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Mirrors > Home > MPE Home > Th. List > decsplit | Structured version Visualization version GIF version |
Description: Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
decsplit0.1 | ⊢ 𝐴 ∈ ℕ0 |
decsplit.2 | ⊢ 𝐵 ∈ ℕ0 |
decsplit.3 | ⊢ 𝐷 ∈ ℕ0 |
decsplit.4 | ⊢ 𝑀 ∈ ℕ0 |
decsplit.5 | ⊢ (𝑀 + 1) = 𝑁 |
decsplit.6 | ⊢ ((𝐴 · (;10↑𝑀)) + 𝐵) = 𝐶 |
Ref | Expression |
---|---|
decsplit | ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ;𝐶𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 12778 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
2 | decsplit0.1 | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
3 | decsplit.4 | . . . . . . . 8 ⊢ 𝑀 ∈ ℕ0 | |
4 | 1, 3 | nn0expcli 14141 | . . . . . . 7 ⊢ (;10↑𝑀) ∈ ℕ0 |
5 | 2, 4 | nn0mulcli 12593 | . . . . . 6 ⊢ (𝐴 · (;10↑𝑀)) ∈ ℕ0 |
6 | 1, 5 | nn0mulcli 12593 | . . . . 5 ⊢ (;10 · (𝐴 · (;10↑𝑀))) ∈ ℕ0 |
7 | 6 | nn0cni 12567 | . . . 4 ⊢ (;10 · (𝐴 · (;10↑𝑀))) ∈ ℂ |
8 | decsplit.2 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
9 | 1, 8 | nn0mulcli 12593 | . . . . 5 ⊢ (;10 · 𝐵) ∈ ℕ0 |
10 | 9 | nn0cni 12567 | . . . 4 ⊢ (;10 · 𝐵) ∈ ℂ |
11 | decsplit.3 | . . . . 5 ⊢ 𝐷 ∈ ℕ0 | |
12 | 11 | nn0cni 12567 | . . . 4 ⊢ 𝐷 ∈ ℂ |
13 | 7, 10, 12 | addassi 11302 | . . 3 ⊢ (((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) + 𝐷) = ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) |
14 | 1 | nn0cni 12567 | . . . . . 6 ⊢ ;10 ∈ ℂ |
15 | 5 | nn0cni 12567 | . . . . . 6 ⊢ (𝐴 · (;10↑𝑀)) ∈ ℂ |
16 | 8 | nn0cni 12567 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
17 | 14, 15, 16 | adddii 11304 | . . . . 5 ⊢ (;10 · ((𝐴 · (;10↑𝑀)) + 𝐵)) = ((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) |
18 | decsplit.6 | . . . . . 6 ⊢ ((𝐴 · (;10↑𝑀)) + 𝐵) = 𝐶 | |
19 | 18 | oveq2i 7461 | . . . . 5 ⊢ (;10 · ((𝐴 · (;10↑𝑀)) + 𝐵)) = (;10 · 𝐶) |
20 | 17, 19 | eqtr3i 2770 | . . . 4 ⊢ ((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) = (;10 · 𝐶) |
21 | 20 | oveq1i 7460 | . . 3 ⊢ (((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) + 𝐷) = ((;10 · 𝐶) + 𝐷) |
22 | 13, 21 | eqtr3i 2770 | . 2 ⊢ ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) = ((;10 · 𝐶) + 𝐷) |
23 | decsplit.5 | . . . . . 6 ⊢ (𝑀 + 1) = 𝑁 | |
24 | 4 | nn0cni 12567 | . . . . . . 7 ⊢ (;10↑𝑀) ∈ ℂ |
25 | 24, 14 | mulcomi 11300 | . . . . . 6 ⊢ ((;10↑𝑀) · ;10) = (;10 · (;10↑𝑀)) |
26 | 1, 3, 23, 25 | numexpp1 17127 | . . . . 5 ⊢ (;10↑𝑁) = (;10 · (;10↑𝑀)) |
27 | 26 | oveq2i 7461 | . . . 4 ⊢ (𝐴 · (;10↑𝑁)) = (𝐴 · (;10 · (;10↑𝑀))) |
28 | 2 | nn0cni 12567 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
29 | 28, 14, 24 | mul12i 11487 | . . . 4 ⊢ (𝐴 · (;10 · (;10↑𝑀))) = (;10 · (𝐴 · (;10↑𝑀))) |
30 | 27, 29 | eqtri 2768 | . . 3 ⊢ (𝐴 · (;10↑𝑁)) = (;10 · (𝐴 · (;10↑𝑀))) |
31 | dfdec10 12763 | . . 3 ⊢ ;𝐵𝐷 = ((;10 · 𝐵) + 𝐷) | |
32 | 30, 31 | oveq12i 7462 | . 2 ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) |
33 | dfdec10 12763 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
34 | 22, 32, 33 | 3eqtr4i 2778 | 1 ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ;𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2108 (class class class)co 7450 0cc0 11186 1c1 11187 + caddc 11189 · cmul 11191 ℕ0cn0 12555 ;cdc 12760 ↑cexp 14114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-z 12642 df-dec 12761 df-uz 12906 df-seq 14055 df-exp 14115 |
This theorem is referenced by: (None) |
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