Theorem 235t711 43505

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 11331 saving the lower level uses of mulcomli 11331 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12941 are added then this proof would benefit more than ex-decpmul 43506.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12490 or 8t7e56 12952. (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 12636	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12637	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12842	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12639	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12842	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12641	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12635	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12842	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2763	. 2 ⊢ ;;711 = ;;711
10		eqid 2763	. . 3 ⊢ ;71 = ;71
11		eqid 2763	. . 3 ⊢ ;23 = ;23
12		8nn0 12642	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2763	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12631	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12431	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12506	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12892	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11515	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12634	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12638	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12640	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12842	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12656	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12450	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12941	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11331	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12508	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12892	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12437	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12942	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11331	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12897	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12441	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11328	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11271	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11515	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7440	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12503	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2786	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11515	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12892	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12444	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12944	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11331	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12500	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12903	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12894	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12884	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12631	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11326	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12512	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12892	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12885	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12898	1 ⊢ (;;235 · ;;711) = ;;;;;167085

This proof depends on syntax axioms:   = wceq 1569  (class class class)co 7430  0cc0 11213  1c1 11214   + caddc 11216   · cmul 11218  2c2 12410  3c3 12411  4c4 12412  5c5 12413  6c6 12414  7c7 12415  8c8 12416  ;cdc 12827

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-er 8731  df-en 8988  df-dom 8989  df-sdom 8990  df-pnf 11358  df-mnf 11359  df-ltxr 11361  df-sub 11556  df-nn 12349  df-2 12418  df-3 12419  df-4 12420  df-5 12421  df-6 12422  df-7 12423  df-8 12424  df-9 12425  df-n0 12620  df-dec 12828

This theorem is used by: (None)