Theorem recexnq 7222

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 7180	. 2 ⊢ Q = ((N × N) / ~Q )
2		oveq1 5789	. . . . 5 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = (𝐴 ·Q 𝑦))
3	2	eqeq1d 2149	. . . 4 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
4	3	anbi2d 460	. . 3 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ((𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ (𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
5	4	exbidv 1798	. 2 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (∃𝑦(𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
6		opelxpi 4579	. . . . . 6 ⊢ ((𝑧 ∈ N ∧ 𝑥 ∈ N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
7	6	ancoms 266	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
8		enqex 7192	. . . . . 6 ⊢ ~Q ∈ V
9	8	ecelqsi 6491	. . . . 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 2234	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → [⟨𝑧, 𝑥⟩] ~Q ∈ Q)
12		mulcompig 7163	. . . . . . 7 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥))
13	12	opeq2d 3720	. . . . . 6 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩ = ⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩)
14	13	eceq1d 6473	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → [⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩] ~Q = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
15		mulclpi 7160	. . . . . 6 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) ∈ N)
16		1qec 7220	. . . . . 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 7205	. . . . . . 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 579	. . . . . 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 395	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
21	14, 17, 20	3eqtr4rd 2184	. . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q)
22	11, 21	jca 304	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ([⟨𝑧, 𝑥⟩] ~Q ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
23		eleq1 2203	. . . . 5 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (𝑦 ∈ Q ↔ [⟨𝑧, 𝑥⟩] ~Q ∈ Q))
24		oveq2 5790	. . . . . 6 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ))
25	24	eqeq1d 2149	. . . . 5 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
26	23, 25	anbi12d 465	. . . 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 2777	. . 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 6524	1 ⊢ (𝐴 ∈ Q → ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ⟨cop 3535   × cxp 4545  (class class class)co 5782  [cec 6435   / cqs 6436  Ncnpi 7104   ·_N cmi 7106   ~_Q ceq 7111  Qcnq 7112  1_Qc1q 7113   ·_Q cmq 7115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-mi 7138  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-mqqs 7182  df-1nqqs 7183

This theorem is referenced by:  recmulnqg  7223  recclnq  7224