Theorem 235t711 39197

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 10650 saving the lower level uses of mulcomli 10650 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12208 are added then this proof would benefit more than ex-decpmul 39198.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 11773 or 8t7e56 12219. (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 11915	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 11916	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12114	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 11918	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12114	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 11920	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 11914	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12114	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2821	. 2 ⊢ ;;711 = ;;711
10		eqid 2821	. . 3 ⊢ ;71 = ;71
11		eqid 2821	. . 3 ⊢ ;23 = ;23
12		8nn0 11921	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2821	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 11910	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 11713	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 11789	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12159	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 10832	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 11913	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 11917	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 11919	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12114	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 11935	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 11732	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12208	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 10650	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 11791	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12159	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 11719	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12209	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 10650	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12164	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 11723	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 10647	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 10595	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 10832	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7166	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 11786	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2844	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 10832	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12159	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 11726	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12211	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 10650	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 11783	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12170	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12161	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12151	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 11910	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulid1i 10645	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 11795	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12159	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12152	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12165	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1537  (class class class)co 7156  0cc0 10537  1c1 10538   + caddc 10540   · cmul 10542  2c2 11693  3c3 11694  4c4 11695  5c5 11696  6c6 11697  7c7 11698  8c8 11699  ;cdc 12099

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-ltxr 10680  df-sub 10872  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-dec 12100

This theorem is referenced by: (None)