Theorem ex-decpmul 39198

Description: Example usage of decpmul 39194. This proof is significantly longer than 235t711 39197. There is more unnecessary carrying compared to 235t711 39197. Although saving 5 visual steps, using mulcomli 10650 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 11915	. . 3 ⊢ 2 ∈ ℕ0
2		3nn0 11916	. . 3 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12114	. 2 ⊢ ;23 ∈ ℕ0
4		5nn0 11918	. 2 ⊢ 5 ∈ ℕ0
5		7nn0 11920	. . 3 ⊢ 7 ∈ ℕ0
6		1nn0 11914	. . 3 ⊢ 1 ∈ ℕ0
7	5, 6	deccl 12114	. 2 ⊢ ;71 ∈ ℕ0
8		eqid 2821	. . 3 ⊢ ;71 = ;71
9		6nn0 11919	. . . . 5 ⊢ 6 ∈ ℕ0
10	6, 9	deccl 12114	. . . 4 ⊢ ;16 ∈ ℕ0
11		eqid 2821	. . . . 5 ⊢ ;23 = ;23
12		4nn0 11917	. . . . . 6 ⊢ 4 ∈ ℕ0
13		7cn 11732	. . . . . . 7 ⊢ 7 ∈ ℂ
14		2cn 11713	. . . . . . 7 ⊢ 2 ∈ ℂ
15		7t2e14 12208	. . . . . . 7 ⊢ (7 · 2) = ;14
16	13, 14, 15	mulcomli 10650	. . . . . 6 ⊢ (2 · 7) = ;14
17		4p2e6 11791	. . . . . 6 ⊢ (4 + 2) = 6
18	6, 12, 1, 16, 17	decaddi 12159	. . . . 5 ⊢ ((2 · 7) + 2) = ;16
19		3cn 11719	. . . . . 6 ⊢ 3 ∈ ℂ
20		7t3e21 12209	. . . . . 6 ⊢ (7 · 3) = ;21
21	13, 19, 20	mulcomli 10650	. . . . 5 ⊢ (3 · 7) = ;21
22	5, 1, 2, 11, 6, 1, 18, 21	decmul1c 12164	. . . 4 ⊢ (;23 · 7) = ;;161
23		1p2e3 11781	. . . 4 ⊢ (1 + 2) = 3
24	10, 6, 1, 22, 23	decaddi 12159	. . 3 ⊢ ((;23 · 7) + 2) = ;;163
25	3	nn0cni 11910	. . . 4 ⊢ ;23 ∈ ℂ
26	25	mulid1i 10645	. . 3 ⊢ (;23 · 1) = ;23
27	3, 5, 6, 8, 2, 1, 24, 26	decmul2c 12165	. 2 ⊢ (;23 · ;71) = ;;;1633
28	2, 4	deccl 12114	. . 3 ⊢ ;35 ∈ ℕ0
29	7	nn0cni 11910	. . . 4 ⊢ ;71 ∈ ℂ
30		5cn 11726	. . . 4 ⊢ 5 ∈ ℂ
31		7t5e35 12211	. . . . 5 ⊢ (7 · 5) = ;35
32	30	mulid2i 10646	. . . . 5 ⊢ (1 · 5) = 5
33	4, 5, 6, 8, 31, 32	decmul1 12163	. . . 4 ⊢ (;71 · 5) = ;;355
34	29, 30, 33	mulcomli 10650	. . 3 ⊢ (5 · ;71) = ;;355
35	28	nn0cni 11910	. . . 4 ⊢ ;35 ∈ ℂ
36		eqid 2821	. . . . 5 ⊢ ;35 = ;35
37		5p2e7 11794	. . . . 5 ⊢ (5 + 2) = 7
38	2, 4, 1, 36, 37	decaddi 12159	. . . 4 ⊢ (;35 + 2) = ;37
39	35, 14, 38	addcomli 10832	. . 3 ⊢ (2 + ;35) = ;37
40		5p3e8 11795	. . . 4 ⊢ (5 + 3) = 8
41	30, 19, 40	addcomli 10832	. . 3 ⊢ (3 + 5) = 8
42	1, 2, 28, 4, 26, 34, 39, 41	decadd 12153	. 2 ⊢ ((;23 · 1) + (5 · ;71)) = ;;378
43	30	mulid1i 10645	. . 3 ⊢ (5 · 1) = 5
44	4	dec0h 12121	. . 3 ⊢ 5 = ;05
45	43, 44	eqtri 2844	. 2 ⊢ (5 · 1) = ;05
46	10, 2	deccl 12114	. . . 4 ⊢ ;;163 ∈ ℕ0
47	46, 2	deccl 12114	. . 3 ⊢ ;;;1633 ∈ ℕ0
48		0nn0 11913	. . 3 ⊢ 0 ∈ ℕ0
49	2, 5	deccl 12114	. . 3 ⊢ ;37 ∈ ℕ0
50		8nn0 11921	. . 3 ⊢ 8 ∈ ℕ0
51		eqid 2821	. . 3 ⊢ ;;;;16330 = ;;;;16330
52		eqid 2821	. . 3 ⊢ ;;378 = ;;378
53		eqid 2821	. . . 4 ⊢ ;;;1633 = ;;;1633
54		eqid 2821	. . . 4 ⊢ ;37 = ;37
55		eqid 2821	. . . . . 6 ⊢ ;;163 = ;;163
56		3p3e6 11790	. . . . . 6 ⊢ (3 + 3) = 6
57	10, 2, 2, 55, 56	decaddi 12159	. . . . 5 ⊢ (;;163 + 3) = ;;166
58		6p1e7 11786	. . . . 5 ⊢ (6 + 1) = 7
59	10, 9, 6, 57, 58	decaddi 12159	. . . 4 ⊢ ((;;163 + 3) + 1) = ;;167
60		7p3e10 12174	. . . . 5 ⊢ (7 + 3) = ;10
61	13, 19, 60	addcomli 10832	. . . 4 ⊢ (3 + 7) = ;10
62	46, 2, 2, 5, 53, 54, 59, 61	decaddc2 12155	. . 3 ⊢ (;;;1633 + ;37) = ;;;1670
63		8cn 11735	. . . 4 ⊢ 8 ∈ ℂ
64	63	addid2i 10828	. . 3 ⊢ (0 + 8) = 8
65	47, 48, 49, 50, 51, 52, 62, 64	decadd 12153	. 2 ⊢ (;;;;16330 + ;;378) = ;;;;16708
66	3, 4, 7, 6, 27, 42, 45, 65, 48, 4	decpmul 39194	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1537  (class class class)co 7156  0cc0 10537  1c1 10538   + caddc 10540   · cmul 10542  2c2 11693  3c3 11694  4c4 11695  5c5 11696  6c6 11697  7c7 11698  8c8 11699  ;cdc 12099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-ltxr 10680  df-sub 10872  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-dec 12100

This theorem is referenced by: (None)