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

Step	Hyp	Ref	Expression
1		2nn0 12527	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12528	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12730	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12530	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12730	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12532	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12526	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 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
14	3	nn0cni 12522	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12325	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12401	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12775	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11444	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12525	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12529	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12531	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12730	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12547	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12344	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12824	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11261	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12403	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12775	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12331	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12825	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11261	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12780	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12335	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11258	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11204	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11444	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7436	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12398	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2756	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11444	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12775	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12338	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12827	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11261	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12395	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12786	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12777	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12767	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12522	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11256	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12407	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12775	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12768	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12781	1 ⊢ (;;235 · ;;711) = ;;;;;167085

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)