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Mirrors > Home > MPE Home > Th. List > decrmac | Structured version Visualization version GIF version |
Description: Perform a multiply-add of two numerals 𝑀 and 𝑁 against a fixed multiplicand 𝑃 (with carry). (Contributed by AV, 16-Sep-2021.) |
Ref | Expression |
---|---|
decrmanc.a | ⊢ 𝐴 ∈ ℕ0 |
decrmanc.b | ⊢ 𝐵 ∈ ℕ0 |
decrmanc.n | ⊢ 𝑁 ∈ ℕ0 |
decrmanc.m | ⊢ 𝑀 = ;𝐴𝐵 |
decrmanc.p | ⊢ 𝑃 ∈ ℕ0 |
decrmac.f | ⊢ 𝐹 ∈ ℕ0 |
decrmac.g | ⊢ 𝐺 ∈ ℕ0 |
decrmac.e | ⊢ ((𝐴 · 𝑃) + 𝐺) = 𝐸 |
decrmac.2 | ⊢ ((𝐵 · 𝑃) + 𝑁) = ;𝐺𝐹 |
Ref | Expression |
---|---|
decrmac | ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decrmanc.a | . 2 ⊢ 𝐴 ∈ ℕ0 | |
2 | decrmanc.b | . 2 ⊢ 𝐵 ∈ ℕ0 | |
3 | 0nn0 11307 | . 2 ⊢ 0 ∈ ℕ0 | |
4 | decrmanc.n | . 2 ⊢ 𝑁 ∈ ℕ0 | |
5 | decrmanc.m | . 2 ⊢ 𝑀 = ;𝐴𝐵 | |
6 | 4 | dec0h 11522 | . 2 ⊢ 𝑁 = ;0𝑁 |
7 | decrmanc.p | . 2 ⊢ 𝑃 ∈ ℕ0 | |
8 | decrmac.f | . 2 ⊢ 𝐹 ∈ ℕ0 | |
9 | decrmac.g | . 2 ⊢ 𝐺 ∈ ℕ0 | |
10 | 9 | nn0cni 11304 | . . . . 5 ⊢ 𝐺 ∈ ℂ |
11 | 10 | addid2i 10224 | . . . 4 ⊢ (0 + 𝐺) = 𝐺 |
12 | 11 | oveq2i 6661 | . . 3 ⊢ ((𝐴 · 𝑃) + (0 + 𝐺)) = ((𝐴 · 𝑃) + 𝐺) |
13 | decrmac.e | . . 3 ⊢ ((𝐴 · 𝑃) + 𝐺) = 𝐸 | |
14 | 12, 13 | eqtri 2644 | . 2 ⊢ ((𝐴 · 𝑃) + (0 + 𝐺)) = 𝐸 |
15 | decrmac.2 | . 2 ⊢ ((𝐵 · 𝑃) + 𝑁) = ;𝐺𝐹 | |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15 | decmac 11566 | 1 ⊢ ((𝑀 · 𝑃) + 𝑁) = ;𝐸𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 (class class class)co 6650 0cc0 9936 + caddc 9939 · cmul 9941 ℕ0cn0 11292 ;cdc 11493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-dec 11494 |
This theorem is referenced by: 2exp16 15797 139prm 15831 163prm 15832 1259lem1 15838 1259lem3 15840 1259lem4 15841 2503lem1 15844 2503lem2 15845 4001lem1 15848 4001lem3 15850 4001prm 15852 log2ub 24676 139prmALT 41511 127prm 41515 |
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