Theorem ex-decpmul 42752

Description: Example usage of decpmul 42734. This proof is significantly longer than 235t711 42751. There is more unnecessary carrying compared to 235t711 42751. Although saving 5 visual steps, using mulcomli 11145 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 12445	. . 3 ⊢ 2 ∈ ℕ0
2		3nn0 12446	. . 3 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12650	. 2 ⊢ ;23 ∈ ℕ0
4		5nn0 12448	. 2 ⊢ 5 ∈ ℕ0
5		7nn0 12450	. . 3 ⊢ 7 ∈ ℕ0
6		1nn0 12444	. . 3 ⊢ 1 ∈ ℕ0
7	5, 6	deccl 12650	. 2 ⊢ ;71 ∈ ℕ0
8		eqid 2737	. . 3 ⊢ ;71 = ;71
9		6nn0 12449	. . . . 5 ⊢ 6 ∈ ℕ0
10	6, 9	deccl 12650	. . . 4 ⊢ ;16 ∈ ℕ0
11		eqid 2737	. . . . 5 ⊢ ;23 = ;23
12		4nn0 12447	. . . . . 6 ⊢ 4 ∈ ℕ0
13		7cn 12266	. . . . . . 7 ⊢ 7 ∈ ℂ
14		2cn 12247	. . . . . . 7 ⊢ 2 ∈ ℂ
15		7t2e14 12744	. . . . . . 7 ⊢ (7 · 2) = ;14
16	13, 14, 15	mulcomli 11145	. . . . . 6 ⊢ (2 · 7) = ;14
17		4p2e6 12320	. . . . . 6 ⊢ (4 + 2) = 6
18	6, 12, 1, 16, 17	decaddi 12695	. . . . 5 ⊢ ((2 · 7) + 2) = ;16
19		3cn 12253	. . . . . 6 ⊢ 3 ∈ ℂ
20		7t3e21 12745	. . . . . 6 ⊢ (7 · 3) = ;21
21	13, 19, 20	mulcomli 11145	. . . . 5 ⊢ (3 · 7) = ;21
22	5, 1, 2, 11, 6, 1, 18, 21	decmul1c 12700	. . . 4 ⊢ (;23 · 7) = ;;161
23		1p2e3 12310	. . . 4 ⊢ (1 + 2) = 3
24	10, 6, 1, 22, 23	decaddi 12695	. . 3 ⊢ ((;23 · 7) + 2) = ;;163
25	3	nn0cni 12440	. . . 4 ⊢ ;23 ∈ ℂ
26	25	mulridi 11140	. . 3 ⊢ (;23 · 1) = ;23
27	3, 5, 6, 8, 2, 1, 24, 26	decmul2c 12701	. 2 ⊢ (;23 · ;71) = ;;;1633
28	2, 4	deccl 12650	. . 3 ⊢ ;35 ∈ ℕ0
29	7	nn0cni 12440	. . . 4 ⊢ ;71 ∈ ℂ
30		5cn 12260	. . . 4 ⊢ 5 ∈ ℂ
31		7t5e35 12747	. . . . 5 ⊢ (7 · 5) = ;35
32	30	mullidi 11141	. . . . 5 ⊢ (1 · 5) = 5
33	4, 5, 6, 8, 31, 32	decmul1 12699	. . . 4 ⊢ (;71 · 5) = ;;355
34	29, 30, 33	mulcomli 11145	. . 3 ⊢ (5 · ;71) = ;;355
35	28	nn0cni 12440	. . . 4 ⊢ ;35 ∈ ℂ
36		eqid 2737	. . . . 5 ⊢ ;35 = ;35
37		5p2e7 12323	. . . . 5 ⊢ (5 + 2) = 7
38	2, 4, 1, 36, 37	decaddi 12695	. . . 4 ⊢ (;35 + 2) = ;37
39	35, 14, 38	addcomli 11329	. . 3 ⊢ (2 + ;35) = ;37
40		5p3e8 12324	. . . 4 ⊢ (5 + 3) = 8
41	30, 19, 40	addcomli 11329	. . 3 ⊢ (3 + 5) = 8
42	1, 2, 28, 4, 26, 34, 39, 41	decadd 12689	. 2 ⊢ ((;23 · 1) + (5 · ;71)) = ;;378
43	30	mulridi 11140	. . 3 ⊢ (5 · 1) = 5
44	4	dec0h 12657	. . 3 ⊢ 5 = ;05
45	43, 44	eqtri 2760	. 2 ⊢ (5 · 1) = ;05
46	10, 2	deccl 12650	. . . 4 ⊢ ;;163 ∈ ℕ0
47	46, 2	deccl 12650	. . 3 ⊢ ;;;1633 ∈ ℕ0
48		0nn0 12443	. . 3 ⊢ 0 ∈ ℕ0
49	2, 5	deccl 12650	. . 3 ⊢ ;37 ∈ ℕ0
50		8nn0 12451	. . 3 ⊢ 8 ∈ ℕ0
51		eqid 2737	. . 3 ⊢ ;;;;16330 = ;;;;16330
52		eqid 2737	. . 3 ⊢ ;;378 = ;;378
53		eqid 2737	. . . 4 ⊢ ;;;1633 = ;;;1633
54		eqid 2737	. . . 4 ⊢ ;37 = ;37
55		eqid 2737	. . . . . 6 ⊢ ;;163 = ;;163
56		3p3e6 12319	. . . . . 6 ⊢ (3 + 3) = 6
57	10, 2, 2, 55, 56	decaddi 12695	. . . . 5 ⊢ (;;163 + 3) = ;;166
58		6p1e7 12315	. . . . 5 ⊢ (6 + 1) = 7
59	10, 9, 6, 57, 58	decaddi 12695	. . . 4 ⊢ ((;;163 + 3) + 1) = ;;167
60		7p3e10 12710	. . . . 5 ⊢ (7 + 3) = ;10
61	13, 19, 60	addcomli 11329	. . . 4 ⊢ (3 + 7) = ;10
62	46, 2, 2, 5, 53, 54, 59, 61	decaddc2 12691	. . 3 ⊢ (;;;1633 + ;37) = ;;;1670
63		8cn 12269	. . . 4 ⊢ 8 ∈ ℂ
64	63	addlidi 11325	. . 3 ⊢ (0 + 8) = 8
65	47, 48, 49, 50, 51, 52, 62, 64	decadd 12689	. 2 ⊢ (;;;;16330 + ;;378) = ;;;;16708
66	3, 4, 7, 6, 27, 42, 45, 65, 48, 4	decpmul 42734	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1542  (class class class)co 7360  0cc0 11029  1c1 11030   + caddc 11032   · cmul 11034  2c2 12227  3c3 12228  4c4 12229  5c5 12230  6c6 12231  7c7 12232  8c8 12233  ;cdc 12635

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-ltxr 11175  df-sub 11370  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-dec 12636

This theorem is referenced by: (None)