![]() |
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 10386 saving the lower level uses of mulcomli 10386 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 11956 are added then this proof would benefit more than ex-decpmul 38158. For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 11517 or 8t7e56 11967. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.) |
Ref | Expression |
---|---|
235t711 | ⊢ (;;235 · ;;711) = ;;;;;167085 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 11661 | . . . 4 ⊢ 2 ∈ ℕ0 | |
2 | 3nn0 11662 | . . . 4 ⊢ 3 ∈ ℕ0 | |
3 | 1, 2 | deccl 11860 | . . 3 ⊢ ;23 ∈ ℕ0 |
4 | 5nn0 11664 | . . 3 ⊢ 5 ∈ ℕ0 | |
5 | 3, 4 | deccl 11860 | . 2 ⊢ ;;235 ∈ ℕ0 |
6 | 7nn0 11666 | . . 3 ⊢ 7 ∈ ℕ0 | |
7 | 1nn0 11660 | . . 3 ⊢ 1 ∈ ℕ0 | |
8 | 6, 7 | deccl 11860 | . 2 ⊢ ;71 ∈ ℕ0 |
9 | eqid 2778 | . 2 ⊢ ;;711 = ;;711 | |
10 | eqid 2778 | . . 3 ⊢ ;71 = ;71 | |
11 | eqid 2778 | . . 3 ⊢ ;23 = ;23 | |
12 | 8nn0 11667 | . . 3 ⊢ 8 ∈ ℕ0 | |
13 | eqid 2778 | . . . 4 ⊢ ;;235 = ;;235 | |
14 | 3 | nn0cni 11655 | . . . . 5 ⊢ ;23 ∈ ℂ |
15 | 2cn 11450 | . . . . 5 ⊢ 2 ∈ ℂ | |
16 | 3p2e5 11533 | . . . . . 6 ⊢ (3 + 2) = 5 | |
17 | 1, 2, 1, 11, 16 | decaddi 11906 | . . . . 5 ⊢ (;23 + 2) = ;25 |
18 | 14, 15, 17 | addcomli 10568 | . . . 4 ⊢ (2 + ;23) = ;25 |
19 | 0nn0 11659 | . . . 4 ⊢ 0 ∈ ℕ0 | |
20 | 4nn0 11663 | . . . 4 ⊢ 4 ∈ ℕ0 | |
21 | 6nn0 11665 | . . . . . 6 ⊢ 6 ∈ ℕ0 | |
22 | 7, 21 | deccl 11860 | . . . . 5 ⊢ ;16 ∈ ℕ0 |
23 | 1, 20 | nn0addcli 11681 | . . . . 5 ⊢ (2 + 4) ∈ ℕ0 |
24 | 7cn 11473 | . . . . . . . 8 ⊢ 7 ∈ ℂ | |
25 | 7t2e14 11956 | . . . . . . . 8 ⊢ (7 · 2) = ;14 | |
26 | 24, 15, 25 | mulcomli 10386 | . . . . . . 7 ⊢ (2 · 7) = ;14 |
27 | 4p2e6 11535 | . . . . . . 7 ⊢ (4 + 2) = 6 | |
28 | 7, 20, 1, 26, 27 | decaddi 11906 | . . . . . 6 ⊢ ((2 · 7) + 2) = ;16 |
29 | 3cn 11456 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
30 | 7t3e21 11957 | . . . . . . 7 ⊢ (7 · 3) = ;21 | |
31 | 24, 29, 30 | mulcomli 10386 | . . . . . 6 ⊢ (3 · 7) = ;21 |
32 | 6, 1, 2, 11, 7, 1, 28, 31 | decmul1c 11912 | . . . . 5 ⊢ (;23 · 7) = ;;161 |
33 | 4cn 11461 | . . . . . . 7 ⊢ 4 ∈ ℂ | |
34 | 15, 33 | addcli 10383 | . . . . . 6 ⊢ (2 + 4) ∈ ℂ |
35 | ax-1cn 10330 | . . . . . 6 ⊢ 1 ∈ ℂ | |
36 | 33, 15, 27 | addcomli 10568 | . . . . . . . 8 ⊢ (2 + 4) = 6 |
37 | 36 | oveq1i 6932 | . . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1) |
38 | 6p1e7 11530 | . . . . . . 7 ⊢ (6 + 1) = 7 | |
39 | 37, 38 | eqtri 2802 | . . . . . 6 ⊢ ((2 + 4) + 1) = 7 |
40 | 34, 35, 39 | addcomli 10568 | . . . . 5 ⊢ (1 + (2 + 4)) = 7 |
41 | 22, 7, 23, 32, 40 | decaddi 11906 | . . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167 |
42 | 5cn 11465 | . . . . . 6 ⊢ 5 ∈ ℂ | |
43 | 7t5e35 11959 | . . . . . 6 ⊢ (7 · 5) = ;35 | |
44 | 24, 42, 43 | mulcomli 10386 | . . . . 5 ⊢ (5 · 7) = ;35 |
45 | 3p1e4 11527 | . . . . 5 ⊢ (3 + 1) = 4 | |
46 | 5p5e10 11918 | . . . . 5 ⊢ (5 + 5) = ;10 | |
47 | 2, 4, 4, 44, 45, 46 | decaddci2 11908 | . . . 4 ⊢ ((5 · 7) + 5) = ;40 |
48 | 3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47 | decmac 11898 | . . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670 |
49 | 5 | nn0cni 11655 | . . . . 5 ⊢ ;;235 ∈ ℂ |
50 | 49 | mulid1i 10381 | . . . 4 ⊢ (;;235 · 1) = ;;235 |
51 | 5p3e8 11539 | . . . 4 ⊢ (5 + 3) = 8 | |
52 | 3, 4, 2, 50, 51 | decaddi 11906 | . . 3 ⊢ ((;;235 · 1) + 3) = ;;238 |
53 | 6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52 | decma2c 11899 | . 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708 |
54 | 5, 8, 7, 9, 4, 3, 53, 50 | decmul2c 11913 | 1 ⊢ (;;235 · ;;711) = ;;;;;167085 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 (class class class)co 6922 0cc0 10272 1c1 10273 + caddc 10275 · cmul 10277 2c2 11430 3c3 11431 4c4 11432 5c5 11433 6c6 11434 7c7 11435 8c8 11436 ;cdc 11845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-dec 11846 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |