Theorem 235t711 42990

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 11218 saving the lower level uses of mulcomli 11218 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12825 are added then this proof would benefit more than ex-decpmul 42991.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12375 or 8t7e56 12836. (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 12521	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12522	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12726	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12524	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12726	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12526	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12520	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12726	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2769	. 2 ⊢ ;;711 = ;;711
10		eqid 2769	. . 3 ⊢ ;71 = ;71
11		eqid 2769	. . 3 ⊢ ;23 = ;23
12		8nn0 12527	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2769	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12516	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12316	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12391	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12776	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11402	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12519	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12523	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12525	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12726	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12541	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12335	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12825	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11218	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12393	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12776	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12322	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12826	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11218	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12781	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12326	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11215	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11158	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11402	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7421	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12388	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2792	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11402	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12776	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12329	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12828	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11218	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12385	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12787	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12778	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12768	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12516	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11213	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12397	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12776	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12769	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12782	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1567  (class class class)co 7411  0cc0 11100  1c1 11101   + caddc 11103   · cmul 11105  2c2 12295  3c3 12296  4c4 12297  5c5 12298  6c6 12299  7c7 12300  8c8 12301  ;cdc 12711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11245  df-mnf 11246  df-ltxr 11248  df-sub 11443  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-dec 12712

This theorem is referenced by: (None)