Theorem ex-decpmul 42294

Description: Example usage of decpmul 42277. This proof is significantly longer than 235t711 42293. There is more unnecessary carrying compared to 235t711 42293. 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 12570	. . 3 ⊢ 2 ∈ ℕ0
2		3nn0 12571	. . 3 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12773	. 2 ⊢ ;23 ∈ ℕ0
4		5nn0 12573	. 2 ⊢ 5 ∈ ℕ0
5		7nn0 12575	. . 3 ⊢ 7 ∈ ℕ0
6		1nn0 12569	. . 3 ⊢ 1 ∈ ℕ0
7	5, 6	deccl 12773	. 2 ⊢ ;71 ∈ ℕ0
8		eqid 2740	. . 3 ⊢ ;71 = ;71
9		6nn0 12574	. . . . 5 ⊢ 6 ∈ ℕ0
10	6, 9	deccl 12773	. . . 4 ⊢ ;16 ∈ ℕ0
11		eqid 2740	. . . . 5 ⊢ ;23 = ;23
12		4nn0 12572	. . . . . 6 ⊢ 4 ∈ ℕ0
13		7cn 12387	. . . . . . 7 ⊢ 7 ∈ ℂ
14		2cn 12368	. . . . . . 7 ⊢ 2 ∈ ℂ
15		7t2e14 12867	. . . . . . 7 ⊢ (7 · 2) = ;14
16	13, 14, 15	mulcomli 11299	. . . . . 6 ⊢ (2 · 7) = ;14
17		4p2e6 12446	. . . . . 6 ⊢ (4 + 2) = 6
18	6, 12, 1, 16, 17	decaddi 12818	. . . . 5 ⊢ ((2 · 7) + 2) = ;16
19		3cn 12374	. . . . . 6 ⊢ 3 ∈ ℂ
20		7t3e21 12868	. . . . . 6 ⊢ (7 · 3) = ;21
21	13, 19, 20	mulcomli 11299	. . . . 5 ⊢ (3 · 7) = ;21
22	5, 1, 2, 11, 6, 1, 18, 21	decmul1c 12823	. . . 4 ⊢ (;23 · 7) = ;;161
23		1p2e3 12436	. . . 4 ⊢ (1 + 2) = 3
24	10, 6, 1, 22, 23	decaddi 12818	. . 3 ⊢ ((;23 · 7) + 2) = ;;163
25	3	nn0cni 12565	. . . 4 ⊢ ;23 ∈ ℂ
26	25	mulridi 11294	. . 3 ⊢ (;23 · 1) = ;23
27	3, 5, 6, 8, 2, 1, 24, 26	decmul2c 12824	. 2 ⊢ (;23 · ;71) = ;;;1633
28	2, 4	deccl 12773	. . 3 ⊢ ;35 ∈ ℕ0
29	7	nn0cni 12565	. . . 4 ⊢ ;71 ∈ ℂ
30		5cn 12381	. . . 4 ⊢ 5 ∈ ℂ
31		7t5e35 12870	. . . . 5 ⊢ (7 · 5) = ;35
32	30	mullidi 11295	. . . . 5 ⊢ (1 · 5) = 5
33	4, 5, 6, 8, 31, 32	decmul1 12822	. . . 4 ⊢ (;71 · 5) = ;;355
34	29, 30, 33	mulcomli 11299	. . 3 ⊢ (5 · ;71) = ;;355
35	28	nn0cni 12565	. . . 4 ⊢ ;35 ∈ ℂ
36		eqid 2740	. . . . 5 ⊢ ;35 = ;35
37		5p2e7 12449	. . . . 5 ⊢ (5 + 2) = 7
38	2, 4, 1, 36, 37	decaddi 12818	. . . 4 ⊢ (;35 + 2) = ;37
39	35, 14, 38	addcomli 11482	. . 3 ⊢ (2 + ;35) = ;37
40		5p3e8 12450	. . . 4 ⊢ (5 + 3) = 8
41	30, 19, 40	addcomli 11482	. . 3 ⊢ (3 + 5) = 8
42	1, 2, 28, 4, 26, 34, 39, 41	decadd 12812	. 2 ⊢ ((;23 · 1) + (5 · ;71)) = ;;378
43	30	mulridi 11294	. . 3 ⊢ (5 · 1) = 5
44	4	dec0h 12780	. . 3 ⊢ 5 = ;05
45	43, 44	eqtri 2768	. 2 ⊢ (5 · 1) = ;05
46	10, 2	deccl 12773	. . . 4 ⊢ ;;163 ∈ ℕ0
47	46, 2	deccl 12773	. . 3 ⊢ ;;;1633 ∈ ℕ0
48		0nn0 12568	. . 3 ⊢ 0 ∈ ℕ0
49	2, 5	deccl 12773	. . 3 ⊢ ;37 ∈ ℕ0
50		8nn0 12576	. . 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 12445	. . . . . 6 ⊢ (3 + 3) = 6
57	10, 2, 2, 55, 56	decaddi 12818	. . . . 5 ⊢ (;;163 + 3) = ;;166
58		6p1e7 12441	. . . . 5 ⊢ (6 + 1) = 7
59	10, 9, 6, 57, 58	decaddi 12818	. . . 4 ⊢ ((;;163 + 3) + 1) = ;;167
60		7p3e10 12833	. . . . 5 ⊢ (7 + 3) = ;10
61	13, 19, 60	addcomli 11482	. . . 4 ⊢ (3 + 7) = ;10
62	46, 2, 2, 5, 53, 54, 59, 61	decaddc2 12814	. . 3 ⊢ (;;;1633 + ;37) = ;;;1670
63		8cn 12390	. . . 4 ⊢ 8 ∈ ℂ
64	63	addlidi 11478	. . 3 ⊢ (0 + 8) = 8
65	47, 48, 49, 50, 51, 52, 62, 64	decadd 12812	. 2 ⊢ (;;;;16330 + ;;378) = ;;;;16708
66	3, 4, 7, 6, 27, 42, 45, 65, 48, 4	decpmul 42277	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1537  (class class class)co 7448  0cc0 11184  1c1 11185   + caddc 11187   · cmul 11189  2c2 12348  3c3 12349  4c4 12350  5c5 12351  6c6 12352  7c7 12353  8c8 12354  ;cdc 12758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-ltxr 11329  df-sub 11522  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-dec 12759

This theorem is referenced by: (None)