Theorem fmtno5fac 43737

Description: The factorisation of the 5 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 22-Jul-2021.)

Assertion

Ref	Expression
fmtno5fac	⊢ (FermatNo‘5) = (;;;;;;6700417 · ;;641)

Proof of Theorem fmtno5fac

Step	Hyp	Ref	Expression
1		4nn0 11910	. . . . . . . . . . 11 ⊢ 4 ∈ ℕ0
2		2nn0 11908	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ0
3	1, 2	deccl 12107	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ0
4		8nn0 11914	. . . . . . . . . 10 ⊢ 8 ∈ ℕ0
5	3, 4	deccl 12107	. . . . . . . . 9 ⊢ ;;428 ∈ ℕ0
6	5, 4	deccl 12107	. . . . . . . 8 ⊢ ;;;4288 ∈ ℕ0
7	6, 2	deccl 12107	. . . . . . 7 ⊢ ;;;;42882 ∈ ℕ0
8		6nn0 11912	. . . . . . 7 ⊢ 6 ∈ ℕ0
9	7, 8	deccl 12107	. . . . . 6 ⊢ ;;;;;428826 ∈ ℕ0
10	9, 8	deccl 12107	. . . . 5 ⊢ ;;;;;;4288266 ∈ ℕ0
11	10, 4	deccl 12107	. . . 4 ⊢ ;;;;;;;42882668 ∈ ℕ0
12	11, 4	deccl 12107	. . 3 ⊢ ;;;;;;;;428826688 ∈ ℕ0
13		0nn0 11906	. . 3 ⊢ 0 ∈ ℕ0
14		7nn0 11913	. . . . . . . 8 ⊢ 7 ∈ ℕ0
15	8, 14	deccl 12107	. . . . . . 7 ⊢ ;67 ∈ ℕ0
16	15, 13	deccl 12107	. . . . . 6 ⊢ ;;670 ∈ ℕ0
17	16, 13	deccl 12107	. . . . 5 ⊢ ;;;6700 ∈ ℕ0
18	17, 1	deccl 12107	. . . 4 ⊢ ;;;;67004 ∈ ℕ0
19		1nn0 11907	. . . 4 ⊢ 1 ∈ ℕ0
20	18, 19	deccl 12107	. . 3 ⊢ ;;;;;670041 ∈ ℕ0
21		fmtno5faclem3 43736	. . . 4 ⊢ (;;;;;;;;402025020 + ;;;;;;;26801668) = ;;;;;;;;428826688
22	21	deceq1i 12099	. . 3 ⊢ ;(;;;;;;;;402025020 + ;;;;;;;26801668)0 = ;;;;;;;;;4288266880
23		eqid 2821	. . 3 ⊢ ;;;;;;6700417 = ;;;;;;6700417
24		eqid 2821	. . . 4 ⊢ ;;;;;;;;428826688 = ;;;;;;;;428826688
25		eqid 2821	. . . 4 ⊢ ;;;;;670041 = ;;;;;670041
26		eqid 2821	. . . . 5 ⊢ ;;;;;;;42882668 = ;;;;;;;42882668
27		eqid 2821	. . . . 5 ⊢ ;;;;67004 = ;;;;67004
28		9nn0 11915	. . . . . . . . . 10 ⊢ 9 ∈ ℕ0
29	3, 28	deccl 12107	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ0
30	29, 1	deccl 12107	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ0
31	30, 28	deccl 12107	. . . . . . 7 ⊢ ;;;;42949 ∈ ℕ0
32	31, 8	deccl 12107	. . . . . 6 ⊢ ;;;;;429496 ∈ ℕ0
33		6p1e7 11779	. . . . . 6 ⊢ (6 + 1) = 7
34		eqid 2821	. . . . . . 7 ⊢ ;;;;;;4288266 = ;;;;;;4288266
35		eqid 2821	. . . . . . 7 ⊢ ;;;6700 = ;;;6700
36		eqid 2821	. . . . . . . 8 ⊢ ;;;;;428826 = ;;;;;428826
37		eqid 2821	. . . . . . . 8 ⊢ ;;670 = ;;670
38		eqid 2821	. . . . . . . . 9 ⊢ ;;;;42882 = ;;;;42882
39		eqid 2821	. . . . . . . . 9 ⊢ ;67 = ;67
40		eqid 2821	. . . . . . . . . 10 ⊢ ;;;4288 = ;;;4288
41		8p1e9 11781	. . . . . . . . . . 11 ⊢ (8 + 1) = 9
42		eqid 2821	. . . . . . . . . . 11 ⊢ ;;428 = ;;428
43	3, 4, 41, 42	decsuc 12123	. . . . . . . . . 10 ⊢ (;;428 + 1) = ;;429
44		8p6e14 12176	. . . . . . . . . 10 ⊢ (8 + 6) = ;14
45	5, 4, 8, 40, 43, 1, 44	decaddci 12153	. . . . . . . . 9 ⊢ (;;;4288 + 6) = ;;;4294
46		7cn 11725	. . . . . . . . . 10 ⊢ 7 ∈ ℂ
47		2cn 11706	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
48		7p2e9 11792	. . . . . . . . . 10 ⊢ (7 + 2) = 9
49	46, 47, 48	addcomli 10826	. . . . . . . . 9 ⊢ (2 + 7) = 9
50	6, 2, 8, 14, 38, 39, 45, 49	decadd 12146	. . . . . . . 8 ⊢ (;;;;42882 + ;67) = ;;;;42949
51		6cn 11722	. . . . . . . . 9 ⊢ 6 ∈ ℂ
52	51	addid1i 10821	. . . . . . . 8 ⊢ (6 + 0) = 6
53	7, 8, 15, 13, 36, 37, 50, 52	decadd 12146	. . . . . . 7 ⊢ (;;;;;428826 + ;;670) = ;;;;;429496
54	9, 8, 16, 13, 34, 35, 53, 52	decadd 12146	. . . . . 6 ⊢ (;;;;;;4288266 + ;;;6700) = ;;;;;;4294966
55	32, 8, 33, 54	decsuc 12123	. . . . 5 ⊢ ((;;;;;;4288266 + ;;;6700) + 1) = ;;;;;;4294967
56		8p4e12 12174	. . . . 5 ⊢ (8 + 4) = ;12
57	10, 4, 17, 1, 26, 27, 55, 2, 56	decaddc 12147	. . . 4 ⊢ (;;;;;;;42882668 + ;;;;67004) = ;;;;;;;42949672
58	11, 4, 18, 19, 24, 25, 57, 41	decadd 12146	. . 3 ⊢ (;;;;;;;;428826688 + ;;;;;670041) = ;;;;;;;;429496729
59	46	addid2i 10822	. . 3 ⊢ (0 + 7) = 7
60	12, 13, 20, 14, 22, 23, 58, 59	decadd 12146	. 2 ⊢ (;(;;;;;;;;402025020 + ;;;;;;;26801668)0 + ;;;;;;6700417) = ;;;;;;;;;4294967297
61	8, 1	deccl 12107	. . 3 ⊢ ;64 ∈ ℕ0
62	20, 14	deccl 12107	. . 3 ⊢ ;;;;;;6700417 ∈ ℕ0
63		fmtno5faclem2 43735	. . . . . 6 ⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502
64	63	eqcomi 2830	. . . . 5 ⊢ ;;;;;;;40202502 = (;;;;;;6700417 · 6)
65		fmtno5faclem1 43734	. . . . . 6 ⊢ (;;;;;;6700417 · 4) = ;;;;;;;26801668
66	65	eqcomi 2830	. . . . 5 ⊢ ;;;;;;;26801668 = (;;;;;;6700417 · 4)
67	8, 1, 62, 64, 66	decmul10add 12161	. . . 4 ⊢ (;;;;;;6700417 · ;64) = (;;;;;;;;402025020 + ;;;;;;;26801668)
68	67	eqcomi 2830	. . 3 ⊢ (;;;;;;;;402025020 + ;;;;;;;26801668) = (;;;;;;6700417 · ;64)
69	62	nn0cni 11903	. . . . 5 ⊢ ;;;;;;6700417 ∈ ℂ
70	69	mulid1i 10639	. . . 4 ⊢ (;;;;;;6700417 · 1) = ;;;;;;6700417
71	70	eqcomi 2830	. . 3 ⊢ ;;;;;;6700417 = (;;;;;;6700417 · 1)
72	61, 19, 62, 68, 71	decmul10add 12161	. 2 ⊢ (;;;;;;6700417 · ;;641) = (;(;;;;;;;;402025020 + ;;;;;;;26801668)0 + ;;;;;;6700417)
73		fmtno5 43712	. 2 ⊢ (FermatNo‘5) = ;;;;;;;;;4294967297
74	60, 72, 73	3eqtr4ri 2855	1 ⊢ (FermatNo‘5) = (;;;;;;6700417 · ;;641)

Syntax hints:   = wceq 1533  ‘cfv 6350  (class class class)co 7150  0cc0 10531  1c1 10532   + caddc 10534   · cmul 10536  2c2 11686  4c4 11688  5c5 11689  6c6 11690  7c7 11691  8c8 11692  9c9 11693  ;cdc 12092  FermatNocfmtno 43682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-seq 13364  df-exp 13424  df-fmtno 43683

This theorem is referenced by:  fmtno5nprm  43738