Theorem 235t711 43362

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 11318 saving the lower level uses of mulcomli 11318 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12928 are added then this proof would benefit more than ex-decpmul 43363.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12477 or 8t7e56 12939. (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 12623	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12624	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12829	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12626	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12829	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12628	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12622	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12829	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2761	. 2 ⊢ ;;711 = ;;711
10		eqid 2761	. . 3 ⊢ ;71 = ;71
11		eqid 2761	. . 3 ⊢ ;23 = ;23
12		8nn0 12629	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2761	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12618	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12418	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12493	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12879	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11502	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12621	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12625	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12627	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12829	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12643	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12437	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12928	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11318	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12495	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12879	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12424	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12929	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11318	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12884	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12428	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11315	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11258	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11502	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7430	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12490	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2784	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11502	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12879	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12431	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12931	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11318	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12487	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12890	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12881	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12871	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12618	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11313	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12499	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12879	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12872	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12885	1 ⊢ (;;235 · ;;711) = ;;;;;167085

This proof depends on syntax axioms:   = wceq 1570  (class class class)co 7420  0cc0 11200  1c1 11201   + caddc 11203   · cmul 11205  2c2 12397  3c3 12398  4c4 12399  5c5 12400  6c6 12401  7c7 12402  8c8 12403  ;cdc 12814

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-ltxr 11348  df-sub 11543  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-dec 12815

This theorem is used by: (None)