Theorem 235t711 43086

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 11213 saving the lower level uses of mulcomli 11213 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12820 are added then this proof would benefit more than ex-decpmul 43087.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12370 or 8t7e56 12831. (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 12516	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12517	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12721	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12519	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12721	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12521	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12515	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12721	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2763	. 2 ⊢ ;;711 = ;;711
10		eqid 2763	. . 3 ⊢ ;71 = ;71
11		eqid 2763	. . 3 ⊢ ;23 = ;23
12		8nn0 12522	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2763	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12511	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12311	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12386	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12771	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11397	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12514	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12518	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12520	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12721	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12536	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12330	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12820	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11213	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12388	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12771	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12317	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12821	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11213	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12776	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12321	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11210	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11153	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11397	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7420	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12383	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2786	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11397	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12771	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12324	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12823	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11213	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12380	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12782	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12773	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12763	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12511	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11208	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12392	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12771	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12764	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12777	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1570  (class class class)co 7410  0cc0 11095  1c1 11096   + caddc 11098   · cmul 11100  2c2 12290  3c3 12291  4c4 12292  5c5 12293  6c6 12294  7c7 12295  8c8 12296  ;cdc 12706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-ltxr 11243  df-sub 11438  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-dec 12707

This theorem is referenced by: (None)