Theorem 235t711 42106

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 11273 saving the lower level uses of mulcomli 11273 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12838 are added then this proof would benefit more than ex-decpmul 42107.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12399 or 8t7e56 12849. (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 12541	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12542	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12744	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12544	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12744	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12546	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12540	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12744	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2726	. 2 ⊢ ;;711 = ;;711
10		eqid 2726	. . 3 ⊢ ;71 = ;71
11		eqid 2726	. . 3 ⊢ ;23 = ;23
12		8nn0 12547	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2726	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12536	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12339	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12415	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12789	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11456	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12539	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12543	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12545	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12744	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12561	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12358	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12838	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11273	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12417	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12789	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12345	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12839	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11273	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12794	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12349	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11270	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11216	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11456	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7434	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12412	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2754	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11456	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12789	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12352	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12841	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11273	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12409	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12800	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12791	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12781	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12536	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11268	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12421	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12789	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12782	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12795	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1534  (class class class)co 7424  0cc0 11158  1c1 11159   + caddc 11161   · cmul 11163  2c2 12319  3c3 12320  4c4 12321  5c5 12322  6c6 12323  7c7 12324  8c8 12325  ;cdc 12729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-ltxr 11303  df-sub 11496  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-dec 12730

This theorem is referenced by: (None)