Theorem 235t711 42354

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 11244 saving the lower level uses of mulcomli 11244 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12817 are added then this proof would benefit more than ex-decpmul 42355.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12375 or 8t7e56 12828. (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 12518	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12519	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12723	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12521	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12723	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12523	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12517	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12723	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2735	. 2 ⊢ ;;711 = ;;711
10		eqid 2735	. . 3 ⊢ ;71 = ;71
11		eqid 2735	. . 3 ⊢ ;23 = ;23
12		8nn0 12524	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2735	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12513	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12315	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12391	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12768	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11427	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12516	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12520	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12522	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12723	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12538	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12334	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12817	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11244	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12393	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12768	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12321	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12818	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11244	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12773	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12325	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11241	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11187	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11427	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7415	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12388	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2758	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11427	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12768	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12328	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12820	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11244	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12385	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12779	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12770	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12760	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12513	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11239	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12397	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12768	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12761	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12774	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1540  (class class class)co 7405  0cc0 11129  1c1 11130   + caddc 11132   · cmul 11134  2c2 12295  3c3 12296  4c4 12297  5c5 12298  6c6 12299  7c7 12300  8c8 12301  ;cdc 12708

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-ltxr 11274  df-sub 11468  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-dec 12709

This theorem is referenced by: (None)