Theorem ex-decpmul 40792

Description: Example usage of decpmul 40788. This proof is significantly longer than 235t711 40791. There is more unnecessary carrying compared to 235t711 40791. Although saving 5 visual steps, using mulcomli 11164 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 12430	. . 3 ⊢ 2 ∈ ℕ0
2		3nn0 12431	. . 3 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12633	. 2 ⊢ ;23 ∈ ℕ0
4		5nn0 12433	. 2 ⊢ 5 ∈ ℕ0
5		7nn0 12435	. . 3 ⊢ 7 ∈ ℕ0
6		1nn0 12429	. . 3 ⊢ 1 ∈ ℕ0
7	5, 6	deccl 12633	. 2 ⊢ ;71 ∈ ℕ0
8		eqid 2736	. . 3 ⊢ ;71 = ;71
9		6nn0 12434	. . . . 5 ⊢ 6 ∈ ℕ0
10	6, 9	deccl 12633	. . . 4 ⊢ ;16 ∈ ℕ0
11		eqid 2736	. . . . 5 ⊢ ;23 = ;23
12		4nn0 12432	. . . . . 6 ⊢ 4 ∈ ℕ0
13		7cn 12247	. . . . . . 7 ⊢ 7 ∈ ℂ
14		2cn 12228	. . . . . . 7 ⊢ 2 ∈ ℂ
15		7t2e14 12727	. . . . . . 7 ⊢ (7 · 2) = ;14
16	13, 14, 15	mulcomli 11164	. . . . . 6 ⊢ (2 · 7) = ;14
17		4p2e6 12306	. . . . . 6 ⊢ (4 + 2) = 6
18	6, 12, 1, 16, 17	decaddi 12678	. . . . 5 ⊢ ((2 · 7) + 2) = ;16
19		3cn 12234	. . . . . 6 ⊢ 3 ∈ ℂ
20		7t3e21 12728	. . . . . 6 ⊢ (7 · 3) = ;21
21	13, 19, 20	mulcomli 11164	. . . . 5 ⊢ (3 · 7) = ;21
22	5, 1, 2, 11, 6, 1, 18, 21	decmul1c 12683	. . . 4 ⊢ (;23 · 7) = ;;161
23		1p2e3 12296	. . . 4 ⊢ (1 + 2) = 3
24	10, 6, 1, 22, 23	decaddi 12678	. . 3 ⊢ ((;23 · 7) + 2) = ;;163
25	3	nn0cni 12425	. . . 4 ⊢ ;23 ∈ ℂ
26	25	mulid1i 11159	. . 3 ⊢ (;23 · 1) = ;23
27	3, 5, 6, 8, 2, 1, 24, 26	decmul2c 12684	. 2 ⊢ (;23 · ;71) = ;;;1633
28	2, 4	deccl 12633	. . 3 ⊢ ;35 ∈ ℕ0
29	7	nn0cni 12425	. . . 4 ⊢ ;71 ∈ ℂ
30		5cn 12241	. . . 4 ⊢ 5 ∈ ℂ
31		7t5e35 12730	. . . . 5 ⊢ (7 · 5) = ;35
32	30	mulid2i 11160	. . . . 5 ⊢ (1 · 5) = 5
33	4, 5, 6, 8, 31, 32	decmul1 12682	. . . 4 ⊢ (;71 · 5) = ;;355
34	29, 30, 33	mulcomli 11164	. . 3 ⊢ (5 · ;71) = ;;355
35	28	nn0cni 12425	. . . 4 ⊢ ;35 ∈ ℂ
36		eqid 2736	. . . . 5 ⊢ ;35 = ;35
37		5p2e7 12309	. . . . 5 ⊢ (5 + 2) = 7
38	2, 4, 1, 36, 37	decaddi 12678	. . . 4 ⊢ (;35 + 2) = ;37
39	35, 14, 38	addcomli 11347	. . 3 ⊢ (2 + ;35) = ;37
40		5p3e8 12310	. . . 4 ⊢ (5 + 3) = 8
41	30, 19, 40	addcomli 11347	. . 3 ⊢ (3 + 5) = 8
42	1, 2, 28, 4, 26, 34, 39, 41	decadd 12672	. 2 ⊢ ((;23 · 1) + (5 · ;71)) = ;;378
43	30	mulid1i 11159	. . 3 ⊢ (5 · 1) = 5
44	4	dec0h 12640	. . 3 ⊢ 5 = ;05
45	43, 44	eqtri 2764	. 2 ⊢ (5 · 1) = ;05
46	10, 2	deccl 12633	. . . 4 ⊢ ;;163 ∈ ℕ0
47	46, 2	deccl 12633	. . 3 ⊢ ;;;1633 ∈ ℕ0
48		0nn0 12428	. . 3 ⊢ 0 ∈ ℕ0
49	2, 5	deccl 12633	. . 3 ⊢ ;37 ∈ ℕ0
50		8nn0 12436	. . 3 ⊢ 8 ∈ ℕ0
51		eqid 2736	. . 3 ⊢ ;;;;16330 = ;;;;16330
52		eqid 2736	. . 3 ⊢ ;;378 = ;;378
53		eqid 2736	. . . 4 ⊢ ;;;1633 = ;;;1633
54		eqid 2736	. . . 4 ⊢ ;37 = ;37
55		eqid 2736	. . . . . 6 ⊢ ;;163 = ;;163
56		3p3e6 12305	. . . . . 6 ⊢ (3 + 3) = 6
57	10, 2, 2, 55, 56	decaddi 12678	. . . . 5 ⊢ (;;163 + 3) = ;;166
58		6p1e7 12301	. . . . 5 ⊢ (6 + 1) = 7
59	10, 9, 6, 57, 58	decaddi 12678	. . . 4 ⊢ ((;;163 + 3) + 1) = ;;167
60		7p3e10 12693	. . . . 5 ⊢ (7 + 3) = ;10
61	13, 19, 60	addcomli 11347	. . . 4 ⊢ (3 + 7) = ;10
62	46, 2, 2, 5, 53, 54, 59, 61	decaddc2 12674	. . 3 ⊢ (;;;1633 + ;37) = ;;;1670
63		8cn 12250	. . . 4 ⊢ 8 ∈ ℂ
64	63	addid2i 11343	. . 3 ⊢ (0 + 8) = 8
65	47, 48, 49, 50, 51, 52, 62, 64	decadd 12672	. 2 ⊢ (;;;;16330 + ;;378) = ;;;;16708
66	3, 4, 7, 6, 27, 42, 45, 65, 48, 4	decpmul 40788	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1541  (class class class)co 7357  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  2c2 12208  3c3 12209  4c4 12210  5c5 12211  6c6 12212  7c7 12213  8c8 12214  ;cdc 12618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-ltxr 11194  df-sub 11387  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-dec 12619

This theorem is referenced by: (None)