Theorem pw2recs 28701

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 7424	. . . . . 6 ⊢ (𝑚 = 0s → (2s↑s𝑚) = (2s↑s 0s ))
2		2no 28682	. . . . . . 7 ⊢ 2s ∈ No
3		exps0 28690	. . . . . . 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 7431	. . . 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 7424	. . . . 5 ⊢ (𝑚 = 𝑛 → (2s↑s𝑚) = (2s↑s𝑛))
10	9	oveq1d 7431	. . . 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 7424	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
14	13	oveq1d 7431	. . . . 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 7424	. . . . 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 7424	. . . . 5 ⊢ (𝑚 = 𝑁 → (2s↑s𝑚) = (2s↑s𝑁))
22	21	oveq1d 7431	. . . 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 28073	. . 3 ⊢ 1s ∈ No
26		mulsrid 28376	. . . 4 ⊢ ( 1s ∈ No → ( 1s ·s 1s ) = 1s )
27	25, 26	ax-mp 5	. . 3 ⊢ ( 1s ·s 1s ) = 1s
28		oveq2 7424	. . . . 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 3579	. . 3 ⊢ (( 1s ∈ No ∧ ( 1s ·s 1s ) = 1s ) → ∃𝑥 ∈ No ( 1s ·s 𝑥) = 1s )
31	25, 27, 30	mp2an 705	. 2 ⊢ ∃𝑥 ∈ No ( 1s ·s 𝑥) = 1s
32		oveq2 7424	. . . . 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 783	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → 𝑥 ∈ No )
35		0no 28072	. . . . . . . 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 28075	. . . . . . . 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 28033	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → { 0s } <<s { 1s })
41	40	cutscld 28046	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ({ 0s } |s { 1s }) ∈ No )
42	34, 41	mulscld 28398	. . . 4 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (𝑥 ·s ({ 0s } |s { 1s })) ∈ No )
43		expsp1 28692	. . . . . . . 8 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s) → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
44	2, 43	mpan 703	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
45	44	adantr 486	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
46	45	oveq1d 7431	. . . . 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 28694	. . . . . . . 8 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s) → (2s↑s𝑛) ∈ No )
48	2, 47	mpan 703	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (2s↑s𝑛) ∈ No )
49	48	adantr 486	. . . . . 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 28431	. . . . 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 785	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ((2s↑s𝑛) ·s 𝑥) = 1s )
53		twocut 28686	. . . . . . . 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 7434	. . . . . 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 3582	. . 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 28596	1 ⊢ (𝑁 ∈ ℕ0s → ∃𝑥 ∈ No ((2s↑s𝑁) ·s 𝑥) = 1s )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  {csn 4587   class class class wbr 5107  (class class class)co 7416   No csur 27874   <s clts 27875   |s ccuts 28022   0_s c0s 28068   1_s c1s 28069   +_s cadds 28222   ·_s cmuls 28369  ℕ_0scn0s 28575  2_sc2s 28673  ↑_scexps 28675

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 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 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-oadd 8462  df-nadd 8657  df-no 27877  df-lts 27878  df-bday 27879  df-les 27979  df-slts 28021  df-cuts 28023  df-0s 28070  df-1s 28071  df-made 28090  df-old 28091  df-left 28093  df-right 28094  df-norec 28201  df-norec2 28212  df-adds 28223  df-negs 28284  df-subs 28285  df-muls 28370  df-seqs 28547  df-n0s 28577  df-nns 28578  df-zs 28642  df-2s 28674  df-exps 28676

This theorem is used by:  pw2divscld  28702  pw2divmulsd  28703  pw2divscan2d  28705  pw2divsassd  28706  pw2ltdivmulsd  28713  pw2ltmuldivs2d  28714  pw2ltdivmuls2d  28720  z12zsodd  28745