Theorem 235t711 40299

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 10968 saving the lower level uses of mulcomli 10968 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12528 are added then this proof would benefit more than ex-decpmul 40300.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12091 or 8t7e56 12539. (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 12233	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12234	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12434	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12236	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12434	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12238	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12232	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12434	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2739	. 2 ⊢ ;;711 = ;;711
10		eqid 2739	. . 3 ⊢ ;71 = ;71
11		eqid 2739	. . 3 ⊢ ;23 = ;23
12		8nn0 12239	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2739	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12228	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12031	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12107	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12479	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11150	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12231	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12235	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12237	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12434	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12253	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12050	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12528	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 10968	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12109	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12479	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12037	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12529	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 10968	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12484	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12041	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 10965	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 10913	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11150	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7278	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12104	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2767	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11150	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12479	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12044	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12531	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 10968	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12101	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12490	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12481	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12471	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12228	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulid1i 10963	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12113	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12479	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12472	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12485	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1541  (class class class)co 7268  0cc0 10855  1c1 10856   + caddc 10858   · cmul 10860  2c2 12011  3c3 12012  4c4 12013  5c5 12014  6c6 12015  7c7 12016  8c8 12017  ;cdc 12419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-resscn 10912  ax-1cn 10913  ax-icn 10914  ax-addcl 10915  ax-addrcl 10916  ax-mulcl 10917  ax-mulrcl 10918  ax-mulcom 10919  ax-addass 10920  ax-mulass 10921  ax-distr 10922  ax-i2m1 10923  ax-1ne0 10924  ax-1rid 10925  ax-rnegex 10926  ax-rrecex 10927  ax-cnre 10928  ax-pre-lttri 10929  ax-pre-lttrn 10930  ax-pre-ltadd 10931

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-er 8472  df-en 8708  df-dom 8709  df-sdom 8710  df-pnf 10995  df-mnf 10996  df-ltxr 10998  df-sub 11190  df-nn 11957  df-2 12019  df-3 12020  df-4 12021  df-5 12022  df-6 12023  df-7 12024  df-8 12025  df-9 12026  df-n0 12217  df-dec 12420

This theorem is referenced by: (None)