Theorem pw2recs 28642

Description: Any power of two has a multiplicative inverse. Note that this theorem does not require the axiom of infinity. (Contributed by Scott Fenton, 5-Sep-2025.)

Assertion

Ref	Expression
pw2recs	⊢ (𝑁 ∈ ℕ0s → ∃𝑥 ∈ No ((2s↑s𝑁) ·s 𝑥) = 1s )

Distinct variable group:   𝑥,𝑁

Proof of Theorem pw2recs

Dummy variables 𝑛 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7420	. . . . . 6 ⊢ (𝑚 = 0s → (2s↑s𝑚) = (2s↑s 0s ))
2		2no 28623	. . . . . . 7 ⊢ 2s ∈ No
3		exps0 28631	. . . . . . 7 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (2s↑s 0s ) = 1s
5	1, 4	eqtrdi 2813	. . . . 5 ⊢ (𝑚 = 0s → (2s↑s𝑚) = 1s )
6	5	oveq1d 7427	. . . 4 ⊢ (𝑚 = 0s → ((2s↑s𝑚) ·s 𝑥) = ( 1s ·s 𝑥))
7	6	eqeq1d 2764	. . 3 ⊢ (𝑚 = 0s → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ( 1s ·s 𝑥) = 1s ))
8	7	rexbidv 3188	. 2 ⊢ (𝑚 = 0s → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ( 1s ·s 𝑥) = 1s ))
9		oveq2 7420	. . . . 5 ⊢ (𝑚 = 𝑛 → (2s↑s𝑚) = (2s↑s𝑛))
10	9	oveq1d 7427	. . . 4 ⊢ (𝑚 = 𝑛 → ((2s↑s𝑚) ·s 𝑥) = ((2s↑s𝑛) ·s 𝑥))
11	10	eqeq1d 2764	. . 3 ⊢ (𝑚 = 𝑛 → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ((2s↑s𝑛) ·s 𝑥) = 1s ))
12	11	rexbidv 3188	. 2 ⊢ (𝑚 = 𝑛 → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ((2s↑s𝑛) ·s 𝑥) = 1s ))
13		oveq2 7420	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
14	13	oveq1d 7427	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → ((2s↑s𝑚) ·s 𝑥) = ((2s↑s(𝑛 +s 1s )) ·s 𝑥))
15	14	eqeq1d 2764	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ((2s↑s(𝑛 +s 1s )) ·s 𝑥) = 1s ))
16	15	rexbidv 3188	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑥) = 1s ))
17		oveq2 7420	. . . . 5 ⊢ (𝑥 = 𝑦 → ((2s↑s(𝑛 +s 1s )) ·s 𝑥) = ((2s↑s(𝑛 +s 1s )) ·s 𝑦))
18	17	eqeq1d 2764	. . . 4 ⊢ (𝑥 = 𝑦 → (((2s↑s(𝑛 +s 1s )) ·s 𝑥) = 1s ↔ ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s ))
19	18	cbvrexvw 3243	. . 3 ⊢ (∃𝑥 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑥) = 1s ↔ ∃𝑦 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s )
20	16, 19	bitrdi 290	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑦 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s ))
21		oveq2 7420	. . . . 5 ⊢ (𝑚 = 𝑁 → (2s↑s𝑚) = (2s↑s𝑁))
22	21	oveq1d 7427	. . . 4 ⊢ (𝑚 = 𝑁 → ((2s↑s𝑚) ·s 𝑥) = ((2s↑s𝑁) ·s 𝑥))
23	22	eqeq1d 2764	. . 3 ⊢ (𝑚 = 𝑁 → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ((2s↑s𝑁) ·s 𝑥) = 1s ))
24	23	rexbidv 3188	. 2 ⊢ (𝑚 = 𝑁 → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ((2s↑s𝑁) ·s 𝑥) = 1s ))
25		1no 28014	. . 3 ⊢ 1s ∈ No
26		mulsrid 28317	. . . 4 ⊢ ( 1s ∈ No → ( 1s ·s 1s ) = 1s )
27	25, 26	ax-mp 5	. . 3 ⊢ ( 1s ·s 1s ) = 1s
28		oveq2 7420	. . . . 5 ⊢ (𝑥 = 1s → ( 1s ·s 𝑥) = ( 1s ·s 1s ))
29	28	eqeq1d 2764	. . . 4 ⊢ (𝑥 = 1s → (( 1s ·s 𝑥) = 1s ↔ ( 1s ·s 1s ) = 1s ))
30	29	rspcev 3580	. . 3 ⊢ (( 1s ∈ No ∧ ( 1s ·s 1s ) = 1s ) → ∃𝑥 ∈ No ( 1s ·s 𝑥) = 1s )
31	25, 27, 30	mp2an 704	. 2 ⊢ ∃𝑥 ∈ No ( 1s ·s 𝑥) = 1s
32		oveq2 7420	. . . . 5 ⊢ (𝑦 = (𝑥 ·s ({ 0s } |s { 1s })) → ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = ((2s↑s(𝑛 +s 1s )) ·s (𝑥 ·s ({ 0s } |s { 1s }))))
33	32	eqeq1d 2764	. . . 4 ⊢ (𝑦 = (𝑥 ·s ({ 0s } |s { 1s })) → (((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s ↔ ((2s↑s(𝑛 +s 1s )) ·s (𝑥 ·s ({ 0s } |s { 1s }))) = 1s ))
34		simprl 782	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → 𝑥 ∈ No )
35		0no 28013	. . . . . . . 8 ⊢ 0s ∈ No
36	35	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → 0s ∈ No )
37	25	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → 1s ∈ No )
38		0lt1s 28016	. . . . . . . 8 ⊢ 0s <s 1s
39	38	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → 0s <s 1s )
40	36, 37, 39	sltssn 27974	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → { 0s } <<s { 1s })
41	40	cutscld 27987	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ({ 0s } |s { 1s }) ∈ No )
42	34, 41	mulscld 28339	. . . 4 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (𝑥 ·s ({ 0s } |s { 1s })) ∈ No )
43		expsp1 28633	. . . . . . . 8 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s) → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
44	2, 43	mpan 702	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
45	44	adantr 485	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
46	45	oveq1d 7427	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ((2s↑s(𝑛 +s 1s )) ·s (𝑥 ·s ({ 0s } |s { 1s }))) = (((2s↑s𝑛) ·s 2s) ·s (𝑥 ·s ({ 0s } |s { 1s }))))
47		expscl 28635	. . . . . . . 8 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s) → (2s↑s𝑛) ∈ No )
48	2, 47	mpan 702	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (2s↑s𝑛) ∈ No )
49	48	adantr 485	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (2s↑s𝑛) ∈ No )
50	2	a1i 11	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → 2s ∈ No )
51	49, 50, 34, 41	muls4d 28372	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (((2s↑s𝑛) ·s 2s) ·s (𝑥 ·s ({ 0s } |s { 1s }))) = (((2s↑s𝑛) ·s 𝑥) ·s (2s ·s ({ 0s } |s { 1s }))))
52		simprr 784	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ((2s↑s𝑛) ·s 𝑥) = 1s )
53		twocut 28627	. . . . . . . 8 ⊢ (2s ·s ({ 0s } |s { 1s })) = 1s
54	53	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (2s ·s ({ 0s } |s { 1s })) = 1s )
55	52, 54	oveq12d 7430	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (((2s↑s𝑛) ·s 𝑥) ·s (2s ·s ({ 0s } |s { 1s }))) = ( 1s ·s 1s ))
56	55, 27	eqtrdi 2813	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (((2s↑s𝑛) ·s 𝑥) ·s (2s ·s ({ 0s } |s { 1s }))) = 1s )
57	46, 51, 56	3eqtrd 2801	. . . 4 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ((2s↑s(𝑛 +s 1s )) ·s (𝑥 ·s ({ 0s } |s { 1s }))) = 1s )
58	33, 42, 57	rspcedvdw 3583	. . 3 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ∃𝑦 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s )
59	58	rexlimdvaa 3166	. 2 ⊢ (𝑛 ∈ ℕ0s → (∃𝑥 ∈ No ((2s↑s𝑛) ·s 𝑥) = 1s → ∃𝑦 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s ))
60	8, 12, 20, 24, 31, 59	n0sind 28537	1 ⊢ (𝑁 ∈ ℕ0s → ∃𝑥 ∈ No ((2s↑s𝑁) ·s 𝑥) = 1s )

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∃wrex 3088  {csn 4588   class class class wbr 5108  (class class class)co 7412   No csur 27815   <s clts 27816   |s ccuts 27963   0_s c0s 28009   1_s c1s 28010   +_s cadds 28163   ·_s cmuls 28310  ℕ_0scn0s 28516  2_sc2s 28614  ↑_scexps 28616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-oadd 8455  df-nadd 8650  df-no 27818  df-lts 27819  df-bday 27820  df-les 27920  df-slts 27962  df-cuts 27964  df-0s 28011  df-1s 28012  df-made 28031  df-old 28032  df-left 28034  df-right 28035  df-norec 28142  df-norec2 28153  df-adds 28164  df-negs 28225  df-subs 28226  df-muls 28311  df-seqs 28488  df-n0s 28518  df-nns 28519  df-zs 28583  df-2s 28615  df-exps 28617

This theorem is used by:  pw2divscld  28643  pw2divmulsd  28644  pw2divscan2d  28646  pw2divsassd  28647  pw2ltdivmulsd  28654  pw2ltmuldivs2d  28655  pw2ltdivmuls2d  28661  z12zsodd  28686