Theorem ex-decpmul 40315

Description: Example usage of decpmul 40311. This proof is significantly longer than 235t711 40314. There is more unnecessary carrying compared to 235t711 40314. Although saving 5 visual steps, using mulcomli 10983 early on increases the compressed proof length. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ex-decpmul	⊢ (;;235 · ;;711) = ;;;;;167085

Proof of Theorem ex-decpmul

Step	Hyp	Ref	Expression
1		2nn0 12248	. . 3 ⊢ 2 ∈ ℕ0
2		3nn0 12249	. . 3 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12449	. 2 ⊢ ;23 ∈ ℕ0
4		5nn0 12251	. 2 ⊢ 5 ∈ ℕ0
5		7nn0 12253	. . 3 ⊢ 7 ∈ ℕ0
6		1nn0 12247	. . 3 ⊢ 1 ∈ ℕ0
7	5, 6	deccl 12449	. 2 ⊢ ;71 ∈ ℕ0
8		eqid 2740	. . 3 ⊢ ;71 = ;71
9		6nn0 12252	. . . . 5 ⊢ 6 ∈ ℕ0
10	6, 9	deccl 12449	. . . 4 ⊢ ;16 ∈ ℕ0
11		eqid 2740	. . . . 5 ⊢ ;23 = ;23
12		4nn0 12250	. . . . . 6 ⊢ 4 ∈ ℕ0
13		7cn 12065	. . . . . . 7 ⊢ 7 ∈ ℂ
14		2cn 12046	. . . . . . 7 ⊢ 2 ∈ ℂ
15		7t2e14 12543	. . . . . . 7 ⊢ (7 · 2) = ;14
16	13, 14, 15	mulcomli 10983	. . . . . 6 ⊢ (2 · 7) = ;14
17		4p2e6 12124	. . . . . 6 ⊢ (4 + 2) = 6
18	6, 12, 1, 16, 17	decaddi 12494	. . . . 5 ⊢ ((2 · 7) + 2) = ;16
19		3cn 12052	. . . . . 6 ⊢ 3 ∈ ℂ
20		7t3e21 12544	. . . . . 6 ⊢ (7 · 3) = ;21
21	13, 19, 20	mulcomli 10983	. . . . 5 ⊢ (3 · 7) = ;21
22	5, 1, 2, 11, 6, 1, 18, 21	decmul1c 12499	. . . 4 ⊢ (;23 · 7) = ;;161
23		1p2e3 12114	. . . 4 ⊢ (1 + 2) = 3
24	10, 6, 1, 22, 23	decaddi 12494	. . 3 ⊢ ((;23 · 7) + 2) = ;;163
25	3	nn0cni 12243	. . . 4 ⊢ ;23 ∈ ℂ
26	25	mulid1i 10978	. . 3 ⊢ (;23 · 1) = ;23
27	3, 5, 6, 8, 2, 1, 24, 26	decmul2c 12500	. 2 ⊢ (;23 · ;71) = ;;;1633
28	2, 4	deccl 12449	. . 3 ⊢ ;35 ∈ ℕ0
29	7	nn0cni 12243	. . . 4 ⊢ ;71 ∈ ℂ
30		5cn 12059	. . . 4 ⊢ 5 ∈ ℂ
31		7t5e35 12546	. . . . 5 ⊢ (7 · 5) = ;35
32	30	mulid2i 10979	. . . . 5 ⊢ (1 · 5) = 5
33	4, 5, 6, 8, 31, 32	decmul1 12498	. . . 4 ⊢ (;71 · 5) = ;;355
34	29, 30, 33	mulcomli 10983	. . 3 ⊢ (5 · ;71) = ;;355
35	28	nn0cni 12243	. . . 4 ⊢ ;35 ∈ ℂ
36		eqid 2740	. . . . 5 ⊢ ;35 = ;35
37		5p2e7 12127	. . . . 5 ⊢ (5 + 2) = 7
38	2, 4, 1, 36, 37	decaddi 12494	. . . 4 ⊢ (;35 + 2) = ;37
39	35, 14, 38	addcomli 11165	. . 3 ⊢ (2 + ;35) = ;37
40		5p3e8 12128	. . . 4 ⊢ (5 + 3) = 8
41	30, 19, 40	addcomli 11165	. . 3 ⊢ (3 + 5) = 8
42	1, 2, 28, 4, 26, 34, 39, 41	decadd 12488	. 2 ⊢ ((;23 · 1) + (5 · ;71)) = ;;378
43	30	mulid1i 10978	. . 3 ⊢ (5 · 1) = 5
44	4	dec0h 12456	. . 3 ⊢ 5 = ;05
45	43, 44	eqtri 2768	. 2 ⊢ (5 · 1) = ;05
46	10, 2	deccl 12449	. . . 4 ⊢ ;;163 ∈ ℕ0
47	46, 2	deccl 12449	. . 3 ⊢ ;;;1633 ∈ ℕ0
48		0nn0 12246	. . 3 ⊢ 0 ∈ ℕ0
49	2, 5	deccl 12449	. . 3 ⊢ ;37 ∈ ℕ0
50		8nn0 12254	. . 3 ⊢ 8 ∈ ℕ0
51		eqid 2740	. . 3 ⊢ ;;;;16330 = ;;;;16330
52		eqid 2740	. . 3 ⊢ ;;378 = ;;378
53		eqid 2740	. . . 4 ⊢ ;;;1633 = ;;;1633
54		eqid 2740	. . . 4 ⊢ ;37 = ;37
55		eqid 2740	. . . . . 6 ⊢ ;;163 = ;;163
56		3p3e6 12123	. . . . . 6 ⊢ (3 + 3) = 6
57	10, 2, 2, 55, 56	decaddi 12494	. . . . 5 ⊢ (;;163 + 3) = ;;166
58		6p1e7 12119	. . . . 5 ⊢ (6 + 1) = 7
59	10, 9, 6, 57, 58	decaddi 12494	. . . 4 ⊢ ((;;163 + 3) + 1) = ;;167
60		7p3e10 12509	. . . . 5 ⊢ (7 + 3) = ;10
61	13, 19, 60	addcomli 11165	. . . 4 ⊢ (3 + 7) = ;10
62	46, 2, 2, 5, 53, 54, 59, 61	decaddc2 12490	. . 3 ⊢ (;;;1633 + ;37) = ;;;1670
63		8cn 12068	. . . 4 ⊢ 8 ∈ ℂ
64	63	addid2i 11161	. . 3 ⊢ (0 + 8) = 8
65	47, 48, 49, 50, 51, 52, 62, 64	decadd 12488	. 2 ⊢ (;;;;16330 + ;;378) = ;;;;16708
66	3, 4, 7, 6, 27, 42, 45, 65, 48, 4	decpmul 40311	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1542  (class class class)co 7269  0cc0 10870  1c1 10871   + caddc 10873   · cmul 10875  2c2 12026  3c3 12027  4c4 12028  5c5 12029  6c6 12030  7c7 12031  8c8 12032  ;cdc 12434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-resscn 10927  ax-1cn 10928  ax-icn 10929  ax-addcl 10930  ax-addrcl 10931  ax-mulcl 10932  ax-mulrcl 10933  ax-mulcom 10934  ax-addass 10935  ax-mulass 10936  ax-distr 10937  ax-i2m1 10938  ax-1ne0 10939  ax-1rid 10940  ax-rnegex 10941  ax-rrecex 10942  ax-cnre 10943  ax-pre-lttri 10944  ax-pre-lttrn 10945  ax-pre-ltadd 10946

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6200  df-ord 6267  df-on 6268  df-lim 6269  df-suc 6270  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7705  df-2nd 7823  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-er 8479  df-en 8715  df-dom 8716  df-sdom 8717  df-pnf 11010  df-mnf 11011  df-ltxr 11013  df-sub 11205  df-nn 11972  df-2 12034  df-3 12035  df-4 12036  df-5 12037  df-6 12038  df-7 12039  df-8 12040  df-9 12041  df-n0 12232  df-dec 12435

This theorem is referenced by: (None)