Theorem 235t711 40514

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 11034 saving the lower level uses of mulcomli 11034 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12596 are added then this proof would benefit more than ex-decpmul 40515.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12158 or 8t7e56 12607. (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 12300	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12301	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12502	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12303	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12502	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12305	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12299	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12502	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2736	. 2 ⊢ ;;711 = ;;711
10		eqid 2736	. . 3 ⊢ ;71 = ;71
11		eqid 2736	. . 3 ⊢ ;23 = ;23
12		8nn0 12306	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2736	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12295	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12098	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12174	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12547	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11217	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12298	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12302	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12304	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12502	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12320	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12117	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12596	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11034	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12176	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12547	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12104	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12597	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11034	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12552	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12108	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11031	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 10979	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11217	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7317	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12171	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2764	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11217	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12547	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12111	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12599	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11034	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12168	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12558	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12549	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12539	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12295	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulid1i 11029	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12180	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12547	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12540	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12553	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1539  (class class class)co 7307  0cc0 10921  1c1 10922   + caddc 10924   · cmul 10926  2c2 12078  3c3 12079  4c4 12080  5c5 12081  6c6 12082  7c7 12083  8c8 12084  ;cdc 12487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-resscn 10978  ax-1cn 10979  ax-icn 10980  ax-addcl 10981  ax-addrcl 10982  ax-mulcl 10983  ax-mulrcl 10984  ax-mulcom 10985  ax-addass 10986  ax-mulass 10987  ax-distr 10988  ax-i2m1 10989  ax-1ne0 10990  ax-1rid 10991  ax-rnegex 10992  ax-rrecex 10993  ax-cnre 10994  ax-pre-lttri 10995  ax-pre-lttrn 10996  ax-pre-ltadd 10997

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-er 8529  df-en 8765  df-dom 8766  df-sdom 8767  df-pnf 11061  df-mnf 11062  df-ltxr 11064  df-sub 11257  df-nn 12024  df-2 12086  df-3 12087  df-4 12088  df-5 12089  df-6 12090  df-7 12091  df-8 12092  df-9 12093  df-n0 12284  df-dec 12488

This theorem is referenced by: (None)