Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dfdec100 | Structured version Visualization version GIF version |
Description: Split the hundreds from a decimal value. (Contributed by Thierry Arnoux, 25-Dec-2021.) |
Ref | Expression |
---|---|
dfdec100.a | ⊢ 𝐴 ∈ ℕ0 |
dfdec100.b | ⊢ 𝐵 ∈ ℕ0 |
dfdec100.c | ⊢ 𝐶 ∈ ℝ |
Ref | Expression |
---|---|
dfdec100 | ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdec10 12325 | . . 3 ⊢ ;𝐵𝐶 = ((;10 · 𝐵) + 𝐶) | |
2 | 1 | oveq2i 7245 | . 2 ⊢ ((;;100 · 𝐴) + ;𝐵𝐶) = ((;;100 · 𝐴) + ((;10 · 𝐵) + 𝐶)) |
3 | 10nn0 12340 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
4 | 3 | dec0u 12343 | . . . . 5 ⊢ (;10 · ;10) = ;;100 |
5 | 3 | nn0cni 12131 | . . . . . 6 ⊢ ;10 ∈ ℂ |
6 | 5, 5 | mulcli 10869 | . . . . 5 ⊢ (;10 · ;10) ∈ ℂ |
7 | 4, 6 | eqeltrri 2837 | . . . 4 ⊢ ;;100 ∈ ℂ |
8 | dfdec100.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
9 | 8 | nn0cni 12131 | . . . 4 ⊢ 𝐴 ∈ ℂ |
10 | 7, 9 | mulcli 10869 | . . 3 ⊢ (;;100 · 𝐴) ∈ ℂ |
11 | dfdec100.b | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
12 | 11 | nn0cni 12131 | . . . 4 ⊢ 𝐵 ∈ ℂ |
13 | 5, 12 | mulcli 10869 | . . 3 ⊢ (;10 · 𝐵) ∈ ℂ |
14 | dfdec100.c | . . . 4 ⊢ 𝐶 ∈ ℝ | |
15 | 14 | recni 10876 | . . 3 ⊢ 𝐶 ∈ ℂ |
16 | 10, 13, 15 | addassi 10872 | . 2 ⊢ (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) = ((;;100 · 𝐴) + ((;10 · 𝐵) + 𝐶)) |
17 | dfdec10 12325 | . . 3 ⊢ ;;𝐴𝐵𝐶 = ((;10 · ;𝐴𝐵) + 𝐶) | |
18 | dfdec10 12325 | . . . . . 6 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
19 | 18 | oveq2i 7245 | . . . . 5 ⊢ (;10 · ;𝐴𝐵) = (;10 · ((;10 · 𝐴) + 𝐵)) |
20 | 5, 9 | mulcli 10869 | . . . . . 6 ⊢ (;10 · 𝐴) ∈ ℂ |
21 | 5, 20, 12 | adddii 10874 | . . . . 5 ⊢ (;10 · ((;10 · 𝐴) + 𝐵)) = ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) |
22 | 5, 5, 9 | mulassi 10873 | . . . . . . 7 ⊢ ((;10 · ;10) · 𝐴) = (;10 · (;10 · 𝐴)) |
23 | 4 | oveq1i 7244 | . . . . . . 7 ⊢ ((;10 · ;10) · 𝐴) = (;;100 · 𝐴) |
24 | 22, 23 | eqtr3i 2769 | . . . . . 6 ⊢ (;10 · (;10 · 𝐴)) = (;;100 · 𝐴) |
25 | 24 | oveq1i 7244 | . . . . 5 ⊢ ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) = ((;;100 · 𝐴) + (;10 · 𝐵)) |
26 | 19, 21, 25 | 3eqtri 2771 | . . . 4 ⊢ (;10 · ;𝐴𝐵) = ((;;100 · 𝐴) + (;10 · 𝐵)) |
27 | 26 | oveq1i 7244 | . . 3 ⊢ ((;10 · ;𝐴𝐵) + 𝐶) = (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) |
28 | 17, 27 | eqtr2i 2768 | . 2 ⊢ (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) = ;;𝐴𝐵𝐶 |
29 | 2, 16, 28 | 3eqtr2ri 2774 | 1 ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 (class class class)co 7234 ℂcc 10756 ℝcr 10757 0cc0 10758 1c1 10759 + caddc 10761 · cmul 10763 ℕ0cn0 12119 ;cdc 12322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-ov 7237 df-om 7666 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-pnf 10898 df-mnf 10899 df-ltxr 10901 df-nn 11860 df-2 11922 df-3 11923 df-4 11924 df-5 11925 df-6 11926 df-7 11927 df-8 11928 df-9 11929 df-n0 12120 df-dec 12323 |
This theorem is referenced by: dpmul100 30922 dpmul1000 30924 dpmul4 30939 |
Copyright terms: Public domain | W3C validator |