Theorem 235t711 42288

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 11124 saving the lower level uses of mulcomli 11124 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12700 are added then this proof would benefit more than ex-decpmul 42289.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12258 or 8t7e56 12711. (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 12401	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12402	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12606	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12404	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12606	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12406	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12400	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12606	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2729	. 2 ⊢ ;;711 = ;;711
10		eqid 2729	. . 3 ⊢ ;71 = ;71
11		eqid 2729	. . 3 ⊢ ;23 = ;23
12		8nn0 12407	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2729	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12396	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12203	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12274	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12651	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11308	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12399	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12403	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12405	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12606	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12421	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12222	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12700	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11124	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12276	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12651	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12209	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12701	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11124	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12656	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12213	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11121	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11067	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11308	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7359	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12271	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2752	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11308	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12651	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12216	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12703	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11124	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12268	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12662	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12653	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12643	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12396	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11119	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12280	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12651	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12644	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12657	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1540  (class class class)co 7349  0cc0 11009  1c1 11010   + caddc 11012   · cmul 11014  2c2 12183  3c3 12184  4c4 12185  5c5 12186  6c6 12187  7c7 12188  8c8 12189  ;cdc 12591

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-ltxr 11154  df-sub 11349  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-dec 12592

This theorem is referenced by: (None)