Theorem pw2recs 28452

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 7368	. . . . . 6 ⊢ (𝑚 = 0s → (2s↑s𝑚) = (2s↑s 0s ))
2		2no 28433	. . . . . . 7 ⊢ 2s ∈ No
3		exps0 28441	. . . . . . 7 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
4	2, 3	ax-mp 5	. . . . . 6 ⊢ (2s↑s 0s ) = 1s
5	1, 4	eqtrdi 2792	. . . . 5 ⊢ (𝑚 = 0s → (2s↑s𝑚) = 1s )
6	5	oveq1d 7375	. . . 4 ⊢ (𝑚 = 0s → ((2s↑s𝑚) ·s 𝑥) = ( 1s ·s 𝑥))
7	6	eqeq1d 2743	. . 3 ⊢ (𝑚 = 0s → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ( 1s ·s 𝑥) = 1s ))
8	7	rexbidv 3165	. 2 ⊢ (𝑚 = 0s → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ( 1s ·s 𝑥) = 1s ))
9		oveq2 7368	. . . . 5 ⊢ (𝑚 = 𝑛 → (2s↑s𝑚) = (2s↑s𝑛))
10	9	oveq1d 7375	. . . 4 ⊢ (𝑚 = 𝑛 → ((2s↑s𝑚) ·s 𝑥) = ((2s↑s𝑛) ·s 𝑥))
11	10	eqeq1d 2743	. . 3 ⊢ (𝑚 = 𝑛 → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ((2s↑s𝑛) ·s 𝑥) = 1s ))
12	11	rexbidv 3165	. 2 ⊢ (𝑚 = 𝑛 → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ((2s↑s𝑛) ·s 𝑥) = 1s ))
13		oveq2 7368	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
14	13	oveq1d 7375	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → ((2s↑s𝑚) ·s 𝑥) = ((2s↑s(𝑛 +s 1s )) ·s 𝑥))
15	14	eqeq1d 2743	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ((2s↑s(𝑛 +s 1s )) ·s 𝑥) = 1s ))
16	15	rexbidv 3165	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑥) = 1s ))
17		oveq2 7368	. . . . 5 ⊢ (𝑥 = 𝑦 → ((2s↑s(𝑛 +s 1s )) ·s 𝑥) = ((2s↑s(𝑛 +s 1s )) ·s 𝑦))
18	17	eqeq1d 2743	. . . 4 ⊢ (𝑥 = 𝑦 → (((2s↑s(𝑛 +s 1s )) ·s 𝑥) = 1s ↔ ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s ))
19	18	cbvrexvw 3220	. . 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 289	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑦 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s ))
21		oveq2 7368	. . . . 5 ⊢ (𝑚 = 𝑁 → (2s↑s𝑚) = (2s↑s𝑁))
22	21	oveq1d 7375	. . . 4 ⊢ (𝑚 = 𝑁 → ((2s↑s𝑚) ·s 𝑥) = ((2s↑s𝑁) ·s 𝑥))
23	22	eqeq1d 2743	. . 3 ⊢ (𝑚 = 𝑁 → (((2s↑s𝑚) ·s 𝑥) = 1s ↔ ((2s↑s𝑁) ·s 𝑥) = 1s ))
24	23	rexbidv 3165	. 2 ⊢ (𝑚 = 𝑁 → (∃𝑥 ∈ No ((2s↑s𝑚) ·s 𝑥) = 1s ↔ ∃𝑥 ∈ No ((2s↑s𝑁) ·s 𝑥) = 1s ))
25		1no 27824	. . 3 ⊢ 1s ∈ No
26		mulsrid 28127	. . . 4 ⊢ ( 1s ∈ No → ( 1s ·s 1s ) = 1s )
27	25, 26	ax-mp 5	. . 3 ⊢ ( 1s ·s 1s ) = 1s
28		oveq2 7368	. . . . 5 ⊢ (𝑥 = 1s → ( 1s ·s 𝑥) = ( 1s ·s 1s ))
29	28	eqeq1d 2743	. . . 4 ⊢ (𝑥 = 1s → (( 1s ·s 𝑥) = 1s ↔ ( 1s ·s 1s ) = 1s ))
30	29	rspcev 3562	. . 3 ⊢ (( 1s ∈ No ∧ ( 1s ·s 1s ) = 1s ) → ∃𝑥 ∈ No ( 1s ·s 𝑥) = 1s )
31	25, 27, 30	mp2an 699	. 2 ⊢ ∃𝑥 ∈ No ( 1s ·s 𝑥) = 1s
32		oveq2 7368	. . . . 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 2743	. . . 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 777	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → 𝑥 ∈ No )
35		0no 27823	. . . . . . . 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 27826	. . . . . . . 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 27784	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → { 0s } <<s { 1s })
41	40	cutscld 27797	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ({ 0s } |s { 1s }) ∈ No )
42	34, 41	mulscld 28149	. . . 4 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (𝑥 ·s ({ 0s } |s { 1s })) ∈ No )
43		expsp1 28443	. . . . . . . 8 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s) → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
44	2, 43	mpan 697	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
45	44	adantr 482	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
46	45	oveq1d 7375	. . . . 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 28445	. . . . . . . 8 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s) → (2s↑s𝑛) ∈ No )
48	2, 47	mpan 697	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (2s↑s𝑛) ∈ No )
49	48	adantr 482	. . . . . 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 28182	. . . . 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 779	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ((2s↑s𝑛) ·s 𝑥) = 1s )
53		twocut 28437	. . . . . . . 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 7378	. . . . . 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 2792	. . . . 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 2780	. . . 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 3565	. . 3 ⊢ ((𝑛 ∈ ℕ0s ∧ (𝑥 ∈ No ∧ ((2s↑s𝑛) ·s 𝑥) = 1s )) → ∃𝑦 ∈ No ((2s↑s(𝑛 +s 1s )) ·s 𝑦) = 1s )
59	58	rexlimdvaa 3143	. 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 28347	1 ⊢ (𝑁 ∈ ℕ0s → ∃𝑥 ∈ No ((2s↑s𝑁) ·s 𝑥) = 1s )

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∃wrex 3065  {csn 4558   class class class wbr 5075  (class class class)co 7360   No csur 27625   <s clts 27626   |s ccuts 27773   0_s c0s 27819   1_s c1s 27820   +_s cadds 27973   ·_s cmuls 28120  ℕ_0scn0s 28326  2_sc2s 28424  ↑_scexps 28426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-nadd 8596  df-no 27628  df-lts 27629  df-bday 27630  df-les 27731  df-slts 27772  df-cuts 27774  df-0s 27821  df-1s 27822  df-made 27841  df-old 27842  df-left 27844  df-right 27845  df-norec 27952  df-norec2 27963  df-adds 27974  df-negs 28035  df-subs 28036  df-muls 28121  df-seqs 28298  df-n0s 28328  df-nns 28329  df-zs 28393  df-2s 28425  df-exps 28427

This theorem is referenced by:  pw2divscld  28453  pw2divmulsd  28454  pw2divscan2d  28456  pw2divsassd  28457  pw2ltdivmulsd  28464  pw2ltmuldivs2d  28465  pw2ltdivmuls2d  28471  z12zsodd  28496