Theorem pw2recs 28757

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 7416	. . . . . 6 ⊢ (𝑚 = 0s → (2s↑s𝑚) = (2s↑s 0s ))
2		2no 28738	. . . . . . 7 ⊢ 2s ∈ No
3		exps0 28746	. . . . . . 7 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (2s↑s 0s ) = 1s
5	1, 4	eqtrdi 2811	. . . . 5 ⊢ (𝑚 = 0s → (2s↑s𝑚) = 1s )
6	5	oveq1d 7423	. . . 4 ⊢ (𝑚 = 0s → ((2s↑s𝑚) ·s 𝑥) = ( 1s ·s 𝑥))
7	6	eqeq1d 2762	. . 3 ⊢ (𝑚 = 0s → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ( 1s ·s 𝑥) = 1s ))
8	7	rexbidv 3186	. 2 ⊢ (𝑚 = 0s → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ( 1s ·s 𝑥) = 1s ))
9		oveq2 7416	. . . . 5 ⊢ (𝑚 = 𝑛 → (2s↑s𝑚) = (2s↑s𝑛))
10	9	oveq1d 7423	. . . 4 ⊢ (𝑚 = 𝑛 → ((2s↑s𝑚) ·s 𝑥) = ((2s↑s𝑛) ·s 𝑥))
11	10	eqeq1d 2762	. . 3 ⊢ (𝑚 = 𝑛 → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ((2s↑s𝑛) ·s 𝑥) = 1s ))
12	11	rexbidv 3186	. 2 ⊢ (𝑚 = 𝑛 → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ((2s↑s𝑛) ·s 𝑥) = 1s ))
13		oveq2 7416	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
14	13	oveq1d 7423	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → ((2s↑s𝑚) ·s 𝑥) = ((2s↑s(𝑛 +s 1s )) ·s 𝑥))
15	14	eqeq1d 2762	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ((2s↑s(𝑛 +s 1s )) ·s 𝑥) = 1s ))
16	15	rexbidv 3186	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑥) = 1s ))
17		oveq2 7416	. . . . 5 ⊢ (𝑥 = 𝑦 → ((2s↑s(𝑛 +s 1s )) ·s 𝑥) = ((2s↑s(𝑛 +s 1s )) ·s 𝑦))
18	17	eqeq1d 2762	. . . 4 ⊢ (𝑥 = 𝑦 → (((2s↑s(𝑛 +s 1s )) ·s 𝑥) = 1s ↔ ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s ))
19	18	cbvrexvw 3241	. . 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 7416	. . . . 5 ⊢ (𝑚 = 𝑁 → (2s↑s𝑚) = (2s↑s𝑁))
22	21	oveq1d 7423	. . . 4 ⊢ (𝑚 = 𝑁 → ((2s↑s𝑚) ·s 𝑥) = ((2s↑s𝑁) ·s 𝑥))
23	22	eqeq1d 2762	. . 3 ⊢ (𝑚 = 𝑁 → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ((2s↑s𝑁) ·s 𝑥) = 1s ))
24	23	rexbidv 3186	. 2 ⊢ (𝑚 = 𝑁 → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ((2s↑s𝑁) ·s 𝑥) = 1s ))
25		1no 28129	. . 3 ⊢ 1s ∈ No
26		mulsrid 28432	. . . 4 ⊢ ( 1s ∈ No → ( 1s ·s 1s ) = 1s )
27	25, 26	ax-mp 5	. . 3 ⊢ ( 1s ·s 1s ) = 1s
28		oveq2 7416	. . . . 5 ⊢ (𝑥 = 1s → ( 1s ·s 𝑥) = ( 1s ·s 1s ))
29	28	eqeq1d 2762	. . . 4 ⊢ (𝑥 = 1s → (( 1s ·s 𝑥) = 1s ↔ ( 1s ·s 1s ) = 1s ))
30	29	rspcev 3576	. . 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 7416	. . . . 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 2762	. . . 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 28128	. . . . . . . 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 28131	. . . . . . . 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 28089	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → { 0s } <<s { 1s })
41	40	cutscld 28102	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ({ 0s } |s { 1s }) ∈ No )
42	34, 41	mulscld 28454	. . . 4 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (𝑥 ·s ({ 0s } |s { 1s })) ∈ No )
43		expsp1 28748	. . . . . . . 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 7423	. . . . 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 28750	. . . . . . . 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 28487	. . . . 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 28742	. . . . . . . 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 7426	. . . . . 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 2811	. . . . 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 2799	. . . 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 3579	. . 3 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ∃𝑦 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s )
59	58	rexlimdvaa 3164	. 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 28652	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 3086  {csn 4583   class class class wbr 5102  (class class class)co 7408   No csur 27930   <s clts 27931   |s ccuts 28078   0_s c0s 28124   1_s c1s 28125   +_s cadds 28278   ·_s cmuls 28425  ℕ_0scn0s 28631  2_sc2s 28729  ↑_scexps 28731

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 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-oadd 8458  df-nadd 8653  df-no 27933  df-lts 27934  df-bday 27935  df-les 28035  df-slts 28077  df-cuts 28079  df-0s 28126  df-1s 28127  df-made 28146  df-old 28147  df-left 28149  df-right 28150  df-norec 28257  df-norec2 28268  df-adds 28279  df-negs 28340  df-subs 28341  df-muls 28426  df-seqs 28603  df-n0s 28633  df-nns 28634  df-zs 28698  df-2s 28730  df-exps 28732

This theorem is used by:  pw2divscld  28758  pw2divmulsd  28759  pw2divscan2d  28761  pw2divsassd  28762  pw2ltdivmulsd  28769  pw2ltmuldivs2d  28770  pw2ltdivmuls2d  28776  z12zsodd  28801