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Theorem 235t711 39311
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 10627 saving the lower level uses of mulcomli 10627 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12185 are added then this proof would benefit more than ex-decpmul 39312.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 11750 or 8t7e56 12196. (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 11892 . . . 4 2 ∈ ℕ0
2 3nn0 11893 . . . 4 3 ∈ ℕ0
31, 2deccl 12091 . . 3 23 ∈ ℕ0
4 5nn0 11895 . . 3 5 ∈ ℕ0
53, 4deccl 12091 . 2 235 ∈ ℕ0
6 7nn0 11897 . . 3 7 ∈ ℕ0
7 1nn0 11891 . . 3 1 ∈ ℕ0
86, 7deccl 12091 . 2 71 ∈ ℕ0
9 eqid 2821 . 2 711 = 711
10 eqid 2821 . . 3 71 = 71
11 eqid 2821 . . 3 23 = 23
12 8nn0 11898 . . 3 8 ∈ ℕ0
13 eqid 2821 . . . 4 235 = 235
143nn0cni 11887 . . . . 5 23 ∈ ℂ
15 2cn 11690 . . . . 5 2 ∈ ℂ
16 3p2e5 11766 . . . . . 6 (3 + 2) = 5
171, 2, 1, 11, 16decaddi 12136 . . . . 5 (23 + 2) = 25
1814, 15, 17addcomli 10809 . . . 4 (2 + 23) = 25
19 0nn0 11890 . . . 4 0 ∈ ℕ0
20 4nn0 11894 . . . 4 4 ∈ ℕ0
21 6nn0 11896 . . . . . 6 6 ∈ ℕ0
227, 21deccl 12091 . . . . 5 16 ∈ ℕ0
231, 20nn0addcli 11912 . . . . 5 (2 + 4) ∈ ℕ0
24 7cn 11709 . . . . . . . 8 7 ∈ ℂ
25 7t2e14 12185 . . . . . . . 8 (7 · 2) = 14
2624, 15, 25mulcomli 10627 . . . . . . 7 (2 · 7) = 14
27 4p2e6 11768 . . . . . . 7 (4 + 2) = 6
287, 20, 1, 26, 27decaddi 12136 . . . . . 6 ((2 · 7) + 2) = 16
29 3cn 11696 . . . . . . 7 3 ∈ ℂ
30 7t3e21 12186 . . . . . . 7 (7 · 3) = 21
3124, 29, 30mulcomli 10627 . . . . . 6 (3 · 7) = 21
326, 1, 2, 11, 7, 1, 28, 31decmul1c 12141 . . . . 5 (23 · 7) = 161
33 4cn 11700 . . . . . . 7 4 ∈ ℂ
3415, 33addcli 10624 . . . . . 6 (2 + 4) ∈ ℂ
35 ax-1cn 10572 . . . . . 6 1 ∈ ℂ
3633, 15, 27addcomli 10809 . . . . . . . 8 (2 + 4) = 6
3736oveq1i 7140 . . . . . . 7 ((2 + 4) + 1) = (6 + 1)
38 6p1e7 11763 . . . . . . 7 (6 + 1) = 7
3937, 38eqtri 2844 . . . . . 6 ((2 + 4) + 1) = 7
4034, 35, 39addcomli 10809 . . . . 5 (1 + (2 + 4)) = 7
4122, 7, 23, 32, 40decaddi 12136 . . . 4 ((23 · 7) + (2 + 4)) = 167
42 5cn 11703 . . . . . 6 5 ∈ ℂ
43 7t5e35 12188 . . . . . 6 (7 · 5) = 35
4424, 42, 43mulcomli 10627 . . . . 5 (5 · 7) = 35
45 3p1e4 11760 . . . . 5 (3 + 1) = 4
46 5p5e10 12147 . . . . 5 (5 + 5) = 10
472, 4, 4, 44, 45, 46decaddci2 12138 . . . 4 ((5 · 7) + 5) = 40
483, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47decmac 12128 . . 3 ((235 · 7) + (2 + 23)) = 1670
495nn0cni 11887 . . . . 5 235 ∈ ℂ
5049mulid1i 10622 . . . 4 (235 · 1) = 235
51 5p3e8 11772 . . . 4 (5 + 3) = 8
523, 4, 2, 50, 51decaddi 12136 . . 3 ((235 · 1) + 3) = 238
536, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52decma2c 12129 . 2 ((235 · 71) + 23) = 16708
545, 8, 7, 9, 4, 3, 53, 50decmul2c 12142 1 (235 · 711) = 167085
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  (class class class)co 7130  0cc0 10514  1c1 10515   + caddc 10517   · cmul 10519  2c2 11670  3c3 11671  4c4 11672  5c5 11673  6c6 11674  7c7 11675  8c8 11676  cdc 12076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-ltxr 10657  df-sub 10849  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-n0 11876  df-dec 12077
This theorem is referenced by: (None)
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