((;10 · ;10) · 𝐴) = ((;10↑2) · 𝐴) |
18 | 14, 17 | eqtri 2821 | . . . . 5 ⊢ (;10 · (;10 · 𝐴)) = ((;10↑2) · 𝐴) |
19 | 18 | oveq1i 7145 | . . . 4 ⊢ ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) = (((;10↑2) · 𝐴) + (;10 · 𝐵)) |
20 | 12, 19 | eqtri 2821 | . . 3 ⊢ (;10 · ;𝐴𝐵) = (((;10↑2) · 𝐴) + (;10 · 𝐵)) |
21 | 20 | oveq1i 7145 | . 2 ⊢ ((;10 · ;𝐴𝐵) + 𝐶) = ((((;10↑2) · 𝐴) + (;10 · 𝐵)) + 𝐶) |
22 | 1, 21 | eqtri 2821 | 1 ⊢ ;;𝐴𝐵𝐶 = ((((;10↑2) · 𝐴) + (;10 · 𝐵)) + 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 2c2 11680 ℕ0cn0 11885 ;cdc 12086 ↑cexp 13425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-seq 13365 df-exp 13426 |
This theorem is referenced by: 3dvds2dec 15674 |
Copyright terms: Public domain | W3C validator |