Theorem 235t711 42502

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 11139 saving the lower level uses of mulcomli 11139 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12714 are added then this proof would benefit more than ex-decpmul 42503.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12273 or 8t7e56 12725. (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 12416	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12417	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12620	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12419	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12620	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12421	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12415	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12620	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2734	. 2 ⊢ ;;711 = ;;711
10		eqid 2734	. . 3 ⊢ ;71 = ;71
11		eqid 2734	. . 3 ⊢ ;23 = ;23
12		8nn0 12422	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2734	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12411	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12218	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12289	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12665	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11323	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12414	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12418	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12420	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12620	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12436	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12237	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12714	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11139	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12291	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12665	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12224	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12715	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11139	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12670	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12228	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11136	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11082	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11323	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7366	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12286	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2757	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11323	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12665	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12231	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12717	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11139	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12283	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12676	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12667	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12657	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12411	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11134	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12295	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12665	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12658	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12671	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1541  (class class class)co 7356  0cc0 11024  1c1 11025   + caddc 11027   · cmul 11029  2c2 12198  3c3 12199  4c4 12200  5c5 12201  6c6 12202  7c7 12203  8c8 12204  ;cdc 12605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-pnf 11166  df-mnf 11167  df-ltxr 11169  df-sub 11364  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-dec 12606

This theorem is referenced by: (None)