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Theorem 235t711 42106
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 11273 saving the lower level uses of mulcomli 11273 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12838 are added then this proof would benefit more than ex-decpmul 42107.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12399 or 8t7e56 12849. (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 12541 . . . 4 2 ∈ ℕ0
2 3nn0 12542 . . . 4 3 ∈ ℕ0
31, 2deccl 12744 . . 3 23 ∈ ℕ0
4 5nn0 12544 . . 3 5 ∈ ℕ0
53, 4deccl 12744 . 2 235 ∈ ℕ0
6 7nn0 12546 . . 3 7 ∈ ℕ0
7 1nn0 12540 . . 3 1 ∈ ℕ0
86, 7deccl 12744 . 2 71 ∈ ℕ0
9 eqid 2726 . 2 711 = 711
10 eqid 2726 . . 3 71 = 71
11 eqid 2726 . . 3 23 = 23
12 8nn0 12547 . . 3 8 ∈ ℕ0
13 eqid 2726 . . . 4 235 = 235
143nn0cni 12536 . . . . 5 23 ∈ ℂ
15 2cn 12339 . . . . 5 2 ∈ ℂ
16 3p2e5 12415 . . . . . 6 (3 + 2) = 5
171, 2, 1, 11, 16decaddi 12789 . . . . 5 (23 + 2) = 25
1814, 15, 17addcomli 11456 . . . 4 (2 + 23) = 25
19 0nn0 12539 . . . 4 0 ∈ ℕ0
20 4nn0 12543 . . . 4 4 ∈ ℕ0
21 6nn0 12545 . . . . . 6 6 ∈ ℕ0
227, 21deccl 12744 . . . . 5 16 ∈ ℕ0
231, 20nn0addcli 12561 . . . . 5 (2 + 4) ∈ ℕ0
24 7cn 12358 . . . . . . . 8 7 ∈ ℂ
25 7t2e14 12838 . . . . . . . 8 (7 · 2) = 14
2624, 15, 25mulcomli 11273 . . . . . . 7 (2 · 7) = 14
27 4p2e6 12417 . . . . . . 7 (4 + 2) = 6
287, 20, 1, 26, 27decaddi 12789 . . . . . 6 ((2 · 7) + 2) = 16
29 3cn 12345 . . . . . . 7 3 ∈ ℂ
30 7t3e21 12839 . . . . . . 7 (7 · 3) = 21
3124, 29, 30mulcomli 11273 . . . . . 6 (3 · 7) = 21
326, 1, 2, 11, 7, 1, 28, 31decmul1c 12794 . . . . 5 (23 · 7) = 161
33 4cn 12349 . . . . . . 7 4 ∈ ℂ
3415, 33addcli 11270 . . . . . 6 (2 + 4) ∈ ℂ
35 ax-1cn 11216 . . . . . 6 1 ∈ ℂ
3633, 15, 27addcomli 11456 . . . . . . . 8 (2 + 4) = 6
3736oveq1i 7434 . . . . . . 7 ((2 + 4) + 1) = (6 + 1)
38 6p1e7 12412 . . . . . . 7 (6 + 1) = 7
3937, 38eqtri 2754 . . . . . 6 ((2 + 4) + 1) = 7
4034, 35, 39addcomli 11456 . . . . 5 (1 + (2 + 4)) = 7
4122, 7, 23, 32, 40decaddi 12789 . . . 4 ((23 · 7) + (2 + 4)) = 167
42 5cn 12352 . . . . . 6 5 ∈ ℂ
43 7t5e35 12841 . . . . . 6 (7 · 5) = 35
4424, 42, 43mulcomli 11273 . . . . 5 (5 · 7) = 35
45 3p1e4 12409 . . . . 5 (3 + 1) = 4
46 5p5e10 12800 . . . . 5 (5 + 5) = 10
472, 4, 4, 44, 45, 46decaddci2 12791 . . . 4 ((5 · 7) + 5) = 40
483, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47decmac 12781 . . 3 ((235 · 7) + (2 + 23)) = 1670
495nn0cni 12536 . . . . 5 235 ∈ ℂ
5049mulridi 11268 . . . 4 (235 · 1) = 235
51 5p3e8 12421 . . . 4 (5 + 3) = 8
523, 4, 2, 50, 51decaddi 12789 . . 3 ((235 · 1) + 3) = 238
536, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52decma2c 12782 . 2 ((235 · 71) + 23) = 16708
545, 8, 7, 9, 4, 3, 53, 50decmul2c 12795 1 (235 · 711) = 167085
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  (class class class)co 7424  0cc0 11158  1c1 11159   + caddc 11161   · cmul 11163  2c2 12319  3c3 12320  4c4 12321  5c5 12322  6c6 12323  7c7 12324  8c8 12325  cdc 12729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  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 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-ltxr 11303  df-sub 11496  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-dec 12730
This theorem is referenced by: (None)
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