Theorem 235t711 43126

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 11235 saving the lower level uses of mulcomli 11235 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12843 are added then this proof would benefit more than ex-decpmul 43127.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12392 or 8t7e56 12854. (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 12538	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12539	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12744	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12541	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12744	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12543	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12537	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12744	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2765	. 2 ⊢ ;;711 = ;;711
10		eqid 2765	. . 3 ⊢ ;71 = ;71
11		eqid 2765	. . 3 ⊢ ;23 = ;23
12		8nn0 12544	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2765	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12533	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12333	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12408	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12794	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11419	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12536	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12540	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12542	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12744	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12558	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12352	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12843	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11235	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12410	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12794	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12339	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12844	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11235	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12799	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12343	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11232	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11175	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11419	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7429	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12405	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2788	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11419	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12794	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12346	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12846	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11235	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12402	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12805	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12796	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12786	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12533	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11230	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12414	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12794	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12787	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12800	1 ⊢ (;;235 · ;;711) = ;;;;;167085

This proof depends on syntax axioms:   = wceq 1570  (class class class)co 7419  0cc0 11117  1c1 11118   + caddc 11120   · cmul 11122  2c2 12312  3c3 12313  4c4 12314  5c5 12315  6c6 12316  7c7 12317  8c8 12318  ;cdc 12729

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-pnf 11262  df-mnf 11263  df-ltxr 11265  df-sub 11460  df-nn 12251  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-7 12325  df-8 12326  df-9 12327  df-n0 12522  df-dec 12730

This theorem is used by: (None)