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Theorem recmulnq 9642
Description: Relationship between reciprocal and multiplication on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
Assertion
Ref Expression
recmulnq (𝐴Q → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))

Proof of Theorem recmulnq
Dummy variables 𝑥 𝑦 𝑠 𝑟 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6098 . . . 4 (*Q𝐴) ∈ V
21a1i 11 . . 3 (𝐴Q → (*Q𝐴) ∈ V)
3 eleq1 2675 . . 3 ((*Q𝐴) = 𝐵 → ((*Q𝐴) ∈ V ↔ 𝐵 ∈ V))
42, 3syl5ibcom 233 . 2 (𝐴Q → ((*Q𝐴) = 𝐵𝐵 ∈ V))
5 id 22 . . . . . . 7 ((𝐴 ·Q 𝐵) = 1Q → (𝐴 ·Q 𝐵) = 1Q)
6 1nq 9606 . . . . . . 7 1QQ
75, 6syl6eqel 2695 . . . . . 6 ((𝐴 ·Q 𝐵) = 1Q → (𝐴 ·Q 𝐵) ∈ Q)
8 mulnqf 9627 . . . . . . . 8 ·Q :(Q × Q)⟶Q
98fdmi 5951 . . . . . . 7 dom ·Q = (Q × Q)
10 0nnq 9602 . . . . . . 7 ¬ ∅ ∈ Q
119, 10ndmovrcl 6695 . . . . . 6 ((𝐴 ·Q 𝐵) ∈ Q → (𝐴Q𝐵Q))
127, 11syl 17 . . . . 5 ((𝐴 ·Q 𝐵) = 1Q → (𝐴Q𝐵Q))
1312simprd 477 . . . 4 ((𝐴 ·Q 𝐵) = 1Q𝐵Q)
14 elex 3184 . . . 4 (𝐵Q𝐵 ∈ V)
1513, 14syl 17 . . 3 ((𝐴 ·Q 𝐵) = 1Q𝐵 ∈ V)
1615a1i 11 . 2 (𝐴Q → ((𝐴 ·Q 𝐵) = 1Q𝐵 ∈ V))
17 oveq1 6534 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·Q 𝑦) = (𝐴 ·Q 𝑦))
1817eqeq1d 2611 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
19 oveq2 6535 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·Q 𝑦) = (𝐴 ·Q 𝐵))
2019eqeq1d 2611 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝐵) = 1Q))
21 nqerid 9611 . . . . . . . . . 10 (𝑥Q → ([Q]‘𝑥) = 𝑥)
22 relxp 5139 . . . . . . . . . . . 12 Rel (N × N)
23 elpqn 9603 . . . . . . . . . . . 12 (𝑥Q𝑥 ∈ (N × N))
24 1st2nd 7082 . . . . . . . . . . . 12 ((Rel (N × N) ∧ 𝑥 ∈ (N × N)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2522, 23, 24sylancr 693 . . . . . . . . . . 11 (𝑥Q𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2625fveq2d 6092 . . . . . . . . . 10 (𝑥Q → ([Q]‘𝑥) = ([Q]‘⟨(1st𝑥), (2nd𝑥)⟩))
2721, 26eqtr3d 2645 . . . . . . . . 9 (𝑥Q𝑥 = ([Q]‘⟨(1st𝑥), (2nd𝑥)⟩))
2827oveq1d 6542 . . . . . . . 8 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = (([Q]‘⟨(1st𝑥), (2nd𝑥)⟩) ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)))
29 mulerpq 9635 . . . . . . . 8 (([Q]‘⟨(1st𝑥), (2nd𝑥)⟩) ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩))
3028, 29syl6eq 2659 . . . . . . 7 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩)))
31 xp1st 7066 . . . . . . . . . . 11 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
3223, 31syl 17 . . . . . . . . . 10 (𝑥Q → (1st𝑥) ∈ N)
33 xp2nd 7067 . . . . . . . . . . 11 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
3423, 33syl 17 . . . . . . . . . 10 (𝑥Q → (2nd𝑥) ∈ N)
35 mulpipq 9618 . . . . . . . . . 10 ((((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N) ∧ ((2nd𝑥) ∈ N ∧ (1st𝑥) ∈ N)) → (⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩) = ⟨((1st𝑥) ·N (2nd𝑥)), ((2nd𝑥) ·N (1st𝑥))⟩)
3632, 34, 34, 32, 35syl22anc 1318 . . . . . . . . 9 (𝑥Q → (⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩) = ⟨((1st𝑥) ·N (2nd𝑥)), ((2nd𝑥) ·N (1st𝑥))⟩)
37 mulcompi 9574 . . . . . . . . . 10 ((2nd𝑥) ·N (1st𝑥)) = ((1st𝑥) ·N (2nd𝑥))
3837opeq2i 4338 . . . . . . . . 9 ⟨((1st𝑥) ·N (2nd𝑥)), ((2nd𝑥) ·N (1st𝑥))⟩ = ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩
3936, 38syl6eq 2659 . . . . . . . 8 (𝑥Q → (⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩) = ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)
4039fveq2d 6092 . . . . . . 7 (𝑥Q → ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩))
41 nqerid 9611 . . . . . . . . 9 (1QQ → ([Q]‘1Q) = 1Q)
426, 41ax-mp 5 . . . . . . . 8 ([Q]‘1Q) = 1Q
43 mulclpi 9571 . . . . . . . . . . 11 (((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N) → ((1st𝑥) ·N (2nd𝑥)) ∈ N)
4432, 34, 43syl2anc 690 . . . . . . . . . 10 (𝑥Q → ((1st𝑥) ·N (2nd𝑥)) ∈ N)
45 1nqenq 9640 . . . . . . . . . 10 (((1st𝑥) ·N (2nd𝑥)) ∈ N → 1Q ~Q ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)
4644, 45syl 17 . . . . . . . . 9 (𝑥Q → 1Q ~Q ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)
47 elpqn 9603 . . . . . . . . . . 11 (1QQ → 1Q ∈ (N × N))
486, 47ax-mp 5 . . . . . . . . . 10 1Q ∈ (N × N)
49 opelxpi 5062 . . . . . . . . . . 11 ((((1st𝑥) ·N (2nd𝑥)) ∈ N ∧ ((1st𝑥) ·N (2nd𝑥)) ∈ N) → ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ∈ (N × N))
5044, 44, 49syl2anc 690 . . . . . . . . . 10 (𝑥Q → ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ∈ (N × N))
51 nqereq 9613 . . . . . . . . . 10 ((1Q ∈ (N × N) ∧ ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ∈ (N × N)) → (1Q ~Q ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ↔ ([Q]‘1Q) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)))
5248, 50, 51sylancr 693 . . . . . . . . 9 (𝑥Q → (1Q ~Q ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ↔ ([Q]‘1Q) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)))
5346, 52mpbid 220 . . . . . . . 8 (𝑥Q → ([Q]‘1Q) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩))
5442, 53syl5reqr 2658 . . . . . . 7 (𝑥Q → ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩) = 1Q)
5530, 40, 543eqtrd 2647 . . . . . 6 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q)
56 fvex 6098 . . . . . . 7 ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) ∈ V
57 oveq2 6535 . . . . . . . 8 (𝑦 = ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) → (𝑥 ·Q 𝑦) = (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)))
5857eqeq1d 2611 . . . . . . 7 (𝑦 = ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q))
5956, 58spcev 3272 . . . . . 6 ((𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q → ∃𝑦(𝑥 ·Q 𝑦) = 1Q)
6055, 59syl 17 . . . . 5 (𝑥Q → ∃𝑦(𝑥 ·Q 𝑦) = 1Q)
61 mulcomnq 9631 . . . . . . 7 (𝑟 ·Q 𝑠) = (𝑠 ·Q 𝑟)
62 mulassnq 9637 . . . . . . 7 ((𝑟 ·Q 𝑠) ·Q 𝑡) = (𝑟 ·Q (𝑠 ·Q 𝑡))
63 mulidnq 9641 . . . . . . 7 (𝑟Q → (𝑟 ·Q 1Q) = 𝑟)
646, 9, 10, 61, 62, 63caovmo 6746 . . . . . 6 ∃*𝑦(𝑥 ·Q 𝑦) = 1Q
65 eu5 2483 . . . . . 6 (∃!𝑦(𝑥 ·Q 𝑦) = 1Q ↔ (∃𝑦(𝑥 ·Q 𝑦) = 1Q ∧ ∃*𝑦(𝑥 ·Q 𝑦) = 1Q))
6664, 65mpbiran2 955 . . . . 5 (∃!𝑦(𝑥 ·Q 𝑦) = 1Q ↔ ∃𝑦(𝑥 ·Q 𝑦) = 1Q)
6760, 66sylibr 222 . . . 4 (𝑥Q → ∃!𝑦(𝑥 ·Q 𝑦) = 1Q)
68 cnvimass 5391 . . . . . . . 8 ( ·Q “ {1Q}) ⊆ dom ·Q
69 df-rq 9595 . . . . . . . 8 *Q = ( ·Q “ {1Q})
709eqcomi 2618 . . . . . . . 8 (Q × Q) = dom ·Q
7168, 69, 703sstr4i 3606 . . . . . . 7 *Q ⊆ (Q × Q)
72 relxp 5139 . . . . . . 7 Rel (Q × Q)
73 relss 5119 . . . . . . 7 (*Q ⊆ (Q × Q) → (Rel (Q × Q) → Rel *Q))
7471, 72, 73mp2 9 . . . . . 6 Rel *Q
7569eleq2i 2679 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ *Q ↔ ⟨𝑥, 𝑦⟩ ∈ ( ·Q “ {1Q}))
76 ffn 5944 . . . . . . . . 9 ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
77 fniniseg 6231 . . . . . . . . 9 ( ·Q Fn (Q × Q) → (⟨𝑥, 𝑦⟩ ∈ ( ·Q “ {1Q}) ↔ (⟨𝑥, 𝑦⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q)))
788, 76, 77mp2b 10 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ ( ·Q “ {1Q}) ↔ (⟨𝑥, 𝑦⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q))
79 ancom 464 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q) ↔ (( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)))
80 ancom 464 . . . . . . . . . 10 ((𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ ((𝑥 ·Q 𝑦) = 1Q𝑥Q))
81 eleq1 2675 . . . . . . . . . . . . . . 15 ((𝑥 ·Q 𝑦) = 1Q → ((𝑥 ·Q 𝑦) ∈ Q ↔ 1QQ))
826, 81mpbiri 246 . . . . . . . . . . . . . 14 ((𝑥 ·Q 𝑦) = 1Q → (𝑥 ·Q 𝑦) ∈ Q)
839, 10ndmovrcl 6695 . . . . . . . . . . . . . 14 ((𝑥 ·Q 𝑦) ∈ Q → (𝑥Q𝑦Q))
8482, 83syl 17 . . . . . . . . . . . . 13 ((𝑥 ·Q 𝑦) = 1Q → (𝑥Q𝑦Q))
85 opelxpi 5062 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q) → ⟨𝑥, 𝑦⟩ ∈ (Q × Q))
8684, 85syl 17 . . . . . . . . . . . 12 ((𝑥 ·Q 𝑦) = 1Q → ⟨𝑥, 𝑦⟩ ∈ (Q × Q))
8784simpld 473 . . . . . . . . . . . 12 ((𝑥 ·Q 𝑦) = 1Q𝑥Q)
8886, 872thd 253 . . . . . . . . . . 11 ((𝑥 ·Q 𝑦) = 1Q → (⟨𝑥, 𝑦⟩ ∈ (Q × Q) ↔ 𝑥Q))
8988pm5.32i 666 . . . . . . . . . 10 (((𝑥 ·Q 𝑦) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)) ↔ ((𝑥 ·Q 𝑦) = 1Q𝑥Q))
90 df-ov 6530 . . . . . . . . . . . 12 (𝑥 ·Q 𝑦) = ( ·Q ‘⟨𝑥, 𝑦⟩)
9190eqeq1i 2614 . . . . . . . . . . 11 ((𝑥 ·Q 𝑦) = 1Q ↔ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q)
9291anbi1i 726 . . . . . . . . . 10 (((𝑥 ·Q 𝑦) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)) ↔ (( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)))
9380, 89, 923bitr2ri 287 . . . . . . . . 9 ((( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)) ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9479, 93bitri 262 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q) ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9575, 78, 943bitri 284 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ *Q ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9695a1i 11 . . . . . 6 (⊤ → (⟨𝑥, 𝑦⟩ ∈ *Q ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)))
9774, 96opabbi2dv 5181 . . . . 5 (⊤ → *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)})
9897trud 1483 . . . 4 *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)}
9918, 20, 67, 98fvopab3g 6172 . . 3 ((𝐴Q𝐵 ∈ V) → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
10099ex 448 . 2 (𝐴Q → (𝐵 ∈ V → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q)))
1014, 16, 100pm5.21ndd 367 1 (𝐴Q → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wtru 1475  wex 1694  wcel 1976  ∃!weu 2457  ∃*wmo 2458  Vcvv 3172  wss 3539  {csn 4124  cop 4130   class class class wbr 4577  {copab 4636   × cxp 5026  ccnv 5027  dom cdm 5028  cima 5031  Rel wrel 5033   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  1st c1st 7034  2nd c2nd 7035  Ncnpi 9522   ·N cmi 9524   ·pQ cmpq 9527   ~Q ceq 9529  Qcnq 9530  1Qc1q 9531  [Q]cerq 9532   ·Q cmq 9534  *Qcrq 9535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-omul 7429  df-er 7606  df-ni 9550  df-mi 9552  df-lti 9553  df-mpq 9587  df-enq 9589  df-nq 9590  df-erq 9591  df-mq 9593  df-1nq 9594  df-rq 9595
This theorem is referenced by:  recidnq  9643  recrecnq  9645  reclem3pr  9727
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