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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 12733 | . . 3 ⊢ ;𝐵𝐶 = ((;10 · 𝐵) + 𝐶) | |
2 | 1 | oveq2i 7441 | . 2 ⊢ ((;;100 · 𝐴) + ;𝐵𝐶) = ((;;100 · 𝐴) + ((;10 · 𝐵) + 𝐶)) |
3 | 10nn0 12748 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
4 | 3 | dec0u 12751 | . . . . 5 ⊢ (;10 · ;10) = ;;100 |
5 | 3 | nn0cni 12535 | . . . . . 6 ⊢ ;10 ∈ ℂ |
6 | 5, 5 | mulcli 11265 | . . . . 5 ⊢ (;10 · ;10) ∈ ℂ |
7 | 4, 6 | eqeltrri 2835 | . . . 4 ⊢ ;;100 ∈ ℂ |
8 | dfdec100.a | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
9 | 8 | nn0cni 12535 | . . . 4 ⊢ 𝐴 ∈ ℂ |
10 | 7, 9 | mulcli 11265 | . . 3 ⊢ (;;100 · 𝐴) ∈ ℂ |
11 | dfdec100.b | . . . . 5 ⊢ 𝐵 ∈ ℕ0 | |
12 | 11 | nn0cni 12535 | . . . 4 ⊢ 𝐵 ∈ ℂ |
13 | 5, 12 | mulcli 11265 | . . 3 ⊢ (;10 · 𝐵) ∈ ℂ |
14 | dfdec100.c | . . . 4 ⊢ 𝐶 ∈ ℝ | |
15 | 14 | recni 11272 | . . 3 ⊢ 𝐶 ∈ ℂ |
16 | 10, 13, 15 | addassi 11268 | . 2 ⊢ (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) = ((;;100 · 𝐴) + ((;10 · 𝐵) + 𝐶)) |
17 | dfdec10 12733 | . . 3 ⊢ ;;𝐴𝐵𝐶 = ((;10 · ;𝐴𝐵) + 𝐶) | |
18 | dfdec10 12733 | . . . . . 6 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
19 | 18 | oveq2i 7441 | . . . . 5 ⊢ (;10 · ;𝐴𝐵) = (;10 · ((;10 · 𝐴) + 𝐵)) |
20 | 5, 9 | mulcli 11265 | . . . . . 6 ⊢ (;10 · 𝐴) ∈ ℂ |
21 | 5, 20, 12 | adddii 11270 | . . . . 5 ⊢ (;10 · ((;10 · 𝐴) + 𝐵)) = ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) |
22 | 5, 5, 9 | mulassi 11269 | . . . . . . 7 ⊢ ((;10 · ;10) · 𝐴) = (;10 · (;10 · 𝐴)) |
23 | 4 | oveq1i 7440 | . . . . . . 7 ⊢ ((;10 · ;10) · 𝐴) = (;;100 · 𝐴) |
24 | 22, 23 | eqtr3i 2764 | . . . . . 6 ⊢ (;10 · (;10 · 𝐴)) = (;;100 · 𝐴) |
25 | 24 | oveq1i 7440 | . . . . 5 ⊢ ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) = ((;;100 · 𝐴) + (;10 · 𝐵)) |
26 | 19, 21, 25 | 3eqtri 2766 | . . . 4 ⊢ (;10 · ;𝐴𝐵) = ((;;100 · 𝐴) + (;10 · 𝐵)) |
27 | 26 | oveq1i 7440 | . . 3 ⊢ ((;10 · ;𝐴𝐵) + 𝐶) = (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) |
28 | 17, 27 | eqtr2i 2763 | . 2 ⊢ (((;;100 · 𝐴) + (;10 · 𝐵)) + 𝐶) = ;;𝐴𝐵𝐶 |
29 | 2, 16, 28 | 3eqtr2ri 2769 | 1 ⊢ ;;𝐴𝐵𝐶 = ((;;100 · 𝐴) + ;𝐵𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 ℕ0cn0 12523 ;cdc 12730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-ltxr 11297 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-dec 12731 |
This theorem is referenced by: dpmul100 32863 dpmul1000 32865 dpmul4 32880 |
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