Theorem 235t711 39485

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 10639 saving the lower level uses of mulcomli 10639 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12195 are added then this proof would benefit more than ex-decpmul 39486.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 11760 or 8t7e56 12206. (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 11902	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 11903	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12101	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 11905	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12101	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 11907	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 11901	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12101	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2798	. 2 ⊢ ;;711 = ;;711
10		eqid 2798	. . 3 ⊢ ;71 = ;71
11		eqid 2798	. . 3 ⊢ ;23 = ;23
12		8nn0 11908	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2798	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 11897	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 11700	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 11776	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12146	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 10821	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 11900	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 11904	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 11906	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12101	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 11922	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 11719	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12195	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 10639	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 11778	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12146	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 11706	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12196	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 10639	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12151	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 11710	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 10636	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 10584	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 10821	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7145	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 11773	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2821	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 10821	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12146	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 11713	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12198	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 10639	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 11770	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12157	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12148	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12138	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 11897	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulid1i 10634	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 11782	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12146	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12139	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12152	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1538  (class class class)co 7135  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531  2c2 11680  3c3 11681  4c4 11682  5c5 11683  6c6 11684  7c7 11685  8c8 11686  ;cdc 12086

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-ltxr 10669  df-sub 10861  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-dec 12087

This theorem is referenced by: (None)