Theorem ex-decpmul 42790

Description: Example usage of decpmul 42772. This proof is significantly longer than 235t711 42789. There is more unnecessary carrying compared to 235t711 42789. Although saving 5 visual steps, using mulcomli 11152 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 12452	. . 3 ⊢ 2 ∈ ℕ0
2		3nn0 12453	. . 3 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12657	. 2 ⊢ ;23 ∈ ℕ0
4		5nn0 12455	. 2 ⊢ 5 ∈ ℕ0
5		7nn0 12457	. . 3 ⊢ 7 ∈ ℕ0
6		1nn0 12451	. . 3 ⊢ 1 ∈ ℕ0
7	5, 6	deccl 12657	. 2 ⊢ ;71 ∈ ℕ0
8		eqid 2740	. . 3 ⊢ ;71 = ;71
9		6nn0 12456	. . . . 5 ⊢ 6 ∈ ℕ0
10	6, 9	deccl 12657	. . . 4 ⊢ ;16 ∈ ℕ0
11		eqid 2740	. . . . 5 ⊢ ;23 = ;23
12		4nn0 12454	. . . . . 6 ⊢ 4 ∈ ℕ0
13		7cn 12273	. . . . . . 7 ⊢ 7 ∈ ℂ
14		2cn 12254	. . . . . . 7 ⊢ 2 ∈ ℂ
15		7t2e14 12751	. . . . . . 7 ⊢ (7 · 2) = ;14
16	13, 14, 15	mulcomli 11152	. . . . . 6 ⊢ (2 · 7) = ;14
17		4p2e6 12327	. . . . . 6 ⊢ (4 + 2) = 6
18	6, 12, 1, 16, 17	decaddi 12702	. . . . 5 ⊢ ((2 · 7) + 2) = ;16
19		3cn 12260	. . . . . 6 ⊢ 3 ∈ ℂ
20		7t3e21 12752	. . . . . 6 ⊢ (7 · 3) = ;21
21	13, 19, 20	mulcomli 11152	. . . . 5 ⊢ (3 · 7) = ;21
22	5, 1, 2, 11, 6, 1, 18, 21	decmul1c 12707	. . . 4 ⊢ (;23 · 7) = ;;161
23		1p2e3 12317	. . . 4 ⊢ (1 + 2) = 3
24	10, 6, 1, 22, 23	decaddi 12702	. . 3 ⊢ ((;23 · 7) + 2) = ;;163
25	3	nn0cni 12447	. . . 4 ⊢ ;23 ∈ ℂ
26	25	mulridi 11147	. . 3 ⊢ (;23 · 1) = ;23
27	3, 5, 6, 8, 2, 1, 24, 26	decmul2c 12708	. 2 ⊢ (;23 · ;71) = ;;;1633
28	2, 4	deccl 12657	. . 3 ⊢ ;35 ∈ ℕ0
29	7	nn0cni 12447	. . . 4 ⊢ ;71 ∈ ℂ
30		5cn 12267	. . . 4 ⊢ 5 ∈ ℂ
31		7t5e35 12754	. . . . 5 ⊢ (7 · 5) = ;35
32	30	mullidi 11148	. . . . 5 ⊢ (1 · 5) = 5
33	4, 5, 6, 8, 31, 32	decmul1 12706	. . . 4 ⊢ (;71 · 5) = ;;355
34	29, 30, 33	mulcomli 11152	. . 3 ⊢ (5 · ;71) = ;;355
35	28	nn0cni 12447	. . . 4 ⊢ ;35 ∈ ℂ
36		eqid 2740	. . . . 5 ⊢ ;35 = ;35
37		5p2e7 12330	. . . . 5 ⊢ (5 + 2) = 7
38	2, 4, 1, 36, 37	decaddi 12702	. . . 4 ⊢ (;35 + 2) = ;37
39	35, 14, 38	addcomli 11336	. . 3 ⊢ (2 + ;35) = ;37
40		5p3e8 12331	. . . 4 ⊢ (5 + 3) = 8
41	30, 19, 40	addcomli 11336	. . 3 ⊢ (3 + 5) = 8
42	1, 2, 28, 4, 26, 34, 39, 41	decadd 12696	. 2 ⊢ ((;23 · 1) + (5 · ;71)) = ;;378
43	30	mulridi 11147	. . 3 ⊢ (5 · 1) = 5
44	4	dec0h 12664	. . 3 ⊢ 5 = ;05
45	43, 44	eqtri 2763	. 2 ⊢ (5 · 1) = ;05
46	10, 2	deccl 12657	. . . 4 ⊢ ;;163 ∈ ℕ0
47	46, 2	deccl 12657	. . 3 ⊢ ;;;1633 ∈ ℕ0
48		0nn0 12450	. . 3 ⊢ 0 ∈ ℕ0
49	2, 5	deccl 12657	. . 3 ⊢ ;37 ∈ ℕ0
50		8nn0 12458	. . 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 12326	. . . . . 6 ⊢ (3 + 3) = 6
57	10, 2, 2, 55, 56	decaddi 12702	. . . . 5 ⊢ (;;163 + 3) = ;;166
58		6p1e7 12322	. . . . 5 ⊢ (6 + 1) = 7
59	10, 9, 6, 57, 58	decaddi 12702	. . . 4 ⊢ ((;;163 + 3) + 1) = ;;167
60		7p3e10 12717	. . . . 5 ⊢ (7 + 3) = ;10
61	13, 19, 60	addcomli 11336	. . . 4 ⊢ (3 + 7) = ;10
62	46, 2, 2, 5, 53, 54, 59, 61	decaddc2 12698	. . 3 ⊢ (;;;1633 + ;37) = ;;;1670
63		8cn 12276	. . . 4 ⊢ 8 ∈ ℂ
64	63	addlidi 11332	. . 3 ⊢ (0 + 8) = 8
65	47, 48, 49, 50, 51, 52, 62, 64	decadd 12696	. 2 ⊢ (;;;;16330 + ;;378) = ;;;;16708
66	3, 4, 7, 6, 27, 42, 45, 65, 48, 4	decpmul 42772	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1547  (class class class)co 7363  0cc0 11036  1c1 11037   + caddc 11039   · cmul 11041  2c2 12234  3c3 12235  4c4 12236  5c5 12237  6c6 12238  7c7 12239  8c8 12240  ;cdc 12642

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-ltxr 11182  df-sub 11377  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-n0 12436  df-dec 12643

This theorem is referenced by: (None)