Theorem 235t711 42293

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 11183 saving the lower level uses of mulcomli 11183 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12758 are added then this proof would benefit more than ex-decpmul 42294.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12316 or 8t7e56 12769. (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 12459	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12460	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12664	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12462	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12664	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12464	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12458	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12664	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2729	. 2 ⊢ ;;711 = ;;711
10		eqid 2729	. . 3 ⊢ ;71 = ;71
11		eqid 2729	. . 3 ⊢ ;23 = ;23
12		8nn0 12465	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2729	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12454	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12261	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12332	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12709	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11366	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12457	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12461	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12463	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12664	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12479	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12280	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12758	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11183	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12334	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12709	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12267	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12759	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11183	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12714	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12271	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11180	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11126	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11366	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7397	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12329	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2752	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11366	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12709	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12274	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12761	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11183	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12326	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12720	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12711	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12701	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12454	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11178	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12338	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12709	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12702	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12715	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1540  (class class class)co 7387  0cc0 11068  1c1 11069   + caddc 11071   · cmul 11073  2c2 12241  3c3 12242  4c4 12243  5c5 12244  6c6 12245  7c7 12246  8c8 12247  ;cdc 12649

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 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144

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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-ltxr 11213  df-sub 11407  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-dec 12650

This theorem is referenced by: (None)