Theorem 235t711 42413

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 11131 saving the lower level uses of mulcomli 11131 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12707 are added then this proof would benefit more than ex-decpmul 42414.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12265 or 8t7e56 12718. (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 12408	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12409	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12613	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12411	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12613	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12413	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12407	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12613	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2733	. 2 ⊢ ;;711 = ;;711
10		eqid 2733	. . 3 ⊢ ;71 = ;71
11		eqid 2733	. . 3 ⊢ ;23 = ;23
12		8nn0 12414	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2733	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12403	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12210	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12281	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12658	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11315	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12406	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12410	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12412	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12613	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12428	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12229	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12707	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11131	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12283	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12658	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12216	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12708	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11131	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12663	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12220	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11128	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11074	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11315	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7365	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12278	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2756	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11315	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12658	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12223	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12710	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11131	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12275	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12669	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12660	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12650	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12403	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11126	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12287	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12658	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12651	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12664	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1541  (class class class)co 7355  0cc0 11016  1c1 11017   + caddc 11019   · cmul 11021  2c2 12190  3c3 12191  4c4 12192  5c5 12193  6c6 12194  7c7 12195  8c8 12196  ;cdc 12598

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-resscn 11073  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-addrcl 11077  ax-mulcl 11078  ax-mulrcl 11079  ax-mulcom 11080  ax-addass 11081  ax-mulass 11082  ax-distr 11083  ax-i2m1 11084  ax-1ne0 11085  ax-1rid 11086  ax-rnegex 11087  ax-rrecex 11088  ax-cnre 11089  ax-pre-lttri 11090  ax-pre-lttrn 11091  ax-pre-ltadd 11092

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-er 8631  df-en 8879  df-dom 8880  df-sdom 8881  df-pnf 11158  df-mnf 11159  df-ltxr 11161  df-sub 11356  df-nn 12136  df-2 12198  df-3 12199  df-4 12200  df-5 12201  df-6 12202  df-7 12203  df-8 12204  df-9 12205  df-n0 12392  df-dec 12599

This theorem is referenced by: (None)