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Theorem recmulnq 9989
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 6343 . . . 4 (*Q𝐴) ∈ V
21a1i 11 . . 3 (𝐴Q → (*Q𝐴) ∈ V)
3 eleq1 2838 . . 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 9953 . . . . . . 7 1QQ
75, 6syl6eqel 2858 . . . . . 6 ((𝐴 ·Q 𝐵) = 1Q → (𝐴 ·Q 𝐵) ∈ Q)
8 mulnqf 9974 . . . . . . . 8 ·Q :(Q × Q)⟶Q
98fdmi 6193 . . . . . . 7 dom ·Q = (Q × Q)
10 0nnq 9949 . . . . . . 7 ¬ ∅ ∈ Q
119, 10ndmovrcl 6968 . . . . . 6 ((𝐴 ·Q 𝐵) ∈ Q → (𝐴Q𝐵Q))
127, 11syl 17 . . . . 5 ((𝐴 ·Q 𝐵) = 1Q → (𝐴Q𝐵Q))
1312simprd 479 . . . 4 ((𝐴 ·Q 𝐵) = 1Q𝐵Q)
14 elex 3364 . . . 4 (𝐵Q𝐵 ∈ V)
1513, 14syl 17 . . 3 ((𝐴 ·Q 𝐵) = 1Q𝐵 ∈ V)
1615a1i 11 . 2 (𝐴Q → ((𝐴 ·Q 𝐵) = 1Q𝐵 ∈ V))
17 oveq1 6801 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·Q 𝑦) = (𝐴 ·Q 𝑦))
1817eqeq1d 2773 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
19 oveq2 6802 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·Q 𝑦) = (𝐴 ·Q 𝐵))
2019eqeq1d 2773 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝐵) = 1Q))
21 nqerid 9958 . . . . . . . . . 10 (𝑥Q → ([Q]‘𝑥) = 𝑥)
22 relxp 5267 . . . . . . . . . . . 12 Rel (N × N)
23 elpqn 9950 . . . . . . . . . . . 12 (𝑥Q𝑥 ∈ (N × N))
24 1st2nd 7364 . . . . . . . . . . . 12 ((Rel (N × N) ∧ 𝑥 ∈ (N × N)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2522, 23, 24sylancr 569 . . . . . . . . . . 11 (𝑥Q𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2625fveq2d 6337 . . . . . . . . . 10 (𝑥Q → ([Q]‘𝑥) = ([Q]‘⟨(1st𝑥), (2nd𝑥)⟩))
2721, 26eqtr3d 2807 . . . . . . . . 9 (𝑥Q𝑥 = ([Q]‘⟨(1st𝑥), (2nd𝑥)⟩))
2827oveq1d 6809 . . . . . . . 8 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = (([Q]‘⟨(1st𝑥), (2nd𝑥)⟩) ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)))
29 mulerpq 9982 . . . . . . . 8 (([Q]‘⟨(1st𝑥), (2nd𝑥)⟩) ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩))
3028, 29syl6eq 2821 . . . . . . 7 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩)))
31 xp1st 7348 . . . . . . . . . . 11 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
3223, 31syl 17 . . . . . . . . . 10 (𝑥Q → (1st𝑥) ∈ N)
33 xp2nd 7349 . . . . . . . . . . 11 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
3423, 33syl 17 . . . . . . . . . 10 (𝑥Q → (2nd𝑥) ∈ N)
35 mulpipq 9965 . . . . . . . . . 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 1477 . . . . . . . . 9 (𝑥Q → (⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩) = ⟨((1st𝑥) ·N (2nd𝑥)), ((2nd𝑥) ·N (1st𝑥))⟩)
37 mulcompi 9921 . . . . . . . . . 10 ((2nd𝑥) ·N (1st𝑥)) = ((1st𝑥) ·N (2nd𝑥))
3837opeq2i 4544 . . . . . . . . 9 ⟨((1st𝑥) ·N (2nd𝑥)), ((2nd𝑥) ·N (1st𝑥))⟩ = ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩
3936, 38syl6eq 2821 . . . . . . . 8 (𝑥Q → (⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩) = ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)
4039fveq2d 6337 . . . . . . 7 (𝑥Q → ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩))
41 nqerid 9958 . . . . . . . . 9 (1QQ → ([Q]‘1Q) = 1Q)
426, 41ax-mp 5 . . . . . . . 8 ([Q]‘1Q) = 1Q
43 mulclpi 9918 . . . . . . . . . . 11 (((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N) → ((1st𝑥) ·N (2nd𝑥)) ∈ N)
4432, 34, 43syl2anc 567 . . . . . . . . . 10 (𝑥Q → ((1st𝑥) ·N (2nd𝑥)) ∈ N)
45 1nqenq 9987 . . . . . . . . . 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 9950 . . . . . . . . . . 11 (1QQ → 1Q ∈ (N × N))
486, 47ax-mp 5 . . . . . . . . . 10 1Q ∈ (N × N)
49 opelxpi 5289 . . . . . . . . . . 11 ((((1st𝑥) ·N (2nd𝑥)) ∈ N ∧ ((1st𝑥) ·N (2nd𝑥)) ∈ N) → ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ∈ (N × N))
5044, 44, 49syl2anc 567 . . . . . . . . . 10 (𝑥Q → ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ∈ (N × N))
51 nqereq 9960 . . . . . . . . . 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 569 . . . . . . . . 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 2820 . . . . . . 7 (𝑥Q → ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩) = 1Q)
5530, 40, 543eqtrd 2809 . . . . . 6 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q)
56 fvex 6343 . . . . . . 7 ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) ∈ V
57 oveq2 6802 . . . . . . . 8 (𝑦 = ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) → (𝑥 ·Q 𝑦) = (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)))
5857eqeq1d 2773 . . . . . . 7 (𝑦 = ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q))
5956, 58spcev 3452 . . . . . 6 ((𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q → ∃𝑦(𝑥 ·Q 𝑦) = 1Q)
6055, 59syl 17 . . . . 5 (𝑥Q → ∃𝑦(𝑥 ·Q 𝑦) = 1Q)
61 mulcomnq 9978 . . . . . . 7 (𝑟 ·Q 𝑠) = (𝑠 ·Q 𝑟)
62 mulassnq 9984 . . . . . . 7 ((𝑟 ·Q 𝑠) ·Q 𝑡) = (𝑟 ·Q (𝑠 ·Q 𝑡))
63 mulidnq 9988 . . . . . . 7 (𝑟Q → (𝑟 ·Q 1Q) = 𝑟)
646, 9, 10, 61, 62, 63caovmo 7019 . . . . . 6 ∃*𝑦(𝑥 ·Q 𝑦) = 1Q
65 eu5 2644 . . . . . 6 (∃!𝑦(𝑥 ·Q 𝑦) = 1Q ↔ (∃𝑦(𝑥 ·Q 𝑦) = 1Q ∧ ∃*𝑦(𝑥 ·Q 𝑦) = 1Q))
6664, 65mpbiran2 683 . . . . 5 (∃!𝑦(𝑥 ·Q 𝑦) = 1Q ↔ ∃𝑦(𝑥 ·Q 𝑦) = 1Q)
6760, 66sylibr 224 . . . 4 (𝑥Q → ∃!𝑦(𝑥 ·Q 𝑦) = 1Q)
68 cnvimass 5627 . . . . . . . 8 ( ·Q “ {1Q}) ⊆ dom ·Q
69 df-rq 9942 . . . . . . . 8 *Q = ( ·Q “ {1Q})
709eqcomi 2780 . . . . . . . 8 (Q × Q) = dom ·Q
7168, 69, 703sstr4i 3794 . . . . . . 7 *Q ⊆ (Q × Q)
72 relxp 5267 . . . . . . 7 Rel (Q × Q)
73 relss 5347 . . . . . . 7 (*Q ⊆ (Q × Q) → (Rel (Q × Q) → Rel *Q))
7471, 72, 73mp2 9 . . . . . 6 Rel *Q
7569eleq2i 2842 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ *Q ↔ ⟨𝑥, 𝑦⟩ ∈ ( ·Q “ {1Q}))
76 ffn 6186 . . . . . . . . 9 ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
77 fniniseg 6482 . . . . . . . . 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 452 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q) ↔ (( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)))
80 ancom 452 . . . . . . . . . 10 ((𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ ((𝑥 ·Q 𝑦) = 1Q𝑥Q))
81 eleq1 2838 . . . . . . . . . . . . . . 15 ((𝑥 ·Q 𝑦) = 1Q → ((𝑥 ·Q 𝑦) ∈ Q ↔ 1QQ))
826, 81mpbiri 248 . . . . . . . . . . . . . 14 ((𝑥 ·Q 𝑦) = 1Q → (𝑥 ·Q 𝑦) ∈ Q)
839, 10ndmovrcl 6968 . . . . . . . . . . . . . 14 ((𝑥 ·Q 𝑦) ∈ Q → (𝑥Q𝑦Q))
8482, 83syl 17 . . . . . . . . . . . . 13 ((𝑥 ·Q 𝑦) = 1Q → (𝑥Q𝑦Q))
85 opelxpi 5289 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q) → ⟨𝑥, 𝑦⟩ ∈ (Q × Q))
8684, 85syl 17 . . . . . . . . . . . 12 ((𝑥 ·Q 𝑦) = 1Q → ⟨𝑥, 𝑦⟩ ∈ (Q × Q))
8784simpld 478 . . . . . . . . . . . 12 ((𝑥 ·Q 𝑦) = 1Q𝑥Q)
8886, 872thd 255 . . . . . . . . . . 11 ((𝑥 ·Q 𝑦) = 1Q → (⟨𝑥, 𝑦⟩ ∈ (Q × Q) ↔ 𝑥Q))
8988pm5.32i 558 . . . . . . . . . 10 (((𝑥 ·Q 𝑦) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)) ↔ ((𝑥 ·Q 𝑦) = 1Q𝑥Q))
90 df-ov 6797 . . . . . . . . . . . 12 (𝑥 ·Q 𝑦) = ( ·Q ‘⟨𝑥, 𝑦⟩)
9190eqeq1i 2776 . . . . . . . . . . 11 ((𝑥 ·Q 𝑦) = 1Q ↔ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q)
9291anbi1i 604 . . . . . . . . . 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 5411 . . . . 5 (⊤ → *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)})
9897trud 1641 . . . 4 *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)}
9918, 20, 67, 98fvopab3g 6420 . . 3 ((𝐴Q𝐵 ∈ V) → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
10099ex 397 . 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 382   = wceq 1631  wtru 1632  wex 1852  wcel 2145  ∃!weu 2618  ∃*wmo 2619  Vcvv 3351  wss 3724  {csn 4317  cop 4323   class class class wbr 4787  {copab 4847   × cxp 5248  ccnv 5249  dom cdm 5250  cima 5253  Rel wrel 5255   Fn wfn 6027  wf 6028  cfv 6032  (class class class)co 6794  1st c1st 7314  2nd c2nd 7315  Ncnpi 9869   ·N cmi 9871   ·pQ cmpq 9874   ~Q ceq 9876  Qcnq 9877  1Qc1q 9878  [Q]cerq 9879   ·Q cmq 9881  *Qcrq 9882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-1st 7316  df-2nd 7317  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-oadd 7718  df-omul 7719  df-er 7897  df-ni 9897  df-mi 9899  df-lti 9900  df-mpq 9934  df-enq 9936  df-nq 9937  df-erq 9938  df-mq 9940  df-1nq 9941  df-rq 9942
This theorem is referenced by:  recidnq  9990  recrecnq  9992  reclem3pr  10074
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