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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0dp2dp | Structured version Visualization version GIF version |
Description: Multiply by 10 a decimal expansion which starts with a zero. (Contributed by Thierry Arnoux, 16-Dec-2021.) |
Ref | Expression |
---|---|
0dp2dp.a | ⊢ 𝐴 ∈ ℕ0 |
0dp2dp.b | ⊢ 𝐵 ∈ ℝ+ |
Ref | Expression |
---|---|
0dp2dp | ⊢ ((0._𝐴𝐵) · ;10) = (𝐴.𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0dp2dp.a | . . . 4 ⊢ 𝐴 ∈ ℕ0 | |
2 | 0dp2dp.b | . . . 4 ⊢ 𝐵 ∈ ℝ+ | |
3 | 0p1e1 11753 | . . . 4 ⊢ (0 + 1) = 1 | |
4 | 0z 11986 | . . . 4 ⊢ 0 ∈ ℤ | |
5 | 1z 12006 | . . . 4 ⊢ 1 ∈ ℤ | |
6 | 1, 2, 3, 4, 5 | dpexpp1 30579 | . . 3 ⊢ ((𝐴.𝐵) · (;10↑0)) = ((0._𝐴𝐵) · (;10↑1)) |
7 | 10nn0 12110 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
8 | 7 | nn0cni 11903 | . . . . 5 ⊢ ;10 ∈ ℂ |
9 | exp0 13427 | . . . . 5 ⊢ (;10 ∈ ℂ → (;10↑0) = 1) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ (;10↑0) = 1 |
11 | 10 | oveq2i 7161 | . . 3 ⊢ ((𝐴.𝐵) · (;10↑0)) = ((𝐴.𝐵) · 1) |
12 | exp1 13429 | . . . . 5 ⊢ (;10 ∈ ℂ → (;10↑1) = ;10) | |
13 | 8, 12 | ax-mp 5 | . . . 4 ⊢ (;10↑1) = ;10 |
14 | 13 | oveq2i 7161 | . . 3 ⊢ ((0._𝐴𝐵) · (;10↑1)) = ((0._𝐴𝐵) · ;10) |
15 | 6, 11, 14 | 3eqtr3ri 2853 | . 2 ⊢ ((0._𝐴𝐵) · ;10) = ((𝐴.𝐵) · 1) |
16 | 1, 2 | rpdpcl 30574 | . . . 4 ⊢ (𝐴.𝐵) ∈ ℝ+ |
17 | rpcn 12393 | . . . 4 ⊢ ((𝐴.𝐵) ∈ ℝ+ → (𝐴.𝐵) ∈ ℂ) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ (𝐴.𝐵) ∈ ℂ |
19 | mulid1 10633 | . . 3 ⊢ ((𝐴.𝐵) ∈ ℂ → ((𝐴.𝐵) · 1) = (𝐴.𝐵)) | |
20 | 18, 19 | ax-mp 5 | . 2 ⊢ ((𝐴.𝐵) · 1) = (𝐴.𝐵) |
21 | 15, 20 | eqtri 2844 | 1 ⊢ ((0._𝐴𝐵) · ;10) = (𝐴.𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 · cmul 10536 ℕ0cn0 11891 ;cdc 12092 ℝ+crp 12383 ↑cexp 13423 _cdp2 30542 .cdp 30559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-dp2 30543 df-dp 30560 |
This theorem is referenced by: hgt750lem 31917 |
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