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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 12691 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
2 | decsplit0.1 | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
3 | decsplit.4 | . . . . . . . 8 ⊢ 𝑀 ∈ ℕ0 | |
4 | 1, 3 | nn0expcli 14050 | . . . . . . 7 ⊢ (;10↑𝑀) ∈ ℕ0 |
5 | 2, 4 | nn0mulcli 12506 | . . . . . 6 ⊢ (𝐴 · (;10↑𝑀)) ∈ ℕ0 |
6 | 1, 5 | nn0mulcli 12506 | . . . . 5 ⊢ (;10 · (𝐴 · (;10↑𝑀))) ∈ ℕ0 |
7 | 6 | nn0cni 12480 | . . . 4 ⊢ (;10 · (𝐴 · (;10↑𝑀))) ∈ ℂ |
8 | decsplit.2 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 | |
9 | 1, 8 | nn0mulcli 12506 | . . . . 5 ⊢ (;10 · 𝐵) ∈ ℕ0 |
10 | 9 | nn0cni 12480 | . . . 4 ⊢ (;10 · 𝐵) ∈ ℂ |
11 | decsplit.3 | . . . . 5 ⊢ 𝐷 ∈ ℕ0 | |
12 | 11 | nn0cni 12480 | . . . 4 ⊢ 𝐷 ∈ ℂ |
13 | 7, 10, 12 | addassi 11220 | . . 3 ⊢ (((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) + 𝐷) = ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) |
14 | 1 | nn0cni 12480 | . . . . . 6 ⊢ ;10 ∈ ℂ |
15 | 5 | nn0cni 12480 | . . . . . 6 ⊢ (𝐴 · (;10↑𝑀)) ∈ ℂ |
16 | 8 | nn0cni 12480 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
17 | 14, 15, 16 | adddii 11222 | . . . . 5 ⊢ (;10 · ((𝐴 · (;10↑𝑀)) + 𝐵)) = ((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) |
18 | decsplit.6 | . . . . . 6 ⊢ ((𝐴 · (;10↑𝑀)) + 𝐵) = 𝐶 | |
19 | 18 | oveq2i 7412 | . . . . 5 ⊢ (;10 · ((𝐴 · (;10↑𝑀)) + 𝐵)) = (;10 · 𝐶) |
20 | 17, 19 | eqtr3i 2754 | . . . 4 ⊢ ((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) = (;10 · 𝐶) |
21 | 20 | oveq1i 7411 | . . 3 ⊢ (((;10 · (𝐴 · (;10↑𝑀))) + (;10 · 𝐵)) + 𝐷) = ((;10 · 𝐶) + 𝐷) |
22 | 13, 21 | eqtr3i 2754 | . 2 ⊢ ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) = ((;10 · 𝐶) + 𝐷) |
23 | decsplit.5 | . . . . . 6 ⊢ (𝑀 + 1) = 𝑁 | |
24 | 4 | nn0cni 12480 | . . . . . . 7 ⊢ (;10↑𝑀) ∈ ℂ |
25 | 24, 14 | mulcomi 11218 | . . . . . 6 ⊢ ((;10↑𝑀) · ;10) = (;10 · (;10↑𝑀)) |
26 | 1, 3, 23, 25 | numexpp1 17009 | . . . . 5 ⊢ (;10↑𝑁) = (;10 · (;10↑𝑀)) |
27 | 26 | oveq2i 7412 | . . . 4 ⊢ (𝐴 · (;10↑𝑁)) = (𝐴 · (;10 · (;10↑𝑀))) |
28 | 2 | nn0cni 12480 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
29 | 28, 14, 24 | mul12i 11405 | . . . 4 ⊢ (𝐴 · (;10 · (;10↑𝑀))) = (;10 · (𝐴 · (;10↑𝑀))) |
30 | 27, 29 | eqtri 2752 | . . 3 ⊢ (𝐴 · (;10↑𝑁)) = (;10 · (𝐴 · (;10↑𝑀))) |
31 | dfdec10 12676 | . . 3 ⊢ ;𝐵𝐷 = ((;10 · 𝐵) + 𝐷) | |
32 | 30, 31 | oveq12i 7413 | . 2 ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ((;10 · (𝐴 · (;10↑𝑀))) + ((;10 · 𝐵) + 𝐷)) |
33 | dfdec10 12676 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
34 | 22, 32, 33 | 3eqtr4i 2762 | 1 ⊢ ((𝐴 · (;10↑𝑁)) + ;𝐵𝐷) = ;𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 (class class class)co 7401 0cc0 11105 1c1 11106 + caddc 11108 · cmul 11110 ℕ0cn0 12468 ;cdc 12673 ↑cexp 14023 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-seq 13963 df-exp 14024 |
This theorem is referenced by: (None) |
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