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Mirrors > Home > MPE Home > Th. List > decmul2c | Structured version Visualization version GIF version |
Description: The product of a numeral with a number (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
decmul1.p | ⊢ 𝑃 ∈ ℕ0 |
decmul1.a | ⊢ 𝐴 ∈ ℕ0 |
decmul1.b | ⊢ 𝐵 ∈ ℕ0 |
decmul1.n | ⊢ 𝑁 = ;𝐴𝐵 |
decmul1.0 | ⊢ 𝐷 ∈ ℕ0 |
decmul1c.e | ⊢ 𝐸 ∈ ℕ0 |
decmul2c.c | ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 |
decmul2c.2 | ⊢ (𝑃 · 𝐵) = ;𝐸𝐷 |
Ref | Expression |
---|---|
decmul2c | ⊢ (𝑃 · 𝑁) = ;𝐶𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 12276 | . . 3 ⊢ ;10 ∈ ℕ0 | |
2 | decmul1.p | . . 3 ⊢ 𝑃 ∈ ℕ0 | |
3 | decmul1.a | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
4 | decmul1.b | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
5 | decmul1.n | . . . 4 ⊢ 𝑁 = ;𝐴𝐵 | |
6 | dfdec10 12261 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
7 | 5, 6 | eqtri 2759 | . . 3 ⊢ 𝑁 = ((;10 · 𝐴) + 𝐵) |
8 | decmul1.0 | . . 3 ⊢ 𝐷 ∈ ℕ0 | |
9 | decmul1c.e | . . 3 ⊢ 𝐸 ∈ ℕ0 | |
10 | decmul2c.c | . . 3 ⊢ ((𝑃 · 𝐴) + 𝐸) = 𝐶 | |
11 | decmul2c.2 | . . . 4 ⊢ (𝑃 · 𝐵) = ;𝐸𝐷 | |
12 | dfdec10 12261 | . . . 4 ⊢ ;𝐸𝐷 = ((;10 · 𝐸) + 𝐷) | |
13 | 11, 12 | eqtri 2759 | . . 3 ⊢ (𝑃 · 𝐵) = ((;10 · 𝐸) + 𝐷) |
14 | 1, 2, 3, 4, 7, 8, 9, 10, 13 | nummul2c 12308 | . 2 ⊢ (𝑃 · 𝑁) = ((;10 · 𝐶) + 𝐷) |
15 | dfdec10 12261 | . 2 ⊢ ;𝐶𝐷 = ((;10 · 𝐶) + 𝐷) | |
16 | 14, 15 | eqtr4i 2762 | 1 ⊢ (𝑃 · 𝑁) = ;𝐶𝐷 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 (class class class)co 7191 0cc0 10694 1c1 10695 + caddc 10697 · cmul 10699 ℕ0cn0 12055 ;cdc 12258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-sub 11029 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-dec 12259 |
This theorem is referenced by: decmulnc 12325 2exp8 16605 2exp16 16607 prmlem2 16636 37prm 16637 1259lem2 16648 1259lem3 16649 1259lem4 16650 1259prm 16652 2503lem1 16653 2503lem2 16654 2503prm 16656 4001lem1 16657 4001lem2 16658 4001lem3 16659 4001prm 16661 log2ublem3 25785 log2ub 25786 birthday 25791 dpmul 30861 420gcd8e4 39697 420lcm8e840 39702 3exp7 39744 3lexlogpow5ineq1 39745 3lexlogpow5ineq5 39751 aks4d1p1 39766 decpmulnc 39963 235t711 39967 ex-decpmul 39968 resqrtvalex 40870 imsqrtvalex 40871 257prm 44629 fmtno4prmfac 44640 fmtno4prmfac193 44641 fmtno4nprmfac193 44642 m11nprm 44669 2exp340mod341 44801 |
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