Theorem 235t711 42791

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 11146 saving the lower level uses of mulcomli 11146 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12745 are added then this proof would benefit more than ex-decpmul 42792.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12303 or 8t7e56 12756. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.)

Assertion

Ref	Expression
235t711	⊢ (;;235 · ;;711) = ;;;;;167085

Proof of Theorem 235t711

Step	Hyp	Ref	Expression
1		2nn0 12446	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12447	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12651	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12449	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12651	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12451	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12445	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12651	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2739	. 2 ⊢ ;;711 = ;;711
10		eqid 2739	. . 3 ⊢ ;71 = ;71
11		eqid 2739	. . 3 ⊢ ;23 = ;23
12		8nn0 12452	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2739	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12441	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12248	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12319	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12696	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11330	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12444	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12448	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12450	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12651	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12466	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12267	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12745	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11146	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12321	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12696	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12254	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12746	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11146	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12701	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12258	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11143	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11088	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11330	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7367	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12316	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2762	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11330	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12696	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12261	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12748	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11146	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12313	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12707	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12698	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12688	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12441	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11141	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12325	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12696	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12689	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12702	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1547  (class class class)co 7357  0cc0 11030  1c1 11031   + caddc 11033   · cmul 11035  2c2 12228  3c3 12229  4c4 12230  5c5 12231  6c6 12232  7c7 12233  8c8 12234  ;cdc 12636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pow 5295  ax-pr 5363  ax-un 7679  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7314  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-er 8634  df-en 8885  df-dom 8886  df-sdom 8887  df-pnf 11173  df-mnf 11174  df-ltxr 11176  df-sub 11371  df-nn 12167  df-2 12236  df-3 12237  df-4 12238  df-5 12239  df-6 12240  df-7 12241  df-8 12242  df-9 12243  df-n0 12430  df-dec 12637

This theorem is referenced by: (None)