Theorem 235t711 42339

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 11270 saving the lower level uses of mulcomli 11270 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12842 are added then this proof would benefit more than ex-decpmul 42340.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12401 or 8t7e56 12853. (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 12543	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12544	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12748	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12546	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12748	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12548	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12542	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12748	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2737	. 2 ⊢ ;;711 = ;;711
10		eqid 2737	. . 3 ⊢ ;71 = ;71
11		eqid 2737	. . 3 ⊢ ;23 = ;23
12		8nn0 12549	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2737	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12538	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12341	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12417	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12793	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11453	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12541	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12545	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12547	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12748	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12563	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12360	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12842	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11270	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12419	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12793	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12347	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12843	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11270	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12798	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12351	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11267	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11213	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11453	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7441	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12414	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2765	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11453	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12793	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12354	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12845	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11270	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12411	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12804	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12795	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12785	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12538	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11265	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12423	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12793	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12786	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12799	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1540  (class class class)co 7431  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  2c2 12321  3c3 12322  4c4 12323  5c5 12324  6c6 12325  7c7 12326  8c8 12327  ;cdc 12733

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-ltxr 11300  df-sub 11494  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-dec 12734

This theorem is referenced by: (None)