Theorem pw2recs 28448

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 7370	. . . . . 6 ⊢ (𝑚 = 0s → (2s↑s𝑚) = (2s↑s 0s ))
2		2no 28429	. . . . . . 7 ⊢ 2s ∈ No
3		exps0 28437	. . . . . . 7 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (2s↑s 0s ) = 1s
5	1, 4	eqtrdi 2788	. . . . 5 ⊢ (𝑚 = 0s → (2s↑s𝑚) = 1s )
6	5	oveq1d 7377	. . . 4 ⊢ (𝑚 = 0s → ((2s↑s𝑚) ·s 𝑥) = ( 1s ·s 𝑥))
7	6	eqeq1d 2739	. . 3 ⊢ (𝑚 = 0s → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ( 1s ·s 𝑥) = 1s ))
8	7	rexbidv 3162	. 2 ⊢ (𝑚 = 0s → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ( 1s ·s 𝑥) = 1s ))
9		oveq2 7370	. . . . 5 ⊢ (𝑚 = 𝑛 → (2s↑s𝑚) = (2s↑s𝑛))
10	9	oveq1d 7377	. . . 4 ⊢ (𝑚 = 𝑛 → ((2s↑s𝑚) ·s 𝑥) = ((2s↑s𝑛) ·s 𝑥))
11	10	eqeq1d 2739	. . 3 ⊢ (𝑚 = 𝑛 → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ((2s↑s𝑛) ·s 𝑥) = 1s ))
12	11	rexbidv 3162	. 2 ⊢ (𝑚 = 𝑛 → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ((2s↑s𝑛) ·s 𝑥) = 1s ))
13		oveq2 7370	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
14	13	oveq1d 7377	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → ((2s↑s𝑚) ·s 𝑥) = ((2s↑s(𝑛 +s 1s )) ·s 𝑥))
15	14	eqeq1d 2739	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ((2s↑s(𝑛 +s 1s )) ·s 𝑥) = 1s ))
16	15	rexbidv 3162	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑥) = 1s ))
17		oveq2 7370	. . . . 5 ⊢ (𝑥 = 𝑦 → ((2s↑s(𝑛 +s 1s )) ·s 𝑥) = ((2s↑s(𝑛 +s 1s )) ·s 𝑦))
18	17	eqeq1d 2739	. . . 4 ⊢ (𝑥 = 𝑦 → (((2s↑s(𝑛 +s 1s )) ·s 𝑥) = 1s ↔ ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s ))
19	18	cbvrexvw 3217	. . 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 287	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑦 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s ))
21		oveq2 7370	. . . . 5 ⊢ (𝑚 = 𝑁 → (2s↑s𝑚) = (2s↑s𝑁))
22	21	oveq1d 7377	. . . 4 ⊢ (𝑚 = 𝑁 → ((2s↑s𝑚) ·s 𝑥) = ((2s↑s𝑁) ·s 𝑥))
23	22	eqeq1d 2739	. . 3 ⊢ (𝑚 = 𝑁 → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ((2s↑s𝑁) ·s 𝑥) = 1s ))
24	23	rexbidv 3162	. 2 ⊢ (𝑚 = 𝑁 → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ((2s↑s𝑁) ·s 𝑥) = 1s ))
25		1no 27820	. . 3 ⊢ 1s ∈ No
26		mulsrid 28123	. . . 4 ⊢ ( 1s ∈ No → ( 1s ·s 1s ) = 1s )
27	25, 26	ax-mp 5	. . 3 ⊢ ( 1s ·s 1s ) = 1s
28		oveq2 7370	. . . . 5 ⊢ (𝑥 = 1s → ( 1s ·s 𝑥) = ( 1s ·s 1s ))
29	28	eqeq1d 2739	. . . 4 ⊢ (𝑥 = 1s → (( 1s ·s 𝑥) = 1s ↔ ( 1s ·s 1s ) = 1s ))
30	29	rspcev 3565	. . 3 ⊢ (( 1s ∈ No ∧ ( 1s ·s 1s ) = 1s ) → ∃𝑥 ∈ No ( 1s ·s 𝑥) = 1s )
31	25, 27, 30	mp2an 693	. 2 ⊢ ∃𝑥 ∈ No ( 1s ·s 𝑥) = 1s
32		oveq2 7370	. . . . 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 2739	. . . 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 771	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → 𝑥 ∈ No )
35		0no 27819	. . . . . . . 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 27822	. . . . . . . 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 27780	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → { 0s } <<s { 1s })
41	40	cutscld 27793	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ({ 0s } |s { 1s }) ∈ No )
42	34, 41	mulscld 28145	. . . 4 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (𝑥 ·s ({ 0s } |s { 1s })) ∈ No )
43		expsp1 28439	. . . . . . . 8 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s) → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
44	2, 43	mpan 691	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
45	44	adantr 480	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
46	45	oveq1d 7377	. . . . 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 28441	. . . . . . . 8 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s) → (2s↑s𝑛) ∈ No )
48	2, 47	mpan 691	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (2s↑s𝑛) ∈ No )
49	48	adantr 480	. . . . . 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 28178	. . . . 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 773	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ((2s↑s𝑛) ·s 𝑥) = 1s )
53		twocut 28433	. . . . . . . 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 7380	. . . . . 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 2788	. . . . 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 2776	. . . 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 3568	. . 3 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ∃𝑦 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s )
59	58	rexlimdvaa 3140	. 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 28343	1 ⊢ (𝑁 ∈ ℕ0s → ∃𝑥 ∈ No ((2s↑s𝑁) ·s 𝑥) = 1s )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {csn 4568   class class class wbr 5086  (class class class)co 7362   No csur 27621   <s clts 27622   |s ccuts 27769   0_s c0s 27815   1_s c1s 27816   +_s cadds 27969   ·_s cmuls 28116  ℕ_0scn0s 28322  2_sc2s 28420  ↑_scexps 28422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-oadd 8404  df-nadd 8597  df-no 27624  df-lts 27625  df-bday 27626  df-les 27727  df-slts 27768  df-cuts 27770  df-0s 27817  df-1s 27818  df-made 27837  df-old 27838  df-left 27840  df-right 27841  df-norec 27948  df-norec2 27959  df-adds 27970  df-negs 28031  df-subs 28032  df-muls 28117  df-seqs 28294  df-n0s 28324  df-nns 28325  df-zs 28389  df-2s 28421  df-exps 28423

This theorem is referenced by:  pw2divscld  28449  pw2divmulsd  28450  pw2divscan2d  28452  pw2divsassd  28453  pw2ltdivmulsd  28460  pw2ltmuldivs2d  28461  pw2ltdivmuls2d  28467  z12zsodd  28492