Theorem 235t711 41745

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 11224 saving the lower level uses of mulcomli 11224 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12787 are added then this proof would benefit more than ex-decpmul 41746.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12348 or 8t7e56 12798. (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 12490	. . . 4 ⊢ 2 ∈ ℕ0
2		3nn0 12491	. . . 4 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12693	. . 3 ⊢ ;23 ∈ ℕ0
4		5nn0 12493	. . 3 ⊢ 5 ∈ ℕ0
5	3, 4	deccl 12693	. 2 ⊢ ;;235 ∈ ℕ0
6		7nn0 12495	. . 3 ⊢ 7 ∈ ℕ0
7		1nn0 12489	. . 3 ⊢ 1 ∈ ℕ0
8	6, 7	deccl 12693	. 2 ⊢ ;71 ∈ ℕ0
9		eqid 2726	. 2 ⊢ ;;711 = ;;711
10		eqid 2726	. . 3 ⊢ ;71 = ;71
11		eqid 2726	. . 3 ⊢ ;23 = ;23
12		8nn0 12496	. . 3 ⊢ 8 ∈ ℕ0
13		eqid 2726	. . . 4 ⊢ ;;235 = ;;235
14	3	nn0cni 12485	. . . . 5 ⊢ ;23 ∈ ℂ
15		2cn 12288	. . . . 5 ⊢ 2 ∈ ℂ
16		3p2e5 12364	. . . . . 6 ⊢ (3 + 2) = 5
17	1, 2, 1, 11, 16	decaddi 12738	. . . . 5 ⊢ (;23 + 2) = ;25
18	14, 15, 17	addcomli 11407	. . . 4 ⊢ (2 + ;23) = ;25
19		0nn0 12488	. . . 4 ⊢ 0 ∈ ℕ0
20		4nn0 12492	. . . 4 ⊢ 4 ∈ ℕ0
21		6nn0 12494	. . . . . 6 ⊢ 6 ∈ ℕ0
22	7, 21	deccl 12693	. . . . 5 ⊢ ;16 ∈ ℕ0
23	1, 20	nn0addcli 12510	. . . . 5 ⊢ (2 + 4) ∈ ℕ0
24		7cn 12307	. . . . . . . 8 ⊢ 7 ∈ ℂ
25		7t2e14 12787	. . . . . . . 8 ⊢ (7 · 2) = ;14
26	24, 15, 25	mulcomli 11224	. . . . . . 7 ⊢ (2 · 7) = ;14
27		4p2e6 12366	. . . . . . 7 ⊢ (4 + 2) = 6
28	7, 20, 1, 26, 27	decaddi 12738	. . . . . 6 ⊢ ((2 · 7) + 2) = ;16
29		3cn 12294	. . . . . . 7 ⊢ 3 ∈ ℂ
30		7t3e21 12788	. . . . . . 7 ⊢ (7 · 3) = ;21
31	24, 29, 30	mulcomli 11224	. . . . . 6 ⊢ (3 · 7) = ;21
32	6, 1, 2, 11, 7, 1, 28, 31	decmul1c 12743	. . . . 5 ⊢ (;23 · 7) = ;;161
33		4cn 12298	. . . . . . 7 ⊢ 4 ∈ ℂ
34	15, 33	addcli 11221	. . . . . 6 ⊢ (2 + 4) ∈ ℂ
35		ax-1cn 11167	. . . . . 6 ⊢ 1 ∈ ℂ
36	33, 15, 27	addcomli 11407	. . . . . . . 8 ⊢ (2 + 4) = 6
37	36	oveq1i 7414	. . . . . . 7 ⊢ ((2 + 4) + 1) = (6 + 1)
38		6p1e7 12361	. . . . . . 7 ⊢ (6 + 1) = 7
39	37, 38	eqtri 2754	. . . . . 6 ⊢ ((2 + 4) + 1) = 7
40	34, 35, 39	addcomli 11407	. . . . 5 ⊢ (1 + (2 + 4)) = 7
41	22, 7, 23, 32, 40	decaddi 12738	. . . 4 ⊢ ((;23 · 7) + (2 + 4)) = ;;167
42		5cn 12301	. . . . . 6 ⊢ 5 ∈ ℂ
43		7t5e35 12790	. . . . . 6 ⊢ (7 · 5) = ;35
44	24, 42, 43	mulcomli 11224	. . . . 5 ⊢ (5 · 7) = ;35
45		3p1e4 12358	. . . . 5 ⊢ (3 + 1) = 4
46		5p5e10 12749	. . . . 5 ⊢ (5 + 5) = ;10
47	2, 4, 4, 44, 45, 46	decaddci2 12740	. . . 4 ⊢ ((5 · 7) + 5) = ;40
48	3, 4, 1, 4, 13, 18, 6, 19, 20, 41, 47	decmac 12730	. . 3 ⊢ ((;;235 · 7) + (2 + ;23)) = ;;;1670
49	5	nn0cni 12485	. . . . 5 ⊢ ;;235 ∈ ℂ
50	49	mulridi 11219	. . . 4 ⊢ (;;235 · 1) = ;;235
51		5p3e8 12370	. . . 4 ⊢ (5 + 3) = 8
52	3, 4, 2, 50, 51	decaddi 12738	. . 3 ⊢ ((;;235 · 1) + 3) = ;;238
53	6, 7, 1, 2, 10, 11, 5, 12, 3, 48, 52	decma2c 12731	. 2 ⊢ ((;;235 · ;71) + ;23) = ;;;;16708
54	5, 8, 7, 9, 4, 3, 53, 50	decmul2c 12744	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1533  (class class class)co 7404  0cc0 11109  1c1 11110   + caddc 11112   · cmul 11114  2c2 12268  3c3 12269  4c4 12270  5c5 12271  6c6 12272  7c7 12273  8c8 12274  ;cdc 12678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-ltxr 11254  df-sub 11447  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-9 12283  df-n0 12474  df-dec 12679

This theorem is referenced by: (None)