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Mathbox for Thierry Arnoux |
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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 12718 | . . 3 ⊢ ;𝐵𝐶 = ((;10 · 𝐵) + 𝐶) | |
2 | 1 | oveq2i 7437 | . 2 ⊢ ((;;100 · 𝐴) + ;𝐵𝐶) = ((;;100 · 𝐴) + ((;10 · 𝐵) + 𝐶)) |
3 | 10nn0 12733 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
4 | 3 | dec0u 12736 | . . . . 5 ⊢ (;10 · ;10) = ;;100 |
5 | 3 | nn0cni 12522 | . . . . . 6 ⊢ ;10 ∈ ℂ |
6 | 5, 5 | mulcli 11259 | . . . . 5 ⊢ (;10 · ;10) ∈ ℂ |
7 | 4, 6 | eqeltrri 2826 | . . . 4 ⊢ ;;100 ∈ ℂ |
8 | dfdec100.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
9 | 8 | nn0cni 12522 | . . . 4 ⊢ 𝐴 ∈ ℂ |
10 | 7, 9 | mulcli 11259 | . . 3 ⊢ (;;100 · 𝐴) ∈ ℂ |
11 | dfdec100.b | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
12 | 11 | nn0cni 12522 | . . . 4 ⊢ 𝐵 ∈ ℂ |
13 | 5, 12 | mulcli 11259 | . . 3 ⊢ (;10 · 𝐵) ∈ ℂ |
14 | dfdec100.c | . . . 4 ⊢ 𝐶 ∈ ℝ | |
15 | 14 | recni 11266 | . . 3 ⊢ 𝐶 ∈ ℂ |
16 | 10, 13, 15 | addassi 11262 | . 2 ⊢ (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) = ((;;100 · 𝐴) + ((;10 · 𝐵) + 𝐶)) |
17 | dfdec10 12718 | . . 3 ⊢ ;;𝐴𝐵𝐶 = ((;10 · ;𝐴𝐵) + 𝐶) | |
18 | dfdec10 12718 | . . . . . 6 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
19 | 18 | oveq2i 7437 | . . . . 5 ⊢ (;10 · ;𝐴𝐵) = (;10 · ((;10 · 𝐴) + 𝐵)) |
20 | 5, 9 | mulcli 11259 | . . . . . 6 ⊢ (;10 · 𝐴) ∈ ℂ |
21 | 5, 20, 12 | adddii 11264 | . . . . 5 ⊢ (;10 · ((;10 · 𝐴) + 𝐵)) = ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) |
22 | 5, 5, 9 | mulassi 11263 | . . . . . . 7 ⊢ ((;10 · ;10) · 𝐴) = (;10 · (;10 · 𝐴)) |
23 | 4 | oveq1i 7436 | . . . . . . 7 ⊢ ((;10 · ;10) · 𝐴) = (;;100 · 𝐴) |
24 | 22, 23 | eqtr3i 2758 | . . . . . 6 ⊢ (;10 · (;10 · 𝐴)) = (;;100 · 𝐴) |
25 | 24 | oveq1i 7436 | . . . . 5 ⊢ ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) = ((;;100 · 𝐴) + (;10 · 𝐵)) |
26 | 19, 21, 25 | 3eqtri 2760 | . . . 4 ⊢ (;10 · ;𝐴𝐵) = ((;;100 · 𝐴) + (;10 · 𝐵)) |
27 | 26 | oveq1i 7436 | . . 3 ⊢ ((;10 · ;𝐴𝐵) + 𝐶) = (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) |
28 | 17, 27 | eqtr2i 2757 | . 2 ⊢ (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) = ;;𝐴𝐵𝐶 |
29 | 2, 16, 28 | 3eqtr2ri 2763 | 1 ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 ℕ0cn0 12510 ;cdc 12715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-ltxr 11291 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-dec 12716 |
This theorem is referenced by: dpmul100 32641 dpmul1000 32643 dpmul4 32658 |
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