Theorem recexnq 6488

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 6446	. 2 ⊢ Q = ((N × N) / ~Q )
2		oveq1 5519	. . . . 5 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = (𝐴 ·Q 𝑦))
3	2	eqeq1d 2048	. . . 4 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
4	3	anbi2d 437	. . 3 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → ((𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ (𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
5	4	exbidv 1706	. 2 ⊢ ([⟨𝑥, 𝑧⟩] ~Q = 𝐴 → (∃𝑦(𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q) ↔ ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
6		opelxpi 4376	. . . . . 6 ⊢ ((𝑧 ∈ N ∧ 𝑥 ∈ N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
7	6	ancoms 255	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ⟨𝑧, 𝑥⟩ ∈ (N × N))
8		enqex 6458	. . . . . 6 ⊢ ~Q ∈ V
9	8	ecelqsi 6160	. . . . 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 2131	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → [⟨𝑧, 𝑥⟩] ~Q ∈ Q)
12		mulcompig 6429	. . . . . . 7 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) = (𝑧 ·N 𝑥))
13	12	opeq2d 3556	. . . . . 6 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩ = ⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩)
14	13	eceq1d 6142	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → [⟨(𝑥 ·N 𝑧), (𝑥 ·N 𝑧)⟩] ~Q = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
15		mulclpi 6426	. . . . . 6 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → (𝑥 ·N 𝑧) ∈ N)
16		1qec 6486	. . . . . 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 6471	. . . . . . 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 523	. . . . . 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 377	. . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = [⟨(𝑥 ·N 𝑧), (𝑧 ·N 𝑥)⟩] ~Q )
21	14, 17, 20	3eqtr4rd 2083	. . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q)
22	11, 21	jca 290	. . 3 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ([⟨𝑧, 𝑥⟩] ~Q ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
23		eleq1 2100	. . . . 5 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (𝑦 ∈ Q ↔ [⟨𝑧, 𝑥⟩] ~Q ∈ Q))
24		oveq2 5520	. . . . . 6 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ))
25	24	eqeq1d 2048	. . . . 5 ⊢ (𝑦 = [⟨𝑧, 𝑥⟩] ~Q → (([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q ↔ ([⟨𝑥, 𝑧⟩] ~Q ·Q [⟨𝑧, 𝑥⟩] ~Q ) = 1Q))
26	23, 25	anbi12d 442	. . . 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 2641	. . 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 56	. 2 ⊢ ((𝑥 ∈ N ∧ 𝑧 ∈ N) → ∃𝑦(𝑦 ∈ Q ∧ ([⟨𝑥, 𝑧⟩] ~Q ·Q 𝑦) = 1Q))
29	1, 5, 28	ecoptocl 6193	1 ⊢ (𝐴 ∈ Q → ∃𝑦(𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ⟨cop 3378   × cxp 4343  (class class class)co 5512  [cec 6104   / cqs 6105  Ncnpi 6370   ·_N cmi 6372   ~_Q ceq 6377  Qcnq 6378  1_Qc1q 6379   ·_Q cmq 6381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-mqqs 6448  df-1nqqs 6449

This theorem is referenced by:  recmulnqg  6489  recclnq  6490