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Theorem 235t711 41898
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.)

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

Proof of Theorem 235t711
StepHypRef Expression
1 2nn0 12527 . . . 4 2 ∈ ℕ0
2 3nn0 12528 . . . 4 3 ∈ ℕ0
31, 2deccl 12730 . . 3 23 ∈ ℕ0
4 5nn0 12530 . . 3 5 ∈ ℕ0
53, 4deccl 12730 . 2 235 ∈ ℕ0
6 7nn0 12532 . . 3 7 ∈ ℕ0
7 1nn0 12526 . . 3 1 ∈ ℕ0
86, 7deccl 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
143nn0cni 12522 . . . . 5 23 ∈ ℂ
15 2cn 12325 . . . . 5 2 ∈ ℂ
16 3p2e5 12401 . . . . . 6 (3 + 2) = 5
171, 2, 1, 11, 16decaddi 12775 . . . . 5 (23 + 2) = 25
1814, 15, 17addcomli 11444 . . . 4 (2 + 23) = 25
19 0nn0 12525 . . . 4 0 ∈ ℕ0
20 4nn0 12529 . . . 4 4 ∈ ℕ0
21 6nn0 12531 . . . . . 6 6 ∈ ℕ0
227, 21deccl 12730 . . . . 5 16 ∈ ℕ0
231, 20nn0addcli 12547 . . . . 5 (2 + 4) ∈ ℕ0
24 7cn 12344 . . . . . . . 8 7 ∈ ℂ
25 7t2e14 12824 . . . . . . . 8 (7 · 2) = 14
2624, 15, 25mulcomli 11261 . . . . . . 7 (2 · 7) = 14
27 4p2e6 12403 . . . . . . 7 (4 + 2) = 6
287, 20, 1, 26, 27decaddi 12775 . . . . . 6 ((2 · 7) + 2) = 16
29 3cn 12331 . . . . . . 7 3 ∈ ℂ
30 7t3e21 12825 . . . . . . 7 (7 · 3) = 21
3124, 29, 30mulcomli 11261 . . . . . 6 (3 · 7) = 21
326, 1, 2, 11, 7, 1, 28, 31decmul1c 12780 . . . . 5 (23 · 7) = 161
33 4cn 12335 . . . . . . 7 4 ∈ ℂ
3415, 33addcli 11258 . . . . . 6 (2 + 4) ∈ ℂ
35 ax-1cn 11204 . . . . . 6 1 ∈ ℂ
3633, 15, 27addcomli 11444 . . . . . . . 8 (2 + 4) = 6
3736oveq1i 7436 . . . . . . 7 ((2 + 4) + 1) = (6 + 1)
38 6p1e7 12398 . . . . . . 7 (6 + 1) = 7
3937, 38eqtri 2756 . . . . . 6 ((2 + 4) + 1) = 7
4034, 35, 39addcomli 11444 . . . . 5 (1 + (2 + 4)) = 7
4122, 7, 23, 32, 40decaddi 12775 . . . 4 ((23 · 7) + (2 + 4)) = 167
42 5cn 12338 . . . . . 6 5 ∈ ℂ
43 7t5e35 12827 . . . . . 6 (7 · 5) = 35
4424, 42, 43mulcomli 11261 . . . . 5 (5 · 7) = 35
45 3p1e4 12395 . . . . 5 (3 + 1) = 4
46 5p5e10 12786 . . . . 5 (5 + 5) = 10
472, 4, 4, 44, 45, 46decaddci2 12777 . . . 4 ((5 · 7) + 5) = 40
483, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47decmac 12767 . . 3 ((235 · 7) + (2 + 23)) = 1670
495nn0cni 12522 . . . . 5 235 ∈ ℂ
5049mulridi 11256 . . . 4 (235 · 1) = 235
51 5p3e8 12407 . . . 4 (5 + 3) = 8
523, 4, 2, 50, 51decaddi 12775 . . 3 ((235 · 1) + 3) = 238
536, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52decma2c 12768 . 2 ((235 · 71) + 23) = 16708
545, 8, 7, 9, 4, 3, 53, 50decmul2c 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)
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