Theorem 235t711 42300

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 11190 saving the lower level uses of mulcomli 11190 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12765 are added then this proof would benefit more than ex-decpmul 42301.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12323 or 8t7e56 12776. (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 12466	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12467	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12671	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12469	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12671	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12471	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12465	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12671	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2730	. 2 ⊢ ;;711 = ;;711
10		eqid 2730	. . 3 ⊢ ;71 = ;71
11		eqid 2730	. . 3 ⊢ ;23 = ;23
12		8nn0 12472	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2730	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12461	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12268	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12339	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12716	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11373	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12464	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12468	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12470	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12671	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12486	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12287	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12765	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11190	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12341	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12716	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12274	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12766	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11190	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12721	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12278	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11187	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11133	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11373	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7400	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12336	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2753	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11373	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12716	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12281	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12768	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11190	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12333	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12727	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12718	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12708	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12461	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11185	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12345	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12716	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12709	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12722	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1540  (class class class)co 7390  0cc0 11075  1c1 11076   + caddc 11078   · cmul 11080  2c2 12248  3c3 12249  4c4 12250  5c5 12251  6c6 12252  7c7 12253  8c8 12254  ;cdc 12656

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-ltxr 11220  df-sub 11414  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-dec 12657

This theorem is referenced by: (None)