fmtno5lem4 - Mathbox for Alexander van der Vekens

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Theorem fmtno5lem4 48366

Description: Lemma 4 for fmtno5 48367. (Contributed by AV, 30-Jul-2021.)

**Assertion**
Ref	Expression
fmtno5lem4	⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296

**Proof of Theorem fmtno5lem4**
Step	Hyp	Ref	Expression
1		6nn0 12540	. . . . . . . 8 ⊢ 6 ∈ ℕ₀
2		5nn0 12539	. . . . . . . 8 ⊢ 5 ∈ ℕ₀
3	1, 2	deccl 12742	. . . . . . 7 ⊢ ;65 ∈ ℕ₀
4	3, 2	deccl 12742	. . . . . 6 ⊢ ;;655 ∈ ℕ₀
5		3nn0 12537	. . . . . 6 ⊢ 3 ∈ ℕ₀
6	4, 5	deccl 12742	. . . . 5 ⊢ ;;;6553 ∈ ℕ₀
7	6, 1	deccl 12742	. . . 4 ⊢ ;;;;65536 ∈ ℕ₀
8	7	nn0cni 12531	. . 3 ⊢ ;;;;65536 ∈ ℂ
9	8	sqvali 14234	. 2 ⊢ (;;;;65536↑2) = (;;;;65536 · ;;;;65536)
10		fmtno5lem1 48363	. . . . . . . . . 10 ⊢ (;;;;65536 · 6) = ;;;;;393216
11	10	eqcomi 2774	. . . . . . . . 9 ⊢ ;;;;;393216 = (;;;;65536 · 6)
12		fmtno5lem2 48364	. . . . . . . . . 10 ⊢ (;;;;65536 · 5) = ;;;;;327680
13	12	eqcomi 2774	. . . . . . . . 9 ⊢ ;;;;;327680 = (;;;;65536 · 5)
14	1, 2, 7, 11, 13	decmul10add 12801	. . . . . . . 8 ⊢ (;;;;65536 · ;65) = (;;;;;;3932160 + ;;;;;327680)
15	14	eqcomi 2774	. . . . . . 7 ⊢ (;;;;;;3932160 + ;;;;;327680) = (;;;;65536 · ;65)
16	3, 2, 7, 15, 13	decmul10add 12801	. . . . . 6 ⊢ (;;;;65536 · ;;655) = (;(;;;;;;3932160 + ;;;;;327680)0 + ;;;;;327680)
17	16	eqcomi 2774	. . . . 5 ⊢ (;(;;;;;;3932160 + ;;;;;327680)0 + ;;;;;327680) = (;;;;65536 · ;;655)
18		fmtno5lem3 48365	. . . . . 6 ⊢ (;;;;65536 · 3) = ;;;;;196608
19	18	eqcomi 2774	. . . . 5 ⊢ ;;;;;196608 = (;;;;65536 · 3)
20	4, 5, 7, 17, 19	decmul10add 12801	. . . 4 ⊢ (;;;;65536 · ;;;6553) = (;(;(;;;;;;3932160 + ;;;;;327680)0 + ;;;;;327680)0 + ;;;;;196608)
21	20	eqcomi 2774	. . 3 ⊢ (;(;(;;;;;;3932160 + ;;;;;327680)0 + ;;;;;327680)0 + ;;;;;196608) = (;;;;65536 · ;;;6553)
22	6, 1, 7, 21, 11	decmul10add 12801	. 2 ⊢ (;;;;65536 · ;;;;65536) = (;(;(;(;;;;;;3932160 + ;;;;;327680)0 + ;;;;;327680)0 + ;;;;;196608)0 + ;;;;;393216)
23		4nn0 12538	. . . . . . . . . . 11 ⊢ 4 ∈ ℕ₀
24		2nn0 12536	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
25	23, 24	deccl 12742	. . . . . . . . . 10 ⊢ ;42 ∈ ℕ₀
26		9nn0 12543	. . . . . . . . . 10 ⊢ 9 ∈ ℕ₀
27	25, 26	deccl 12742	. . . . . . . . 9 ⊢ ;;429 ∈ ℕ₀
28	27, 23	deccl 12742	. . . . . . . 8 ⊢ ;;;4294 ∈ ℕ₀
29	28, 2	deccl 12742	. . . . . . 7 ⊢ ;;;;42945 ∈ ℕ₀
30		7nn0 12541	. . . . . . 7 ⊢ 7 ∈ ℕ₀
31	29, 30	deccl 12742	. . . . . 6 ⊢ ;;;;;429457 ∈ ℕ₀
32	31, 23	deccl 12742	. . . . 5 ⊢ ;;;;;;4294574 ∈ ℕ₀
33		0nn0 12534	. . . . 5 ⊢ 0 ∈ ℕ₀
34	32, 33	deccl 12742	. . . 4 ⊢ ;;;;;;;42945740 ∈ ℕ₀
35		8nn0 12542	. . . 4 ⊢ 8 ∈ ℕ₀
36	34, 35	deccl 12742	. . 3 ⊢ ;;;;;;;;429457408 ∈ ℕ₀
37	5, 26	deccl 12742	. . . . . 6 ⊢ ;39 ∈ ℕ₀
38	37, 5	deccl 12742	. . . . 5 ⊢ ;;393 ∈ ℕ₀
39	38, 24	deccl 12742	. . . 4 ⊢ ;;;3932 ∈ ℕ₀
40		1nn0 12535	. . . 4 ⊢ 1 ∈ ℕ₀
41	39, 40	deccl 12742	. . 3 ⊢ ;;;;39321 ∈ ℕ₀
42	27, 24	deccl 12742	. . . . . . . . 9 ⊢ ;;;4292 ∈ ℕ₀
43	42, 1	deccl 12742	. . . . . . . 8 ⊢ ;;;;42926 ∈ ℕ₀
44	43, 33	deccl 12742	. . . . . . 7 ⊢ ;;;;;429260 ∈ ℕ₀
45	44, 35	deccl 12742	. . . . . 6 ⊢ ;;;;;;4292608 ∈ ℕ₀
46	45, 33	deccl 12742	. . . . 5 ⊢ ;;;;;;;42926080 ∈ ℕ₀
47	40, 26	deccl 12742	. . . . . . . 8 ⊢ ;19 ∈ ℕ₀
48	47, 1	deccl 12742	. . . . . . 7 ⊢ ;;196 ∈ ℕ₀
49	48, 1	deccl 12742	. . . . . 6 ⊢ ;;;1966 ∈ ℕ₀
50	49, 33	deccl 12742	. . . . 5 ⊢ ;;;;19660 ∈ ℕ₀
51	25, 2	deccl 12742	. . . . . . . . . . 11 ⊢ ;;425 ∈ ℕ₀
52	51, 26	deccl 12742	. . . . . . . . . 10 ⊢ ;;;4259 ∈ ℕ₀
53	52, 35	deccl 12742	. . . . . . . . 9 ⊢ ;;;;42598 ∈ ℕ₀
54	53, 23	deccl 12742	. . . . . . . 8 ⊢ ;;;;;425984 ∈ ℕ₀
55	54, 33	deccl 12742	. . . . . . 7 ⊢ ;;;;;;4259840 ∈ ℕ₀
56	5, 24	deccl 12742	. . . . . . . . . 10 ⊢ ;32 ∈ ℕ₀
57	56, 30	deccl 12742	. . . . . . . . 9 ⊢ ;;327 ∈ ℕ₀
58	57, 1	deccl 12742	. . . . . . . 8 ⊢ ;;;3276 ∈ ℕ₀
59	58, 35	deccl 12742	. . . . . . 7 ⊢ ;;;;32768 ∈ ℕ₀
60	41, 1	deccl 12742	. . . . . . . . 9 ⊢ ;;;;;393216 ∈ ℕ₀
61		eqid 2765	. . . . . . . . 9 ⊢ ;;;;;;3932160 = ;;;;;;3932160
62		eqid 2765	. . . . . . . . 9 ⊢ ;;;;;327680 = ;;;;;327680
63		eqid 2765	. . . . . . . . . 10 ⊢ ;;;;;393216 = ;;;;;393216
64		eqid 2765	. . . . . . . . . 10 ⊢ ;;;;32768 = ;;;;32768
65		7p1e8 12404	. . . . . . . . . . 11 ⊢ (7 + 1) = 8
66		eqid 2765	. . . . . . . . . . . 12 ⊢ ;;;;39321 = ;;;;39321
67		eqid 2765	. . . . . . . . . . . 12 ⊢ ;;;3276 = ;;;3276
68		eqid 2765	. . . . . . . . . . . . 13 ⊢ ;;;3932 = ;;;3932
69		eqid 2765	. . . . . . . . . . . . 13 ⊢ ;;327 = ;;327
70		eqid 2765	. . . . . . . . . . . . . 14 ⊢ ;;393 = ;;393
71		eqid 2765	. . . . . . . . . . . . . 14 ⊢ ;32 = ;32
72		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ ;39 = ;39
73		3p1e4 12400	. . . . . . . . . . . . . . 15 ⊢ (3 + 1) = 4
74		9p3e12 12820	. . . . . . . . . . . . . . 15 ⊢ (9 + 3) = ;12
75	5, 26, 5, 72, 73, 24, 74	decaddci 12793	. . . . . . . . . . . . . 14 ⊢ (;39 + 3) = ;42
76		3p2e5 12406	. . . . . . . . . . . . . 14 ⊢ (3 + 2) = 5
77	37, 5, 5, 24, 70, 71, 75, 76	decadd 12786	. . . . . . . . . . . . 13 ⊢ (;;393 + ;32) = ;;425
78		7cn 12350	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℂ
79		2cn 12331	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℂ
80		7p2e9 12416	. . . . . . . . . . . . . 14 ⊢ (7 + 2) = 9
81	78, 79, 80	addcomli 11417	. . . . . . . . . . . . 13 ⊢ (2 + 7) = 9
82	38, 24, 56, 30, 68, 69, 77, 81	decadd 12786	. . . . . . . . . . . 12 ⊢ (;;;3932 + ;;327) = ;;;4259
83		6cn 12347	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℂ
84		ax-1cn 11173	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
85		6p1e7 12403	. . . . . . . . . . . . 13 ⊢ (6 + 1) = 7
86	83, 84, 85	addcomli 11417	. . . . . . . . . . . 12 ⊢ (1 + 6) = 7
87	39, 40, 57, 1, 66, 67, 82, 86	decadd 12786	. . . . . . . . . . 11 ⊢ (;;;;39321 + ;;;3276) = ;;;;42597
88	52, 30, 65, 87	decsuc 12763	. . . . . . . . . 10 ⊢ ((;;;;39321 + ;;;3276) + 1) = ;;;;42598
89		8cn 12353	. . . . . . . . . . 11 ⊢ 8 ∈ ℂ
90		8p6e14 12816	. . . . . . . . . . 11 ⊢ (8 + 6) = ;14
91	89, 83, 90	addcomli 11417	. . . . . . . . . 10 ⊢ (6 + 8) = ;14
92	41, 1, 58, 35, 63, 64, 88, 23, 91	decaddc 12787	. . . . . . . . 9 ⊢ (;;;;;393216 + ;;;;32768) = ;;;;;425984
93		00id 11400	. . . . . . . . 9 ⊢ (0 + 0) = 0
94	60, 33, 59, 33, 61, 62, 92, 93	decadd 12786	. . . . . . . 8 ⊢ (;;;;;;3932160 + ;;;;;327680) = ;;;;;;4259840
95	94	deceq1i 12734	. . . . . . 7 ⊢ ;(;;;;;;3932160 + ;;;;;327680)0 = ;;;;;;;42598400
96		eqid 2765	. . . . . . . 8 ⊢ ;;;;;;4259840 = ;;;;;;4259840
97		eqid 2765	. . . . . . . . 9 ⊢ ;;;;;425984 = ;;;;;425984
98		5p1e6 12402	. . . . . . . . . 10 ⊢ (5 + 1) = 6
99		eqid 2765	. . . . . . . . . . 11 ⊢ ;;;;42598 = ;;;;42598
100		1p1e2 12379	. . . . . . . . . . . 12 ⊢ (1 + 1) = 2
101		eqid 2765	. . . . . . . . . . . . 13 ⊢ ;;;4259 = ;;;4259
102		8p1e9 12405	. . . . . . . . . . . . . 14 ⊢ (8 + 1) = 9
103		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ ;;425 = ;;425
104		5p3e8 12412	. . . . . . . . . . . . . . 15 ⊢ (5 + 3) = 8
105	25, 2, 5, 103, 104	decaddi 12792	. . . . . . . . . . . . . 14 ⊢ (;;425 + 3) = ;;428
106	25, 35, 102, 105	decsuc 12763	. . . . . . . . . . . . 13 ⊢ ((;;425 + 3) + 1) = ;;429
107		9p2e11 12819	. . . . . . . . . . . . 13 ⊢ (9 + 2) = ;11
108	51, 26, 5, 24, 101, 71, 106, 40, 107	decaddc 12787	. . . . . . . . . . . 12 ⊢ (;;;4259 + ;32) = ;;;4291
109	27, 40, 100, 108	decsuc 12763	. . . . . . . . . . 11 ⊢ ((;;;4259 + ;32) + 1) = ;;;4292
110		8p7e15 12817	. . . . . . . . . . 11 ⊢ (8 + 7) = ;15
111	52, 35, 56, 30, 99, 69, 109, 2, 110	decaddc 12787	. . . . . . . . . 10 ⊢ (;;;;42598 + ;;327) = ;;;;42925
112	42, 2, 98, 111	decsuc 12763	. . . . . . . . 9 ⊢ ((;;;;42598 + ;;327) + 1) = ;;;;42926
113		4cn 12341	. . . . . . . . . 10 ⊢ 4 ∈ ℂ
114		6p4e10 12804	. . . . . . . . . 10 ⊢ (6 + 4) = ;10
115	83, 113, 114	addcomli 11417	. . . . . . . . 9 ⊢ (4 + 6) = ;10
116	53, 23, 57, 1, 97, 67, 112, 33, 115	decaddc 12787	. . . . . . . 8 ⊢ (;;;;;425984 + ;;;3276) = ;;;;;429260
117	89	addlidi 11413	. . . . . . . 8 ⊢ (0 + 8) = 8
118	54, 33, 58, 35, 96, 64, 116, 117	decadd 12786	. . . . . . 7 ⊢ (;;;;;;4259840 + ;;;;32768) = ;;;;;;4292608
119	55, 33, 59, 33, 95, 62, 118, 93	decadd 12786	. . . . . 6 ⊢ (;(;;;;;;3932160 + ;;;;;327680)0 + ;;;;;327680) = ;;;;;;;42926080
120	119	deceq1i 12734	. . . . 5 ⊢ ;(;(;;;;;;3932160 + ;;;;;327680)0 + ;;;;;327680)0 = ;;;;;;;;429260800
121		eqid 2765	. . . . 5 ⊢ ;;;;;196608 = ;;;;;196608
122	45, 49	decaddm10 12791	. . . . . 6 ⊢ (;;;;;;;42926080 + ;;;;19660) = ;(;;;;;;4292608 + ;;;1966)0
123		eqid 2765	. . . . . . . 8 ⊢ ;;;;;;4292608 = ;;;;;;4292608
124		eqid 2765	. . . . . . . 8 ⊢ ;;;1966 = ;;;1966
125		eqid 2765	. . . . . . . . . 10 ⊢ ;;;;;429260 = ;;;;;429260
126		eqid 2765	. . . . . . . . . 10 ⊢ ;;196 = ;;196
127		eqid 2765	. . . . . . . . . . 11 ⊢ ;;;;42926 = ;;;;42926
128		eqid 2765	. . . . . . . . . . 11 ⊢ ;19 = ;19
129		2p1e3 12397	. . . . . . . . . . . . 13 ⊢ (2 + 1) = 3
130		eqid 2765	. . . . . . . . . . . . 13 ⊢ ;;;4292 = ;;;4292
131	27, 24, 129, 130	decsuc 12763	. . . . . . . . . . . 12 ⊢ (;;;4292 + 1) = ;;;4293
132	27, 5, 73, 131	decsuc 12763	. . . . . . . . . . 11 ⊢ ((;;;4292 + 1) + 1) = ;;;4294
133		9cn 12356	. . . . . . . . . . . 12 ⊢ 9 ∈ ℂ
134		9p6e15 12823	. . . . . . . . . . . 12 ⊢ (9 + 6) = ;15
135	133, 83, 134	addcomli 11417	. . . . . . . . . . 11 ⊢ (6 + 9) = ;15
136	42, 1, 40, 26, 127, 128, 132, 2, 135	decaddc 12787	. . . . . . . . . 10 ⊢ (;;;;42926 + ;19) = ;;;;42945
137	83	addlidi 11413	. . . . . . . . . 10 ⊢ (0 + 6) = 6
138	43, 33, 47, 1, 125, 126, 136, 137	decadd 12786	. . . . . . . . 9 ⊢ (;;;;;429260 + ;;196) = ;;;;;429456
139	29, 1, 85, 138	decsuc 12763	. . . . . . . 8 ⊢ ((;;;;;429260 + ;;196) + 1) = ;;;;;429457
140	44, 35, 48, 1, 123, 124, 139, 23, 90	decaddc 12787	. . . . . . 7 ⊢ (;;;;;;4292608 + ;;;1966) = ;;;;;;4294574
141	140	deceq1i 12734	. . . . . 6 ⊢ ;(;;;;;;4292608 + ;;;1966)0 = ;;;;;;;42945740
142	122, 141	eqtri 2788	. . . . 5 ⊢ (;;;;;;;42926080 + ;;;;19660) = ;;;;;;;42945740
143	46, 33, 50, 35, 120, 121, 142, 117	decadd 12786	. . . 4 ⊢ (;(;(;;;;;;3932160 + ;;;;;327680)0 + ;;;;;327680)0 + ;;;;;196608) = ;;;;;;;;429457408
144	143	deceq1i 12734	. . 3 ⊢ ;(;(;(;;;;;;3932160 + ;;;;;327680)0 + ;;;;;327680)0 + ;;;;;196608)0 = ;;;;;;;;;4294574080
145		eqid 2765	. . . 4 ⊢ ;;;;;;;;429457408 = ;;;;;;;;429457408
146		eqid 2765	. . . . 5 ⊢ ;;;;;;;42945740 = ;;;;;;;42945740
147		eqid 2765	. . . . . 6 ⊢ ;;;;;;4294574 = ;;;;;;4294574
148		eqid 2765	. . . . . . 7 ⊢ ;;;;;429457 = ;;;;;429457
149		eqid 2765	. . . . . . . . 9 ⊢ ;;;;42945 = ;;;;42945
150	28, 2, 5, 149, 104	decaddi 12792	. . . . . . . 8 ⊢ (;;;;42945 + 3) = ;;;;42948
151	28, 35, 102, 150	decsuc 12763	. . . . . . 7 ⊢ ((;;;;42945 + 3) + 1) = ;;;;42949
152		9p7e16 12824	. . . . . . . 8 ⊢ (9 + 7) = ;16
153	133, 78, 152	addcomli 11417	. . . . . . 7 ⊢ (7 + 9) = ;16
154	29, 30, 5, 26, 148, 72, 151, 1, 153	decaddc 12787	. . . . . 6 ⊢ (;;;;;429457 + ;39) = ;;;;;429496
155		4p3e7 12409	. . . . . 6 ⊢ (4 + 3) = 7
156	31, 23, 37, 5, 147, 70, 154, 155	decadd 12786	. . . . 5 ⊢ (;;;;;;4294574 + ;;393) = ;;;;;;4294967
157	79	addlidi 11413	. . . . 5 ⊢ (0 + 2) = 2
158	32, 33, 38, 24, 146, 68, 156, 157	decadd 12786	. . . 4 ⊢ (;;;;;;;42945740 + ;;;3932) = ;;;;;;;42949672
159	34, 35, 39, 40, 145, 66, 158, 102	decadd 12786	. . 3 ⊢ (;;;;;;;;429457408 + ;;;;39321) = ;;;;;;;;429496729
160	36, 33, 41, 1, 144, 63, 159, 137	decadd 12786	. 2 ⊢ (;(;(;(;;;;;;3932160 + ;;;;;327680)0 + ;;;;;327680)0 + ;;;;;196608)0 + ;;;;;393216) = ;;;;;;;;;4294967296
161	9, 22, 160	3eqtri 2792	1 ⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296

Colors of variables: wff setvar class

This proof depends on syntax axioms: = wceq 1570 (class class class)co 7419 0cc0 11115 1c1 11116 + caddc 11118 · cmul 11120 2c2 12310 3c3 12311 4c4 12312 5c5 12313 6c6 12314 7c7 12315 8c8 12316 9c9 12317 cdc 12727 ↑cexp 14115

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-seq 14056 df-exp 14116

This theorem is used by: fmtno5 48367

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