Theorem 235t711 42679

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 11153 saving the lower level uses of mulcomli 11153 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12728 are added then this proof would benefit more than ex-decpmul 42680.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12287 or 8t7e56 12739. (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 12430	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12431	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12634	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12433	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12634	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12435	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12429	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12634	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2737	. 2 ⊢ ;;711 = ;;711
10		eqid 2737	. . 3 ⊢ ;71 = ;71
11		eqid 2737	. . 3 ⊢ ;23 = ;23
12		8nn0 12436	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2737	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12425	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12232	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12303	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12679	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11337	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12428	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12432	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12434	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12634	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12450	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12251	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12728	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11153	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12305	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12679	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12238	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12729	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11153	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12684	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12242	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11150	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11096	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11337	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7378	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12300	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2760	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11337	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12679	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12245	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12731	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11153	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12297	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12690	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12681	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12671	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12425	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11148	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12309	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12679	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12672	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12685	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1542  (class class class)co 7368  0cc0 11038  1c1 11039   + caddc 11041   · cmul 11043  2c2 12212  3c3 12213  4c4 12214  5c5 12215  6c6 12216  7c7 12217  8c8 12218  ;cdc 12619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-ltxr 11183  df-sub 11378  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-dec 12620

This theorem is referenced by: (None)