Theorem 235t711 41200

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 11219 saving the lower level uses of mulcomli 11219 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12782 are added then this proof would benefit more than ex-decpmul 41201.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12343 or 8t7e56 12793. (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 12485	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12486	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12688	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12488	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12688	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12490	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12484	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12688	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2732	. 2 ⊢ ;;711 = ;;711
10		eqid 2732	. . 3 ⊢ ;71 = ;71
11		eqid 2732	. . 3 ⊢ ;23 = ;23
12		8nn0 12491	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2732	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12480	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12283	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12359	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12733	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11402	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12483	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12487	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12489	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12688	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12505	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12302	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12782	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11219	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12361	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12733	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12289	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12783	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11219	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12738	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12293	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11216	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11164	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11402	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7415	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12356	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2760	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11402	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12733	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12296	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12785	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11219	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12353	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12744	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12735	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12725	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12480	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11214	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12365	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12733	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12726	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12739	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1541  (class class class)co 7405  0cc0 11106  1c1 11107   + caddc 11109   · cmul 11111  2c2 12263  3c3 12264  4c4 12265  5c5 12266  6c6 12267  7c7 12268  8c8 12269  ;cdc 12673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-ltxr 11249  df-sub 11442  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-dec 12674

This theorem is referenced by: (None)