Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  235t711 Structured version   Visualization version   GIF version

Theorem 235t711 42339
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 11270 saving the lower level uses of mulcomli 11270 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12842 are added then this proof would benefit more than ex-decpmul 42340.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12401 or 8t7e56 12853. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.)

Assertion
Ref Expression
235t711 (235 · 711) = 167085

Proof of Theorem 235t711
StepHypRef Expression
1 2nn0 12543 . . . 4 2 ∈ ℕ0
2 3nn0 12544 . . . 4 3 ∈ ℕ0
31, 2deccl 12748 . . 3 23 ∈ ℕ0
4 5nn0 12546 . . 3 5 ∈ ℕ0
53, 4deccl 12748 . 2 235 ∈ ℕ0
6 7nn0 12548 . . 3 7 ∈ ℕ0
7 1nn0 12542 . . 3 1 ∈ ℕ0
86, 7deccl 12748 . 2 71 ∈ ℕ0
9 eqid 2737 . 2 711 = 711
10 eqid 2737 . . 3 71 = 71
11 eqid 2737 . . 3 23 = 23
12 8nn0 12549 . . 3 8 ∈ ℕ0
13 eqid 2737 . . . 4 235 = 235
143nn0cni 12538 . . . . 5 23 ∈ ℂ
15 2cn 12341 . . . . 5 2 ∈ ℂ
16 3p2e5 12417 . . . . . 6 (3 + 2) = 5
171, 2, 1, 11, 16decaddi 12793 . . . . 5 (23 + 2) = 25
1814, 15, 17addcomli 11453 . . . 4 (2 + 23) = 25
19 0nn0 12541 . . . 4 0 ∈ ℕ0
20 4nn0 12545 . . . 4 4 ∈ ℕ0
21 6nn0 12547 . . . . . 6 6 ∈ ℕ0
227, 21deccl 12748 . . . . 5 16 ∈ ℕ0
231, 20nn0addcli 12563 . . . . 5 (2 + 4) ∈ ℕ0
24 7cn 12360 . . . . . . . 8 7 ∈ ℂ
25 7t2e14 12842 . . . . . . . 8 (7 · 2) = 14
2624, 15, 25mulcomli 11270 . . . . . . 7 (2 · 7) = 14
27 4p2e6 12419 . . . . . . 7 (4 + 2) = 6
287, 20, 1, 26, 27decaddi 12793 . . . . . 6 ((2 · 7) + 2) = 16
29 3cn 12347 . . . . . . 7 3 ∈ ℂ
30 7t3e21 12843 . . . . . . 7 (7 · 3) = 21
3124, 29, 30mulcomli 11270 . . . . . 6 (3 · 7) = 21
326, 1, 2, 11, 7, 1, 28, 31decmul1c 12798 . . . . 5 (23 · 7) = 161
33 4cn 12351 . . . . . . 7 4 ∈ ℂ
3415, 33addcli 11267 . . . . . 6 (2 + 4) ∈ ℂ
35 ax-1cn 11213 . . . . . 6 1 ∈ ℂ
3633, 15, 27addcomli 11453 . . . . . . . 8 (2 + 4) = 6
3736oveq1i 7441 . . . . . . 7 ((2 + 4) + 1) = (6 + 1)
38 6p1e7 12414 . . . . . . 7 (6 + 1) = 7
3937, 38eqtri 2765 . . . . . 6 ((2 + 4) + 1) = 7
4034, 35, 39addcomli 11453 . . . . 5 (1 + (2 + 4)) = 7
4122, 7, 23, 32, 40decaddi 12793 . . . 4 ((23 · 7) + (2 + 4)) = 167
42 5cn 12354 . . . . . 6 5 ∈ ℂ
43 7t5e35 12845 . . . . . 6 (7 · 5) = 35
4424, 42, 43mulcomli 11270 . . . . 5 (5 · 7) = 35
45 3p1e4 12411 . . . . 5 (3 + 1) = 4
46 5p5e10 12804 . . . . 5 (5 + 5) = 10
472, 4, 4, 44, 45, 46decaddci2 12795 . . . 4 ((5 · 7) + 5) = 40
483, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47decmac 12785 . . 3 ((235 · 7) + (2 + 23)) = 1670
495nn0cni 12538 . . . . 5 235 ∈ ℂ
5049mulridi 11265 . . . 4 (235 · 1) = 235
51 5p3e8 12423 . . . 4 (5 + 3) = 8
523, 4, 2, 50, 51decaddi 12793 . . 3 ((235 · 1) + 3) = 238
536, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52decma2c 12786 . 2 ((235 · 71) + 23) = 16708
545, 8, 7, 9, 4, 3, 53, 50decmul2c 12799 1 (235 · 711) = 167085
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  (class class class)co 7431  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  2c2 12321  3c3 12322  4c4 12323  5c5 12324  6c6 12325  7c7 12326  8c8 12327  cdc 12733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-ltxr 11300  df-sub 11494  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-dec 12734
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator