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Mirrors > Home > MPE Home > Th. List > 3dec | Structured version Visualization version GIF version |
Description: A "decimal constructor" which is used to build up "decimal integers" or "numeric terms" in base 10 with 3 "digits". (Contributed by AV, 14-Jun-2021.) (Revised by AV, 1-Aug-2021.) |
Ref | Expression |
---|---|
3dec.a | ⊢ 𝐴 ∈ ℕ0 |
3dec.b | ⊢ 𝐵 ∈ ℕ0 |
Ref | Expression |
---|---|
3dec | ⊢ ;;𝐴𝐵𝐶 = ((((;10↑2) · 𝐴) + (;10 · 𝐵)) + 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdec10 12621 | . 2 ⊢ ;;𝐴𝐵𝐶 = ((;10 · ;𝐴𝐵) + 𝐶) | |
2 | dfdec10 12621 | . . . . . 6 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
3 | 2 | oveq2i 7368 | . . . . 5 ⊢ (;10 · ;𝐴𝐵) = (;10 · ((;10 · 𝐴) + 𝐵)) |
4 | 10nn 12634 | . . . . . . 7 ⊢ ;10 ∈ ℕ | |
5 | 4 | nncni 12163 | . . . . . 6 ⊢ ;10 ∈ ℂ |
6 | 3dec.a | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
7 | 6 | nn0cni 12425 | . . . . . . 7 ⊢ 𝐴 ∈ ℂ |
8 | 5, 7 | mulcli 11162 | . . . . . 6 ⊢ (;10 · 𝐴) ∈ ℂ |
9 | 3dec.b | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
10 | 9 | nn0cni 12425 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
11 | 5, 8, 10 | adddii 11167 | . . . . 5 ⊢ (;10 · ((;10 · 𝐴) + 𝐵)) = ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) |
12 | 3, 11 | eqtri 2764 | . . . 4 ⊢ (;10 · ;𝐴𝐵) = ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) |
13 | 5, 5, 7 | mulassi 11166 | . . . . . . 7 ⊢ ((;10 · ;10) · 𝐴) = (;10 · (;10 · 𝐴)) |
14 | 13 | eqcomi 2745 | . . . . . 6 ⊢ (;10 · (;10 · 𝐴)) = ((;10 · ;10) · 𝐴) |
15 | 5 | sqvali 14084 | . . . . . . . 8 ⊢ (;10↑2) = (;10 · ;10) |
16 | 15 | eqcomi 2745 | . . . . . . 7 ⊢ (;10 · ;10) = (;10↑2) |
17 | 16 | oveq1i 7367 | . . . . . 6 ⊢ ((;10 · ;10) · 𝐴) = ((;10↑2) · 𝐴) |
18 | 14, 17 | eqtri 2764 | . . . . 5 ⊢ (;10 · (;10 · 𝐴)) = ((;10↑2) · 𝐴) |
19 | 18 | oveq1i 7367 | . . . 4 ⊢ ((;10 · (;10 · 𝐴)) + (;10 · 𝐵)) = (((;10↑2) · 𝐴) + (;10 · 𝐵)) |
20 | 12, 19 | eqtri 2764 | . . 3 ⊢ (;10 · ;𝐴𝐵) = (((;10↑2) · 𝐴) + (;10 · 𝐵)) |
21 | 20 | oveq1i 7367 | . 2 ⊢ ((;10 · ;𝐴𝐵) + 𝐶) = ((((;10↑2) · 𝐴) + (;10 · 𝐵)) + 𝐶) |
22 | 1, 21 | eqtri 2764 | 1 ⊢ ;;𝐴𝐵𝐶 = ((((;10↑2) · 𝐴) + (;10 · 𝐵)) + 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 (class class class)co 7357 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 2c2 12208 ℕ0cn0 12413 ;cdc 12618 ↑cexp 13967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-seq 13907 df-exp 13968 |
This theorem is referenced by: 3dvds2dec 16215 |
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