Theorem recexnq 6580

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 6538	. 2 ⊢ Q = ((N × N) / ~Q )
2		oveq1 5539	. . . . 5 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = (𝐴 ·Q 𝑦))
3	2	eqeq1d 2089	. . . 4 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
4	3	anbi2d 451	. . 3 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ((𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ (𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
5	4	exbidv 1746	. 2 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (∃𝑦(𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
6		opelxpi 4394	. . . . . 6 ⊢ ((𝑧 ∈ N ∧ 𝑥 ∈ N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
7	6	ancoms 264	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
8		enqex 6550	. . . . . 6 ⊢ ~Q ∈ V
9	8	ecelqsi 6183	. . . . 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	syl6eleqr 2172	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → [⟨𝑧, 𝑥⟩] ~Q ∈ Q)
12		mulcompig 6521	. . . . . . 7 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥))
13	12	opeq2d 3577	. . . . . 6 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩ = ⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩)
14	13	eceq1d 6165	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → [⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩] ~Q = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
15		mulclpi 6518	. . . . . 6 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) ∈ N)
16		1qec 6578	. . . . . 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 6563	. . . . . . 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 553	. . . . . 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 389	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
21	14, 17, 20	3eqtr4rd 2124	. . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q)
22	11, 21	jca 300	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ([⟨𝑧, 𝑥⟩] ~Q ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
23		eleq1 2141	. . . . 5 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (𝑦 ∈ Q ↔ [⟨𝑧, 𝑥⟩] ~Q ∈ Q))
24		oveq2 5540	. . . . . 6 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ))
25	24	eqeq1d 2089	. . . . 5 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
26	23, 25	anbi12d 456	. . . 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 2686	. . 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 61	. 2 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ∃𝑦(𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q))
29	1, 5, 28	ecoptocl 6216	1 ⊢ (𝐴 ∈ Q → ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ⟨cop 3401   × cxp 4361  (class class class)co 5532  [cec 6127   / cqs 6128  Ncnpi 6462   ·_N cmi 6464   ~_Q ceq 6469  Qcnq 6470  1_Qc1q 6471   ·_Q cmq 6473

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-mi 6496  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-mqqs 6540  df-1nqqs 6541

This theorem is referenced by:  recmulnqg  6581  recclnq  6582