Theorem recexnq 7452

Description: Existence of positive fraction reciprocal. (Contributed by Jim Kingdon, 20-Sep-2019.)

Assertion

Ref	Expression
recexnq	⊢ (𝐴 ∈ Q → ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q))

Distinct variable group:   𝑦,𝐴

Proof of Theorem recexnq

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nqqs 7410	. 2 ⊢ Q = ((N × N) / ~Q )
2		oveq1 5926	. . . . 5 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = (𝐴 ·Q 𝑦))
3	2	eqeq1d 2202	. . . 4 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
4	3	anbi2d 464	. . 3 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ((𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ (𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
5	4	exbidv 1836	. 2 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (∃𝑦(𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
6		opelxpi 4692	. . . . . 6 ⊢ ((𝑧 ∈ N ∧ 𝑥 ∈ N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
7	6	ancoms 268	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
8		enqex 7422	. . . . . 6 ⊢ ~Q ∈ V
9	8	ecelqsi 6645	. . . . 5 ⊢ (⟨𝑧, 𝑥⟩ ∈ (N × N) → [⟨𝑧, 𝑥⟩] ~Q ∈ ((N × N) / ~Q ))
10	7, 9	syl 14	. . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → [⟨𝑧, 𝑥⟩] ~Q ∈ ((N × N) / ~Q ))
11	10, 1	eleqtrrdi 2287	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → [⟨𝑧, 𝑥⟩] ~Q ∈ Q)
12		mulcompig 7393	. . . . . . 7 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥))
13	12	opeq2d 3812	. . . . . 6 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩ = ⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩)
14	13	eceq1d 6625	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → [⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩] ~Q = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
15		mulclpi 7390	. . . . . 6 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) ∈ N)
16		1qec 7450	. . . . . 6 ⊢ ((𝑥 ·N 𝑧) ∈ N → 1Q = [⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩] ~Q )
17	15, 16	syl 14	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → 1Q = [⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩] ~Q )
18		mulpipqqs 7435	. . . . . . 7 ⊢ (((𝑥 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑧 ∈ N ∧ 𝑥 ∈ N)) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
19	18	an42s 589	. . . . . 6 ⊢ (((𝑥 ∈ N ∧ 𝑧 ∈ N) ∧ (𝑥 ∈ N ∧ 𝑧 ∈ N)) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
20	19	anidms 397	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
21	14, 17, 20	3eqtr4rd 2237	. . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q)
22	11, 21	jca 306	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ([⟨𝑧, 𝑥⟩] ~Q ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
23		eleq1 2256	. . . . 5 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (𝑦 ∈ Q ↔ [⟨𝑧, 𝑥⟩] ~Q ∈ Q))
24		oveq2 5927	. . . . . 6 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ))
25	24	eqeq1d 2202	. . . . 5 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
26	23, 25	anbi12d 473	. . . 4 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ((𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ([⟨𝑧, 𝑥⟩] ~Q ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q)))
27	26	spcegv 2849	. . 3 ⊢ ([⟨𝑧, 𝑥⟩] ~Q ∈ Q → (([⟨𝑧, 𝑥⟩] ~Q ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q) → ∃𝑦(𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q)))
28	11, 22, 27	sylc 62	. 2 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ∃𝑦(𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q))
29	1, 5, 28	ecoptocl 6678	1 ⊢ (𝐴 ∈ Q → ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ⟨cop 3622   × cxp 4658  (class class class)co 5919  [cec 6587   / cqs 6588  Ncnpi 7334   ·_N cmi 7336   ~_Q ceq 7341  Qcnq 7342  1_Qc1q 7343   ·_Q cmq 7345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-mi 7368  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-mqqs 7412  df-1nqqs 7413

This theorem is referenced by:  recmulnqg  7453  recclnq  7454