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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 12680 | . . 3 ⊢ ;𝐵𝐶 = ((;10 · 𝐵) + 𝐶) | |
2 | 1 | oveq2i 7420 | . 2 ⊢ ((;;100 · 𝐴) + ;𝐵𝐶) = ((;;100 · 𝐴) + ((;10 · 𝐵) + 𝐶)) |
3 | 10nn0 12695 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
4 | 3 | dec0u 12698 | . . . . 5 ⊢ (;10 · ;10) = ;;100 |
5 | 3 | nn0cni 12484 | . . . . . 6 ⊢ ;10 ∈ ℂ |
6 | 5, 5 | mulcli 11221 | . . . . 5 ⊢ (;10 · ;10) ∈ ℂ |
7 | 4, 6 | eqeltrri 2831 | . . . 4 ⊢ ;;100 ∈ ℂ |
8 | dfdec100.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
9 | 8 | nn0cni 12484 | . . . 4 ⊢ 𝐴 ∈ ℂ |
10 | 7, 9 | mulcli 11221 | . . 3 ⊢ (;;100 · 𝐴) ∈ ℂ |
11 | dfdec100.b | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
12 | 11 | nn0cni 12484 | . . . 4 ⊢ 𝐵 ∈ ℂ |
13 | 5, 12 | mulcli 11221 | . . 3 ⊢ (;10 · 𝐵) ∈ ℂ |
14 | dfdec100.c | . . . 4 ⊢ 𝐶 ∈ ℝ | |
15 | 14 | recni 11228 | . . 3 ⊢ 𝐶 ∈ ℂ |
16 | 10, 13, 15 | addassi 11224 | . 2 ⊢ (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) = ((;;100 · 𝐴) + ((;10 · 𝐵) + 𝐶)) |
17 | dfdec10 12680 | . . 3 ⊢ ;;𝐴𝐵𝐶 = ((;10 · ;𝐴𝐵) + 𝐶) | |
18 | dfdec10 12680 | . . . . . 6 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
19 | 18 | oveq2i 7420 | . . . . 5 ⊢ (;10 · ;𝐴𝐵) = (;10 · ((;10 · 𝐴) + 𝐵)) |
20 | 5, 9 | mulcli 11221 | . . . . . 6 ⊢ (;10 · 𝐴) ∈ ℂ |
21 | 5, 20, 12 | adddii 11226 | . . . . 5 ⊢ (;10 · ((;10 · 𝐴) + 𝐵)) = ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) |
22 | 5, 5, 9 | mulassi 11225 | . . . . . . 7 ⊢ ((;10 · ;10) · 𝐴) = (;10 · (;10 · 𝐴)) |
23 | 4 | oveq1i 7419 | . . . . . . 7 ⊢ ((;10 · ;10) · 𝐴) = (;;100 · 𝐴) |
24 | 22, 23 | eqtr3i 2763 | . . . . . 6 ⊢ (;10 · (;10 · 𝐴)) = (;;100 · 𝐴) |
25 | 24 | oveq1i 7419 | . . . . 5 ⊢ ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) = ((;;100 · 𝐴) + (;10 · 𝐵)) |
26 | 19, 21, 25 | 3eqtri 2765 | . . . 4 ⊢ (;10 · ;𝐴𝐵) = ((;;100 · 𝐴) + (;10 · 𝐵)) |
27 | 26 | oveq1i 7419 | . . 3 ⊢ ((;10 · ;𝐴𝐵) + 𝐶) = (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) |
28 | 17, 27 | eqtr2i 2762 | . 2 ⊢ (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) = ;;𝐴𝐵𝐶 |
29 | 2, 16, 28 | 3eqtr2ri 2768 | 1 ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 ℕ0cn0 12472 ;cdc 12677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-dec 12678 |
This theorem is referenced by: dpmul100 32063 dpmul1000 32065 dpmul4 32080 |
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