Theorem recmulnqg 7453

Description: Relationship between reciprocal and multiplication on positive fractions. (Contributed by Jim Kingdon, 19-Sep-2019.)

Assertion

Ref	Expression
recmulnqg	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))

Proof of Theorem recmulnqg

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

Step	Hyp	Ref	Expression
1		oveq1 5926	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·Q 𝑦) = (𝐴 ·Q 𝑦))
2	1	eqeq1d 2202	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
3	2	anbi2d 464	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ (𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q)))
4		eleq1 2256	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ Q ↔ 𝐵 ∈ Q))
5		oveq2 5927	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·Q 𝑦) = (𝐴 ·Q 𝐵))
6	5	eqeq1d 2202	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝐵) = 1Q))
7	4, 6	anbi12d 473	. . 3 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ Q ∧ (𝐴 ·Q 𝑦) = 1Q) ↔ (𝐵 ∈ Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
8		recexnq 7452	. . . 4 ⊢ (𝑥 ∈ Q → ∃𝑦(𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9		1nq 7428	. . . . 5 ⊢ 1Q ∈ Q
10		mulcomnqg 7445	. . . . 5 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑧 ·Q 𝑤) = (𝑤 ·Q 𝑧))
11		mulassnqg 7446	. . . . 5 ⊢ ((𝑧 ∈ Q ∧ 𝑤 ∈ Q ∧ 𝑣 ∈ Q) → ((𝑧 ·Q 𝑤) ·Q 𝑣) = (𝑧 ·Q (𝑤 ·Q 𝑣)))
12		mulidnq 7451	. . . . 5 ⊢ (𝑧 ∈ Q → (𝑧 ·Q 1Q) = 𝑧)
13	9, 10, 11, 12	caovimo 6114	. . . 4 ⊢ (𝑥 ∈ Q → ∃*𝑦(𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q))
14		eu5 2089	. . . 4 ⊢ (∃!𝑦(𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ (∃𝑦(𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q) ∧ ∃*𝑦(𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q)))
15	8, 13, 14	sylanbrc 417	. . 3 ⊢ (𝑥 ∈ Q → ∃!𝑦(𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q))
16		df-rq 7414	. . . 4 ⊢ *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q)}
17		3anass 984	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ (𝑥 ∈ Q ∧ (𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q)))
18	17	opabbii 4097	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ Q ∧ (𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q))}
19	16, 18	eqtri 2214	. . 3 ⊢ *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ Q ∧ (𝑦 ∈ Q ∧ (𝑥 ·Q 𝑦) = 1Q))}
20	3, 7, 15, 19	fvopab3g 5631	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) = 𝐵 ↔ (𝐵 ∈ Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
21		ibar 301	. . 3 ⊢ (𝐵 ∈ Q → ((𝐴 ·Q 𝐵) = 1Q ↔ (𝐵 ∈ Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
22	21	adantl 277	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((𝐴 ·Q 𝐵) = 1Q ↔ (𝐵 ∈ Q ∧ (𝐴 ·Q 𝐵) = 1Q)))
23	20, 22	bitr4d 191	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃wex 1503  ∃!weu 2042  ∃*wmo 2043   ∈ wcel 2164  {copab 4090  ‘cfv 5255  (class class class)co 5919  Qcnq 7342  1_Qc1q 7343   ·_Q cmq 7345  *_Qcrq 7346

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  df-rq 7414

This theorem is referenced by:  recclnq  7454  recidnq  7455  recrecnq  7456  recexprlem1ssl  7695  recexprlem1ssu  7696