Theorem ex-decpmul 43331

Description: Example usage of decpmul 43313. This proof is significantly longer than 235t711 43330. There is more unnecessary carrying compared to 235t711 43330. Although saving 5 visual steps, using mulcomli 11299 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 12604	. . 3 ⊢ 2 ∈ ℕ0
2		3nn0 12605	. . 3 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12810	. 2 ⊢ ;23 ∈ ℕ0
4		5nn0 12607	. 2 ⊢ 5 ∈ ℕ0
5		7nn0 12609	. . 3 ⊢ 7 ∈ ℕ0
6		1nn0 12603	. . 3 ⊢ 1 ∈ ℕ0
7	5, 6	deccl 12810	. 2 ⊢ ;71 ∈ ℕ0
8		eqid 2761	. . 3 ⊢ ;71 = ;71
9		6nn0 12608	. . . . 5 ⊢ 6 ∈ ℕ0
10	6, 9	deccl 12810	. . . 4 ⊢ ;16 ∈ ℕ0
11		eqid 2761	. . . . 5 ⊢ ;23 = ;23
12		4nn0 12606	. . . . . 6 ⊢ 4 ∈ ℕ0
13		7cn 12418	. . . . . . 7 ⊢ 7 ∈ ℂ
14		2cn 12399	. . . . . . 7 ⊢ 2 ∈ ℂ
15		7t2e14 12909	. . . . . . 7 ⊢ (7 · 2) = ;14
16	13, 14, 15	mulcomli 11299	. . . . . 6 ⊢ (2 · 7) = ;14
17		4p2e6 12476	. . . . . 6 ⊢ (4 + 2) = 6
18	6, 12, 1, 16, 17	decaddi 12860	. . . . 5 ⊢ ((2 · 7) + 2) = ;16
19		3cn 12405	. . . . . 6 ⊢ 3 ∈ ℂ
20		7t3e21 12910	. . . . . 6 ⊢ (7 · 3) = ;21
21	13, 19, 20	mulcomli 11299	. . . . 5 ⊢ (3 · 7) = ;21
22	5, 1, 2, 11, 6, 1, 18, 21	decmul1c 12865	. . . 4 ⊢ (;23 · 7) = ;;161
23		1p2e3 12466	. . . 4 ⊢ (1 + 2) = 3
24	10, 6, 1, 22, 23	decaddi 12860	. . 3 ⊢ ((;23 · 7) + 2) = ;;163
25	3	nn0cni 12599	. . . 4 ⊢ ;23 ∈ ℂ
26	25	mulridi 11294	. . 3 ⊢ (;23 · 1) = ;23
27	3, 5, 6, 8, 2, 1, 24, 26	decmul2c 12866	. 2 ⊢ (;23 · ;71) = ;;;1633
28	2, 4	deccl 12810	. . 3 ⊢ ;35 ∈ ℕ0
29	7	nn0cni 12599	. . . 4 ⊢ ;71 ∈ ℂ
30		5cn 12412	. . . 4 ⊢ 5 ∈ ℂ
31		7t5e35 12912	. . . . 5 ⊢ (7 · 5) = ;35
32	30	mullidi 11295	. . . . 5 ⊢ (1 · 5) = 5
33	4, 5, 6, 8, 31, 32	decmul1 12864	. . . 4 ⊢ (;71 · 5) = ;;355
34	29, 30, 33	mulcomli 11299	. . 3 ⊢ (5 · ;71) = ;;355
35	28	nn0cni 12599	. . . 4 ⊢ ;35 ∈ ℂ
36		eqid 2761	. . . . 5 ⊢ ;35 = ;35
37		5p2e7 12479	. . . . 5 ⊢ (5 + 2) = 7
38	2, 4, 1, 36, 37	decaddi 12860	. . . 4 ⊢ (;35 + 2) = ;37
39	35, 14, 38	addcomli 11483	. . 3 ⊢ (2 + ;35) = ;37
40		5p3e8 12480	. . . 4 ⊢ (5 + 3) = 8
41	30, 19, 40	addcomli 11483	. . 3 ⊢ (3 + 5) = 8
42	1, 2, 28, 4, 26, 34, 39, 41	decadd 12854	. 2 ⊢ ((;23 · 1) + (5 · ;71)) = ;;378
43	30	mulridi 11294	. . 3 ⊢ (5 · 1) = 5
44	4	dec0h 12822	. . 3 ⊢ 5 = ;05
45	43, 44	eqtri 2784	. 2 ⊢ (5 · 1) = ;05
46	10, 2	deccl 12810	. . . 4 ⊢ ;;163 ∈ ℕ0
47	46, 2	deccl 12810	. . 3 ⊢ ;;;1633 ∈ ℕ0
48		0nn0 12602	. . 3 ⊢ 0 ∈ ℕ0
49	2, 5	deccl 12810	. . 3 ⊢ ;37 ∈ ℕ0
50		8nn0 12610	. . 3 ⊢ 8 ∈ ℕ0
51		eqid 2761	. . 3 ⊢ ;;;;16330 = ;;;;16330
52		eqid 2761	. . 3 ⊢ ;;378 = ;;378
53		eqid 2761	. . . 4 ⊢ ;;;1633 = ;;;1633
54		eqid 2761	. . . 4 ⊢ ;37 = ;37
55		eqid 2761	. . . . . 6 ⊢ ;;163 = ;;163
56		3p3e6 12475	. . . . . 6 ⊢ (3 + 3) = 6
57	10, 2, 2, 55, 56	decaddi 12860	. . . . 5 ⊢ (;;163 + 3) = ;;166
58		6p1e7 12471	. . . . 5 ⊢ (6 + 1) = 7
59	10, 9, 6, 57, 58	decaddi 12860	. . . 4 ⊢ ((;;163 + 3) + 1) = ;;167
60		7p3e10 12875	. . . . 5 ⊢ (7 + 3) = ;10
61	13, 19, 60	addcomli 11483	. . . 4 ⊢ (3 + 7) = ;10
62	46, 2, 2, 5, 53, 54, 59, 61	decaddc2 12856	. . 3 ⊢ (;;;1633 + ;37) = ;;;1670
63		8cn 12421	. . . 4 ⊢ 8 ∈ ℂ
64	63	addlidi 11479	. . 3 ⊢ (0 + 8) = 8
65	47, 48, 49, 50, 51, 52, 62, 64	decadd 12854	. 2 ⊢ (;;;;16330 + ;;378) = ;;;;16708
66	3, 4, 7, 6, 27, 42, 45, 65, 48, 4	decpmul 43313	1 ⊢ (;;235 · ;;711) = ;;;;;167085

This proof depends on syntax axioms:   = wceq 1570  (class class class)co 7412  0cc0 11181  1c1 11182   + caddc 11184   · cmul 11186  2c2 12378  3c3 12379  4c4 12380  5c5 12381  6c6 12382  7c7 12383  8c8 12384  ;cdc 12795

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-ltxr 11329  df-sub 11524  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-7 12391  df-8 12392  df-9 12393  df-n0 12588  df-dec 12796

This theorem is used by: (None)