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Theorem 235t711 38157
 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.)
Assertion
Ref Expression
235t711 (235 · 711) = 167085

Proof of Theorem 235t711
StepHypRef Expression
1 2nn0 11661 . . . 4 2 ∈ ℕ0
2 3nn0 11662 . . . 4 3 ∈ ℕ0
31, 2deccl 11860 . . 3 23 ∈ ℕ0
4 5nn0 11664 . . 3 5 ∈ ℕ0
53, 4deccl 11860 . 2 235 ∈ ℕ0
6 7nn0 11666 . . 3 7 ∈ ℕ0
7 1nn0 11660 . . 3 1 ∈ ℕ0
86, 7deccl 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
143nn0cni 11655 . . . . 5 23 ∈ ℂ
15 2cn 11450 . . . . 5 2 ∈ ℂ
16 3p2e5 11533 . . . . . 6 (3 + 2) = 5
171, 2, 1, 11, 16decaddi 11906 . . . . 5 (23 + 2) = 25
1814, 15, 17addcomli 10568 . . . 4 (2 + 23) = 25
19 0nn0 11659 . . . 4 0 ∈ ℕ0
20 4nn0 11663 . . . 4 4 ∈ ℕ0
21 6nn0 11665 . . . . . 6 6 ∈ ℕ0
227, 21deccl 11860 . . . . 5 16 ∈ ℕ0
231, 20nn0addcli 11681 . . . . 5 (2 + 4) ∈ ℕ0
24 7cn 11473 . . . . . . . 8 7 ∈ ℂ
25 7t2e14 11956 . . . . . . . 8 (7 · 2) = 14
2624, 15, 25mulcomli 10386 . . . . . . 7 (2 · 7) = 14
27 4p2e6 11535 . . . . . . 7 (4 + 2) = 6
287, 20, 1, 26, 27decaddi 11906 . . . . . 6 ((2 · 7) + 2) = 16
29 3cn 11456 . . . . . . 7 3 ∈ ℂ
30 7t3e21 11957 . . . . . . 7 (7 · 3) = 21
3124, 29, 30mulcomli 10386 . . . . . 6 (3 · 7) = 21
326, 1, 2, 11, 7, 1, 28, 31decmul1c 11912 . . . . 5 (23 · 7) = 161
33 4cn 11461 . . . . . . 7 4 ∈ ℂ
3415, 33addcli 10383 . . . . . 6 (2 + 4) ∈ ℂ
35 ax-1cn 10330 . . . . . 6 1 ∈ ℂ
3633, 15, 27addcomli 10568 . . . . . . . 8 (2 + 4) = 6
3736oveq1i 6932 . . . . . . 7 ((2 + 4) + 1) = (6 + 1)
38 6p1e7 11530 . . . . . . 7 (6 + 1) = 7
3937, 38eqtri 2802 . . . . . 6 ((2 + 4) + 1) = 7
4034, 35, 39addcomli 10568 . . . . 5 (1 + (2 + 4)) = 7
4122, 7, 23, 32, 40decaddi 11906 . . . 4 ((23 · 7) + (2 + 4)) = 167
42 5cn 11465 . . . . . 6 5 ∈ ℂ
43 7t5e35 11959 . . . . . 6 (7 · 5) = 35
4424, 42, 43mulcomli 10386 . . . . 5 (5 · 7) = 35
45 3p1e4 11527 . . . . 5 (3 + 1) = 4
46 5p5e10 11918 . . . . 5 (5 + 5) = 10
472, 4, 4, 44, 45, 46decaddci2 11908 . . . 4 ((5 · 7) + 5) = 40
483, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47decmac 11898 . . 3 ((235 · 7) + (2 + 23)) = 1670
495nn0cni 11655 . . . . 5 235 ∈ ℂ
5049mulid1i 10381 . . . 4 (235 · 1) = 235
51 5p3e8 11539 . . . 4 (5 + 3) = 8
523, 4, 2, 50, 51decaddi 11906 . . 3 ((235 · 1) + 3) = 238
536, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52decma2c 11899 . 2 ((235 · 71) + 23) = 16708
545, 8, 7, 9, 4, 3, 53, 50decmul2c 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)
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