Theorem 235t711 42408

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 11121 saving the lower level uses of mulcomli 11121 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12697 are added then this proof would benefit more than ex-decpmul 42409.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12255 or 8t7e56 12708. (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 12398	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12399	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12603	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12401	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12603	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12403	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12397	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12603	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2731	. 2 ⊢ ;;711 = ;;711
10		eqid 2731	. . 3 ⊢ ;71 = ;71
11		eqid 2731	. . 3 ⊢ ;23 = ;23
12		8nn0 12404	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2731	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12393	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12200	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12271	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12648	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11305	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12396	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12400	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12402	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12603	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12418	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12219	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12697	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11121	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12273	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12648	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12206	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12698	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11121	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12653	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12210	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11118	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11064	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11305	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7356	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12268	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2754	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11305	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12648	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12213	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12700	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11121	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12265	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12659	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12650	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12640	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12393	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11116	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12277	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12648	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12641	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12654	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1541  (class class class)co 7346  0cc0 11006  1c1 11007   + caddc 11009   · cmul 11011  2c2 12180  3c3 12181  4c4 12182  5c5 12183  6c6 12184  7c7 12185  8c8 12186  ;cdc 12588

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-ltxr 11151  df-sub 11346  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-dec 12589

This theorem is referenced by: (None)