Theorem 235t711 42286

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 11159 saving the lower level uses of mulcomli 11159 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12734 are added then this proof would benefit more than ex-decpmul 42287.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12292 or 8t7e56 12745. (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 12435	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12436	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12640	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12438	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12640	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12440	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12434	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12640	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2729	. 2 ⊢ ;;711 = ;;711
10		eqid 2729	. . 3 ⊢ ;71 = ;71
11		eqid 2729	. . 3 ⊢ ;23 = ;23
12		8nn0 12441	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2729	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12430	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12237	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12308	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12685	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11342	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12433	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12437	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12439	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12640	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12455	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12256	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12734	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11159	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12310	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12685	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12243	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12735	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11159	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12690	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12247	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11156	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11102	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11342	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7379	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12305	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2752	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11342	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12685	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12250	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12737	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11159	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12302	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12696	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12687	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12677	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12430	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11154	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12314	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12685	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12678	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12691	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1540  (class class class)co 7369  0cc0 11044  1c1 11045   + caddc 11047   · cmul 11049  2c2 12217  3c3 12218  4c4 12219  5c5 12220  6c6 12221  7c7 12222  8c8 12223  ;cdc 12625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-ltxr 11189  df-sub 11383  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-dec 12626

This theorem is referenced by: (None)