Theorem recexnq 7747

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 7705	. 2 ⊢ Q = ((N × N) / ~Q )
2		oveq1 6082	. . . . 5 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = (𝐴 ·Q 𝑦))
3	2	eqeq1d 2247	. . . 4 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
4	3	anbi2d 468	. . 3 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ((𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ (𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
5	4	exbidv 1878	. 2 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (∃𝑦(𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
6		opelxpi 4801	. . . . . 6 ⊢ ((𝑧 ∈ N ∧ 𝑥 ∈ N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
7	6	ancoms 268	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
8		enqex 7717	. . . . . 6 ⊢ ~Q ∈ V
9	8	ecelqsi 6853	. . . . 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 2332	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → [⟨𝑧, 𝑥⟩] ~Q ∈ Q)
12		mulcompig 7688	. . . . . . 7 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥))
13	12	opeq2d 3906	. . . . . 6 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩ = ⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩)
14	13	eceq1d 6833	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → [⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩] ~Q = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
15		mulclpi 7685	. . . . . 6 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) ∈ N)
16		1qec 7745	. . . . . 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 7730	. . . . . . 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 597	. . . . . 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 401	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
21	14, 17, 20	3eqtr4rd 2282	. . . 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 2301	. . . . 5 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (𝑦 ∈ Q ↔ [⟨𝑧, 𝑥⟩] ~Q ∈ Q))
24		oveq2 6083	. . . . . 6 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ))
25	24	eqeq1d 2247	. . . . 5 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
26	23, 25	anbi12d 477	. . . 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 2913	. . 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 6886	1 ⊢ (𝐴 ∈ Q → ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ⟨cop 3708   × cxp 4767  (class class class)co 6075  [cec 6795   / cqs 6796  Ncnpi 7629   ·_N cmi 7631   ~_Q ceq 7636  Qcnq 7637  1_Qc1q 7638   ·_Q cmq 7640

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-mi 7663  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-mqqs 7707  df-1nqqs 7708

This theorem is referenced by:  recmulnqg  7748  recclnq  7749