Theorem 235t711 38157

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 10386 saving the lower level uses of mulcomli 10386 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 11956 are added then this proof would benefit more than ex-decpmul 38158.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 11517 or 8t7e56 11967. (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 11661	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 11662	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 11860	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 11664	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 11860	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 11666	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 11660	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 11860	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2778	. 2 ⊢ ;;711 = ;;711
10		eqid 2778	. . 3 ⊢ ;71 = ;71
11		eqid 2778	. . 3 ⊢ ;23 = ;23
12		8nn0 11667	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2778	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 11655	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 11450	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 11533	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 11906	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 10568	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 11659	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 11663	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 11665	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 11860	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 11681	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 11473	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 11956	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 10386	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 11535	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 11906	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 11456	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 11957	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 10386	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 11912	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 11461	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 10383	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 10330	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 10568	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 6932	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 11530	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2802	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 10568	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 11906	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 11465	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 11959	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 10386	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 11527	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 11918	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 11908	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 11898	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 11655	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulid1i 10381	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 11539	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 11906	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 11899	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 11913	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1601  (class class class)co 6922  0cc0 10272  1c1 10273   + caddc 10275   · cmul 10277  2c2 11430  3c3 11431  4c4 11432  5c5 11433  6c6 11434  7c7 11435  8c8 11436  ;cdc 11845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-ltxr 10416  df-sub 10608  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-dec 11846

This theorem is referenced by: (None)