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Mirrors > Home > MPE Home > Th. List > decmul10add | Structured version Visualization version GIF version |
Description: A multiplication of a number and a numeral expressed as addition with first summand as multiple of 10. (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decmul10add.1 | ⊢ 𝐴 ∈ ℕ0 |
decmul10add.2 | ⊢ 𝐵 ∈ ℕ0 |
decmul10add.3 | ⊢ 𝑀 ∈ ℕ0 |
decmul10add.4 | ⊢ 𝐸 = (𝑀 · 𝐴) |
decmul10add.5 | ⊢ 𝐹 = (𝑀 · 𝐵) |
Ref | Expression |
---|---|
decmul10add | ⊢ (𝑀 · ;𝐴𝐵) = (;𝐸0 + 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdec10 12104 | . . 3 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
2 | 1 | oveq2i 7169 | . 2 ⊢ (𝑀 · ;𝐴𝐵) = (𝑀 · ((;10 · 𝐴) + 𝐵)) |
3 | decmul10add.3 | . . . 4 ⊢ 𝑀 ∈ ℕ0 | |
4 | 3 | nn0cni 11912 | . . 3 ⊢ 𝑀 ∈ ℂ |
5 | 10nn0 12119 | . . . . 5 ⊢ ;10 ∈ ℕ0 | |
6 | decmul10add.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
7 | 5, 6 | nn0mulcli 11938 | . . . 4 ⊢ (;10 · 𝐴) ∈ ℕ0 |
8 | 7 | nn0cni 11912 | . . 3 ⊢ (;10 · 𝐴) ∈ ℂ |
9 | decmul10add.2 | . . . 4 ⊢ 𝐵 ∈ ℕ0 | |
10 | 9 | nn0cni 11912 | . . 3 ⊢ 𝐵 ∈ ℂ |
11 | 4, 8, 10 | adddii 10655 | . 2 ⊢ (𝑀 · ((;10 · 𝐴) + 𝐵)) = ((𝑀 · (;10 · 𝐴)) + (𝑀 · 𝐵)) |
12 | 5 | nn0cni 11912 | . . . . 5 ⊢ ;10 ∈ ℂ |
13 | 6 | nn0cni 11912 | . . . . 5 ⊢ 𝐴 ∈ ℂ |
14 | 4, 12, 13 | mul12i 10837 | . . . 4 ⊢ (𝑀 · (;10 · 𝐴)) = (;10 · (𝑀 · 𝐴)) |
15 | 3, 6 | nn0mulcli 11938 | . . . . 5 ⊢ (𝑀 · 𝐴) ∈ ℕ0 |
16 | 15 | dec0u 12122 | . . . 4 ⊢ (;10 · (𝑀 · 𝐴)) = ;(𝑀 · 𝐴)0 |
17 | decmul10add.4 | . . . . . 6 ⊢ 𝐸 = (𝑀 · 𝐴) | |
18 | 17 | eqcomi 2832 | . . . . 5 ⊢ (𝑀 · 𝐴) = 𝐸 |
19 | 18 | deceq1i 12108 | . . . 4 ⊢ ;(𝑀 · 𝐴)0 = ;𝐸0 |
20 | 14, 16, 19 | 3eqtri 2850 | . . 3 ⊢ (𝑀 · (;10 · 𝐴)) = ;𝐸0 |
21 | decmul10add.5 | . . . 4 ⊢ 𝐹 = (𝑀 · 𝐵) | |
22 | 21 | eqcomi 2832 | . . 3 ⊢ (𝑀 · 𝐵) = 𝐹 |
23 | 20, 22 | oveq12i 7170 | . 2 ⊢ ((𝑀 · (;10 · 𝐴)) + (𝑀 · 𝐵)) = (;𝐸0 + 𝐹) |
24 | 2, 11, 23 | 3eqtri 2850 | 1 ⊢ (𝑀 · ;𝐴𝐵) = (;𝐸0 + 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 ℕ0cn0 11900 ;cdc 12101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-dec 12102 |
This theorem is referenced by: fmtno5lem4 43725 fmtno4prmfac 43741 fmtno5fac 43751 |
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