Theorem 235t711 40569

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 11077 saving the lower level uses of mulcomli 11077 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12639 are added then this proof would benefit more than ex-decpmul 40570.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12201 or 8t7e56 12650. (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 12343	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12344	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12545	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12346	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12545	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12348	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12342	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12545	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2736	. 2 ⊢ ;;711 = ;;711
10		eqid 2736	. . 3 ⊢ ;71 = ;71
11		eqid 2736	. . 3 ⊢ ;23 = ;23
12		8nn0 12349	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2736	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12338	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12141	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12217	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12590	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11260	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12341	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12345	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12347	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12545	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12363	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12160	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12639	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11077	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12219	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12590	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12147	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12640	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11077	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12595	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12151	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11074	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11022	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11260	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7339	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12214	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2764	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11260	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12590	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12154	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12642	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11077	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12211	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12601	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12592	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12582	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12338	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulid1i 11072	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12223	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12590	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12583	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12596	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1540  (class class class)co 7329  0cc0 10964  1c1 10965   + caddc 10967   · cmul 10969  2c2 12121  3c3 12122  4c4 12123  5c5 12124  6c6 12125  7c7 12126  8c8 12127  ;cdc 12530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-er 8561  df-en 8797  df-dom 8798  df-sdom 8799  df-pnf 11104  df-mnf 11105  df-ltxr 11107  df-sub 11300  df-nn 12067  df-2 12129  df-3 12130  df-4 12131  df-5 12132  df-6 12133  df-7 12134  df-8 12135  df-9 12136  df-n0 12327  df-dec 12531

This theorem is referenced by: (None)