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Theorem recmulnq 9824
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 6239 . . . 4 (*Q𝐴) ∈ V
21a1i 11 . . 3 (𝐴Q → (*Q𝐴) ∈ V)
3 eleq1 2718 . . 3 ((*Q𝐴) = 𝐵 → ((*Q𝐴) ∈ V ↔ 𝐵 ∈ V))
42, 3syl5ibcom 235 . 2 (𝐴Q → ((*Q𝐴) = 𝐵𝐵 ∈ V))
5 id 22 . . . . . . 7 ((𝐴 ·Q 𝐵) = 1Q → (𝐴 ·Q 𝐵) = 1Q)
6 1nq 9788 . . . . . . 7 1QQ
75, 6syl6eqel 2738 . . . . . 6 ((𝐴 ·Q 𝐵) = 1Q → (𝐴 ·Q 𝐵) ∈ Q)
8 mulnqf 9809 . . . . . . . 8 ·Q :(Q × Q)⟶Q
98fdmi 6090 . . . . . . 7 dom ·Q = (Q × Q)
10 0nnq 9784 . . . . . . 7 ¬ ∅ ∈ Q
119, 10ndmovrcl 6862 . . . . . 6 ((𝐴 ·Q 𝐵) ∈ Q → (𝐴Q𝐵Q))
127, 11syl 17 . . . . 5 ((𝐴 ·Q 𝐵) = 1Q → (𝐴Q𝐵Q))
1312simprd 478 . . . 4 ((𝐴 ·Q 𝐵) = 1Q𝐵Q)
14 elex 3243 . . . 4 (𝐵Q𝐵 ∈ V)
1513, 14syl 17 . . 3 ((𝐴 ·Q 𝐵) = 1Q𝐵 ∈ V)
1615a1i 11 . 2 (𝐴Q → ((𝐴 ·Q 𝐵) = 1Q𝐵 ∈ V))
17 oveq1 6697 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·Q 𝑦) = (𝐴 ·Q 𝑦))
1817eqeq1d 2653 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
19 oveq2 6698 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·Q 𝑦) = (𝐴 ·Q 𝐵))
2019eqeq1d 2653 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝐵) = 1Q))
21 nqerid 9793 . . . . . . . . . 10 (𝑥Q → ([Q]‘𝑥) = 𝑥)
22 relxp 5160 . . . . . . . . . . . 12 Rel (N × N)
23 elpqn 9785 . . . . . . . . . . . 12 (𝑥Q𝑥 ∈ (N × N))
24 1st2nd 7258 . . . . . . . . . . . 12 ((Rel (N × N) ∧ 𝑥 ∈ (N × N)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2522, 23, 24sylancr 696 . . . . . . . . . . 11 (𝑥Q𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2625fveq2d 6233 . . . . . . . . . 10 (𝑥Q → ([Q]‘𝑥) = ([Q]‘⟨(1st𝑥), (2nd𝑥)⟩))
2721, 26eqtr3d 2687 . . . . . . . . 9 (𝑥Q𝑥 = ([Q]‘⟨(1st𝑥), (2nd𝑥)⟩))
2827oveq1d 6705 . . . . . . . 8 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = (([Q]‘⟨(1st𝑥), (2nd𝑥)⟩) ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)))
29 mulerpq 9817 . . . . . . . 8 (([Q]‘⟨(1st𝑥), (2nd𝑥)⟩) ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩))
3028, 29syl6eq 2701 . . . . . . 7 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩)))
31 xp1st 7242 . . . . . . . . . . 11 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
3223, 31syl 17 . . . . . . . . . 10 (𝑥Q → (1st𝑥) ∈ N)
33 xp2nd 7243 . . . . . . . . . . 11 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
3423, 33syl 17 . . . . . . . . . 10 (𝑥Q → (2nd𝑥) ∈ N)
35 mulpipq 9800 . . . . . . . . . 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 1367 . . . . . . . . 9 (𝑥Q → (⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩) = ⟨((1st𝑥) ·N (2nd𝑥)), ((2nd𝑥) ·N (1st𝑥))⟩)
37 mulcompi 9756 . . . . . . . . . 10 ((2nd𝑥) ·N (1st𝑥)) = ((1st𝑥) ·N (2nd𝑥))
3837opeq2i 4437 . . . . . . . . 9 ⟨((1st𝑥) ·N (2nd𝑥)), ((2nd𝑥) ·N (1st𝑥))⟩ = ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩
3936, 38syl6eq 2701 . . . . . . . 8 (𝑥Q → (⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩) = ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)
4039fveq2d 6233 . . . . . . 7 (𝑥Q → ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩))
41 nqerid 9793 . . . . . . . . 9 (1QQ → ([Q]‘1Q) = 1Q)
426, 41ax-mp 5 . . . . . . . 8 ([Q]‘1Q) = 1Q
43 mulclpi 9753 . . . . . . . . . . 11 (((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N) → ((1st𝑥) ·N (2nd𝑥)) ∈ N)
4432, 34, 43syl2anc 694 . . . . . . . . . 10 (𝑥Q → ((1st𝑥) ·N (2nd𝑥)) ∈ N)
45 1nqenq 9822 . . . . . . . . . 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 9785 . . . . . . . . . . 11 (1QQ → 1Q ∈ (N × N))
486, 47ax-mp 5 . . . . . . . . . 10 1Q ∈ (N × N)
49 opelxpi 5182 . . . . . . . . . . 11 ((((1st𝑥) ·N (2nd𝑥)) ∈ N ∧ ((1st𝑥) ·N (2nd𝑥)) ∈ N) → ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ∈ (N × N))
5044, 44, 49syl2anc 694 . . . . . . . . . 10 (𝑥Q → ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ∈ (N × N))
51 nqereq 9795 . . . . . . . . . 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 696 . . . . . . . . 9 (𝑥Q → (1Q ~Q ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ↔ ([Q]‘1Q) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)))
5346, 52mpbid 222 . . . . . . . 8 (𝑥Q → ([Q]‘1Q) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩))
5442, 53syl5reqr 2700 . . . . . . 7 (𝑥Q → ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩) = 1Q)
5530, 40, 543eqtrd 2689 . . . . . 6 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q)
56 fvex 6239 . . . . . . 7 ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) ∈ V
57 oveq2 6698 . . . . . . . 8 (𝑦 = ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) → (𝑥 ·Q 𝑦) = (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)))
5857eqeq1d 2653 . . . . . . 7 (𝑦 = ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q))
5956, 58spcev 3331 . . . . . 6 ((𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q → ∃𝑦(𝑥 ·Q 𝑦) = 1Q)
6055, 59syl 17 . . . . 5 (𝑥Q → ∃𝑦(𝑥 ·Q 𝑦) = 1Q)
61 mulcomnq 9813 . . . . . . 7 (𝑟 ·Q 𝑠) = (𝑠 ·Q 𝑟)
62 mulassnq 9819 . . . . . . 7 ((𝑟 ·Q 𝑠) ·Q 𝑡) = (𝑟 ·Q (𝑠 ·Q 𝑡))
63 mulidnq 9823 . . . . . . 7 (𝑟Q → (𝑟 ·Q 1Q) = 𝑟)
646, 9, 10, 61, 62, 63caovmo 6913 . . . . . 6 ∃*𝑦(𝑥 ·Q 𝑦) = 1Q
65 eu5 2524 . . . . . 6 (∃!𝑦(𝑥 ·Q 𝑦) = 1Q ↔ (∃𝑦(𝑥 ·Q 𝑦) = 1Q ∧ ∃*𝑦(𝑥 ·Q 𝑦) = 1Q))
6664, 65mpbiran2 974 . . . . 5 (∃!𝑦(𝑥 ·Q 𝑦) = 1Q ↔ ∃𝑦(𝑥 ·Q 𝑦) = 1Q)
6760, 66sylibr 224 . . . 4 (𝑥Q → ∃!𝑦(𝑥 ·Q 𝑦) = 1Q)
68 cnvimass 5520 . . . . . . . 8 ( ·Q “ {1Q}) ⊆ dom ·Q
69 df-rq 9777 . . . . . . . 8 *Q = ( ·Q “ {1Q})
709eqcomi 2660 . . . . . . . 8 (Q × Q) = dom ·Q
7168, 69, 703sstr4i 3677 . . . . . . 7 *Q ⊆ (Q × Q)
72 relxp 5160 . . . . . . 7 Rel (Q × Q)
73 relss 5240 . . . . . . 7 (*Q ⊆ (Q × Q) → (Rel (Q × Q) → Rel *Q))
7471, 72, 73mp2 9 . . . . . 6 Rel *Q
7569eleq2i 2722 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ *Q ↔ ⟨𝑥, 𝑦⟩ ∈ ( ·Q “ {1Q}))
76 ffn 6083 . . . . . . . . 9 ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
77 fniniseg 6378 . . . . . . . . 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 465 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q) ↔ (( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)))
80 ancom 465 . . . . . . . . . 10 ((𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ ((𝑥 ·Q 𝑦) = 1Q𝑥Q))
81 eleq1 2718 . . . . . . . . . . . . . . 15 ((𝑥 ·Q 𝑦) = 1Q → ((𝑥 ·Q 𝑦) ∈ Q ↔ 1QQ))
826, 81mpbiri 248 . . . . . . . . . . . . . 14 ((𝑥 ·Q 𝑦) = 1Q → (𝑥 ·Q 𝑦) ∈ Q)
839, 10ndmovrcl 6862 . . . . . . . . . . . . . 14 ((𝑥 ·Q 𝑦) ∈ Q → (𝑥Q𝑦Q))
8482, 83syl 17 . . . . . . . . . . . . 13 ((𝑥 ·Q 𝑦) = 1Q → (𝑥Q𝑦Q))
85 opelxpi 5182 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q) → ⟨𝑥, 𝑦⟩ ∈ (Q × Q))
8684, 85syl 17 . . . . . . . . . . . 12 ((𝑥 ·Q 𝑦) = 1Q → ⟨𝑥, 𝑦⟩ ∈ (Q × Q))
8784simpld 474 . . . . . . . . . . . 12 ((𝑥 ·Q 𝑦) = 1Q𝑥Q)
8886, 872thd 255 . . . . . . . . . . 11 ((𝑥 ·Q 𝑦) = 1Q → (⟨𝑥, 𝑦⟩ ∈ (Q × Q) ↔ 𝑥Q))
8988pm5.32i 670 . . . . . . . . . 10 (((𝑥 ·Q 𝑦) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)) ↔ ((𝑥 ·Q 𝑦) = 1Q𝑥Q))
90 df-ov 6693 . . . . . . . . . . . 12 (𝑥 ·Q 𝑦) = ( ·Q ‘⟨𝑥, 𝑦⟩)
9190eqeq1i 2656 . . . . . . . . . . 11 ((𝑥 ·Q 𝑦) = 1Q ↔ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q)
9291anbi1i 731 . . . . . . . . . 10 (((𝑥 ·Q 𝑦) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)) ↔ (( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)))
9380, 89, 923bitr2ri 289 . . . . . . . . 9 ((( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)) ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9479, 93bitri 264 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q) ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9575, 78, 943bitri 286 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ *Q ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9695a1i 11 . . . . . 6 (⊤ → (⟨𝑥, 𝑦⟩ ∈ *Q ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)))
9774, 96opabbi2dv 5304 . . . . 5 (⊤ → *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)})
9897trud 1533 . . . 4 *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)}
9918, 20, 67, 98fvopab3g 6316 . . 3 ((𝐴Q𝐵 ∈ V) → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
10099ex 449 . 2 (𝐴Q → (𝐵 ∈ V → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q)))
1014, 16, 100pm5.21ndd 368 1 (𝐴Q → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wtru 1524  wex 1744  wcel 2030  ∃!weu 2498  ∃*wmo 2499  Vcvv 3231  wss 3607  {csn 4210  cop 4216   class class class wbr 4685  {copab 4745   × cxp 5141  ccnv 5142  dom cdm 5143  cima 5146  Rel wrel 5148   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Ncnpi 9704   ·N cmi 9706   ·pQ cmpq 9709   ~Q ceq 9711  Qcnq 9712  1Qc1q 9713  [Q]cerq 9714   ·Q cmq 9716  *Qcrq 9717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ni 9732  df-mi 9734  df-lti 9735  df-mpq 9769  df-enq 9771  df-nq 9772  df-erq 9773  df-mq 9775  df-1nq 9776  df-rq 9777
This theorem is referenced by:  recidnq  9825  recrecnq  9827  reclem3pr  9909
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