Theorem ex-decpmul 42424

Description: Example usage of decpmul 42406. This proof is significantly longer than 235t711 42423. There is more unnecessary carrying compared to 235t711 42423. Although saving 5 visual steps, using mulcomli 11128 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 12405	. . 3 ⊢ 2 ∈ ℕ0
2		3nn0 12406	. . 3 ⊢ 3 ∈ ℕ0
3	1, 2	deccl 12609	. 2 ⊢ ;23 ∈ ℕ0
4		5nn0 12408	. 2 ⊢ 5 ∈ ℕ0
5		7nn0 12410	. . 3 ⊢ 7 ∈ ℕ0
6		1nn0 12404	. . 3 ⊢ 1 ∈ ℕ0
7	5, 6	deccl 12609	. 2 ⊢ ;71 ∈ ℕ0
8		eqid 2733	. . 3 ⊢ ;71 = ;71
9		6nn0 12409	. . . . 5 ⊢ 6 ∈ ℕ0
10	6, 9	deccl 12609	. . . 4 ⊢ ;16 ∈ ℕ0
11		eqid 2733	. . . . 5 ⊢ ;23 = ;23
12		4nn0 12407	. . . . . 6 ⊢ 4 ∈ ℕ0
13		7cn 12226	. . . . . . 7 ⊢ 7 ∈ ℂ
14		2cn 12207	. . . . . . 7 ⊢ 2 ∈ ℂ
15		7t2e14 12703	. . . . . . 7 ⊢ (7 · 2) = ;14
16	13, 14, 15	mulcomli 11128	. . . . . 6 ⊢ (2 · 7) = ;14
17		4p2e6 12280	. . . . . 6 ⊢ (4 + 2) = 6
18	6, 12, 1, 16, 17	decaddi 12654	. . . . 5 ⊢ ((2 · 7) + 2) = ;16
19		3cn 12213	. . . . . 6 ⊢ 3 ∈ ℂ
20		7t3e21 12704	. . . . . 6 ⊢ (7 · 3) = ;21
21	13, 19, 20	mulcomli 11128	. . . . 5 ⊢ (3 · 7) = ;21
22	5, 1, 2, 11, 6, 1, 18, 21	decmul1c 12659	. . . 4 ⊢ (;23 · 7) = ;;161
23		1p2e3 12270	. . . 4 ⊢ (1 + 2) = 3
24	10, 6, 1, 22, 23	decaddi 12654	. . 3 ⊢ ((;23 · 7) + 2) = ;;163
25	3	nn0cni 12400	. . . 4 ⊢ ;23 ∈ ℂ
26	25	mulridi 11123	. . 3 ⊢ (;23 · 1) = ;23
27	3, 5, 6, 8, 2, 1, 24, 26	decmul2c 12660	. 2 ⊢ (;23 · ;71) = ;;;1633
28	2, 4	deccl 12609	. . 3 ⊢ ;35 ∈ ℕ0
29	7	nn0cni 12400	. . . 4 ⊢ ;71 ∈ ℂ
30		5cn 12220	. . . 4 ⊢ 5 ∈ ℂ
31		7t5e35 12706	. . . . 5 ⊢ (7 · 5) = ;35
32	30	mullidi 11124	. . . . 5 ⊢ (1 · 5) = 5
33	4, 5, 6, 8, 31, 32	decmul1 12658	. . . 4 ⊢ (;71 · 5) = ;;355
34	29, 30, 33	mulcomli 11128	. . 3 ⊢ (5 · ;71) = ;;355
35	28	nn0cni 12400	. . . 4 ⊢ ;35 ∈ ℂ
36		eqid 2733	. . . . 5 ⊢ ;35 = ;35
37		5p2e7 12283	. . . . 5 ⊢ (5 + 2) = 7
38	2, 4, 1, 36, 37	decaddi 12654	. . . 4 ⊢ (;35 + 2) = ;37
39	35, 14, 38	addcomli 11312	. . 3 ⊢ (2 + ;35) = ;37
40		5p3e8 12284	. . . 4 ⊢ (5 + 3) = 8
41	30, 19, 40	addcomli 11312	. . 3 ⊢ (3 + 5) = 8
42	1, 2, 28, 4, 26, 34, 39, 41	decadd 12648	. 2 ⊢ ((;23 · 1) + (5 · ;71)) = ;;378
43	30	mulridi 11123	. . 3 ⊢ (5 · 1) = 5
44	4	dec0h 12616	. . 3 ⊢ 5 = ;05
45	43, 44	eqtri 2756	. 2 ⊢ (5 · 1) = ;05
46	10, 2	deccl 12609	. . . 4 ⊢ ;;163 ∈ ℕ0
47	46, 2	deccl 12609	. . 3 ⊢ ;;;1633 ∈ ℕ0
48		0nn0 12403	. . 3 ⊢ 0 ∈ ℕ0
49	2, 5	deccl 12609	. . 3 ⊢ ;37 ∈ ℕ0
50		8nn0 12411	. . 3 ⊢ 8 ∈ ℕ0
51		eqid 2733	. . 3 ⊢ ;;;;16330 = ;;;;16330
52		eqid 2733	. . 3 ⊢ ;;378 = ;;378
53		eqid 2733	. . . 4 ⊢ ;;;1633 = ;;;1633
54		eqid 2733	. . . 4 ⊢ ;37 = ;37
55		eqid 2733	. . . . . 6 ⊢ ;;163 = ;;163
56		3p3e6 12279	. . . . . 6 ⊢ (3 + 3) = 6
57	10, 2, 2, 55, 56	decaddi 12654	. . . . 5 ⊢ (;;163 + 3) = ;;166
58		6p1e7 12275	. . . . 5 ⊢ (6 + 1) = 7
59	10, 9, 6, 57, 58	decaddi 12654	. . . 4 ⊢ ((;;163 + 3) + 1) = ;;167
60		7p3e10 12669	. . . . 5 ⊢ (7 + 3) = ;10
61	13, 19, 60	addcomli 11312	. . . 4 ⊢ (3 + 7) = ;10
62	46, 2, 2, 5, 53, 54, 59, 61	decaddc2 12650	. . 3 ⊢ (;;;1633 + ;37) = ;;;1670
63		8cn 12229	. . . 4 ⊢ 8 ∈ ℂ
64	63	addlidi 11308	. . 3 ⊢ (0 + 8) = 8
65	47, 48, 49, 50, 51, 52, 62, 64	decadd 12648	. 2 ⊢ (;;;;16330 + ;;378) = ;;;;16708
66	3, 4, 7, 6, 27, 42, 45, 65, 48, 4	decpmul 42406	1 ⊢ (;;235 · ;;711) = ;;;;;167085

Syntax hints:   = wceq 1541  (class class class)co 7352  0cc0 11013  1c1 11014   + caddc 11016   · cmul 11018  2c2 12187  3c3 12188  4c4 12189  5c5 12190  6c6 12191  7c7 12192  8c8 12193  ;cdc 12594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-pnf 11155  df-mnf 11156  df-ltxr 11158  df-sub 11353  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-dec 12595

This theorem is referenced by: (None)