![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > 235t711 | Structured version Visualization version GIF version |
Description: Calculate a product by
long multiplication as a base comparison with other
multiplication algorithms.
Conveniently, 711 has two ones which greatly simplifies calculations like 235 · 1. There isn't a higher level mulcomli 11261 saving the lower level uses of mulcomli 11261 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12824 are added then this proof would benefit more than ex-decpmul 41899. For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12385 or 8t7e56 12835. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.) |
Ref | Expression |
---|---|
235t711 | ⊢ (;;235 · ;;711) = ;;;;;167085 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 12527 | . . . 4 ⊢ 2 ∈ ℕ0 | |
2 | 3nn0 12528 | . . . 4 ⊢ 3 ∈ ℕ0 | |
3 | 1, 2 | deccl 12730 | . . 3 ⊢ ;23 ∈ ℕ0 |
4 | 5nn0 12530 | . . 3 ⊢ 5 ∈ ℕ0 | |
5 | 3, 4 | deccl 12730 | . 2 ⊢ ;;235 ∈ ℕ0 |
6 | 7nn0 12532 | . . 3 ⊢ 7 ∈ ℕ0 | |
7 | 1nn0 12526 | . . 3 ⊢ 1 ∈ ℕ0 | |
8 | 6, 7 | deccl 12730 | . 2 ⊢ ;71 ∈ ℕ0 |
9 | eqid 2728 | . 2 ⊢ ;;711 = ;;711 | |
10 | eqid 2728 | . . 3 ⊢ ;71 = ;71 | |
11 | eqid 2728 | . . 3 ⊢ ;23 = ;23 | |
12 | 8nn0 12533 | . . 3 ⊢ 8 ∈ ℕ0 | |
13 | eqid 2728 | . . . 4 ⊢ ;;235 = ;;235 | |
14 | 3 | nn0cni 12522 | . . . . 5 ⊢ ;23 ∈ ℂ |
15 | 2cn 12325 | . . . . 5 ⊢ 2 ∈ ℂ | |
16 | 3p2e5 12401 | . . . . . 6 ⊢ (3 + 2) = 5 | |
17 | 1, 2, 1, 11, 16 | decaddi 12775 | . . . . 5 ⊢ (;23 + 2) = ;25 |
18 | 14, 15, 17 | addcomli 11444 | . . . 4 ⊢ (2 + ;23) = ;25 |
19 | 0nn0 12525 | . . . 4 ⊢ 0 ∈ ℕ0 | |
20 | 4nn0 12529 | . . . 4 ⊢ 4 ∈ ℕ0 | |
21 | 6nn0 12531 | . . . . . 6 ⊢ 6 ∈ ℕ0 | |
22 | 7, 21 | deccl 12730 | . . . . 5 ⊢ ;16 ∈ ℕ0 |
23 | 1, 20 | nn0addcli 12547 | . . . . 5 ⊢ (2 + 4) ∈ ℕ0 |
24 | 7cn 12344 | . . . . . . . 8 ⊢ 7 ∈ ℂ | |
25 | 7t2e14 12824 | . . . . . . . 8 ⊢ (7 · 2) = ;14 | |
26 | 24, 15, 25 | mulcomli 11261 | . . . . . . 7 ⊢ (2 · 7) = ;14 |
27 | 4p2e6 12403 | . . . . . . 7 ⊢ (4 + 2) = 6 | |
28 | 7, 20, 1, 26, 27 | decaddi 12775 | . . . . . 6 ⊢ ((2 · 7) + 2) = ;16 |
29 | 3cn 12331 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
30 | 7t3e21 12825 | . . . . . . 7 ⊢ (7 · 3) = ;21 | |
31 | 24, 29, 30 | mulcomli 11261 | . . . . . 6 ⊢ (3 · 7) = ;21 |
32 | 6, 1, 2, 11, 7, 1, 28, 31 | decmul1c 12780 | . . . . 5 ⊢ (;23 · 7) = ;;161 |
33 | 4cn 12335 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
34 | 15, 33 | addcli 11258 | . . . . . 6 ⊢ (2 + 4) ∈ ℂ |
35 | ax-1cn 11204 | . . . . . 6 ⊢ 1 ∈ ℂ | |
36 | 33, 15, 27 | addcomli 11444 | . . . . . . . 8 ⊢ (2 + 4) = 6 |
37 | 36 | oveq1i 7436 | . . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1) |
38 | 6p1e7 12398 | . . . . . . 7 ⊢ (6 + 1) = 7 | |
39 | 37, 38 | eqtri 2756 | . . . . . 6 ⊢ ((2 + 4) + 1) = 7 |
40 | 34, 35, 39 | addcomli 11444 | . . . . 5 ⊢ (1 + (2 + 4)) = 7 |
41 | 22, 7, 23, 32, 40 | decaddi 12775 | . . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167 |
42 | 5cn 12338 | . . . . . 6 ⊢ 5 ∈ ℂ | |
43 | 7t5e35 12827 | . . . . . 6 ⊢ (7 · 5) = ;35 | |
44 | 24, 42, 43 | mulcomli 11261 | . . . . 5 ⊢ (5 · 7) = ;35 |
45 | 3p1e4 12395 | . . . . 5 ⊢ (3 + 1) = 4 | |
46 | 5p5e10 12786 | . . . . 5 ⊢ (5 + 5) = ;10 | |
47 | 2, 4, 4, 44, 45, 46 | decaddci2 12777 | . . . 4 ⊢ ((5 · 7) + 5) = ;40 |
48 | 3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47 | decmac 12767 | . . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670 |
49 | 5 | nn0cni 12522 | . . . . 5 ⊢ ;;235 ∈ ℂ |
50 | 49 | mulridi 11256 | . . . 4 ⊢ (;;235 · 1) = ;;235 |
51 | 5p3e8 12407 | . . . 4 ⊢ (5 + 3) = 8 | |
52 | 3, 4, 2, 50, 51 | decaddi 12775 | . . 3 ⊢ ((;;235 · 1) + 3) = ;;238 |
53 | 6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52 | decma2c 12768 | . 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708 |
54 | 5, 8, 7, 9, 4, 3, 53, 50 | decmul2c 12781 | 1 ⊢ (;;235 · ;;711) = ;;;;;167085 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 (class class class)co 7426 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 2c2 12305 3c3 12306 4c4 12307 5c5 12308 6c6 12309 7c7 12310 8c8 12311 ;cdc 12715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-ltxr 11291 df-sub 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-dec 12716 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |