Theorem 235t711 42915

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 11192 saving the lower level uses of mulcomli 11192 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12803 are added then this proof would benefit more than ex-decpmul 42916.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12353 or 8t7e56 12814. (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 12499	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12500	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12704	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12502	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12704	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12504	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12498	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12704	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2763	. 2 ⊢ ;;711 = ;;711
10		eqid 2763	. . 3 ⊢ ;71 = ;71
11		eqid 2763	. . 3 ⊢ ;23 = ;23
12		8nn0 12505	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2763	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12494	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12294	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12369	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12754	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11376	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12497	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12501	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12503	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12704	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12519	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12313	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12803	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11192	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12371	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12754	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12300	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12804	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11192	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12759	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12304	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11189	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11132	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11376	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7407	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12366	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2786	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11376	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12754	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12307	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12806	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11192	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12363	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12765	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12756	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12746	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12494	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11187	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12375	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12754	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12747	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12760	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1561  (class class class)co 7397  0cc0 11074  1c1 11075   + caddc 11077   · cmul 11079  2c2 12273  3c3 12274  4c4 12275  5c5 12276  6c6 12277  7c7 12278  8c8 12279  ;cdc 12689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-er 8679  df-en 8929  df-dom 8930  df-sdom 8931  df-pnf 11219  df-mnf 11220  df-ltxr 11222  df-sub 11417  df-nn 12212  df-2 12281  df-3 12282  df-4 12283  df-5 12284  df-6 12285  df-7 12286  df-8 12287  df-9 12288  df-n0 12483  df-dec 12690

This theorem is referenced by: (None)