Theorem 235t711 42293

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 11299 saving the lower level uses of mulcomli 11299 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12867 are added then this proof would benefit more than ex-decpmul 42294.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12428 or 8t7e56 12878. (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 12570	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12571	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12773	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12573	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12773	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12575	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12569	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12773	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2740	. 2 ⊢ ;;711 = ;;711
10		eqid 2740	. . 3 ⊢ ;71 = ;71
11		eqid 2740	. . 3 ⊢ ;23 = ;23
12		8nn0 12576	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2740	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12565	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12368	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12444	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12818	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11482	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12568	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12572	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12574	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12773	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12590	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12387	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12867	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11299	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12446	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12818	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12374	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12868	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11299	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12823	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12378	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11296	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11242	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11482	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7458	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12441	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2768	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11482	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12818	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12381	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12870	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11299	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12438	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12829	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12820	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12810	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12565	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11294	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12450	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12818	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12811	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12824	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1537  (class class class)co 7448  0cc0 11184  1c1 11185   + caddc 11187   · cmul 11189  2c2 12348  3c3 12349  4c4 12350  5c5 12351  6c6 12352  7c7 12353  8c8 12354  ;cdc 12758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-ltxr 11329  df-sub 11522  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-dec 12759

This theorem is referenced by: (None)