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Theorem recmulnq 11004
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 6919 . . . 4 (*Q𝐴) ∈ V
21a1i 11 . . 3 (𝐴Q → (*Q𝐴) ∈ V)
3 eleq1 2829 . . 3 ((*Q𝐴) = 𝐵 → ((*Q𝐴) ∈ V ↔ 𝐵 ∈ V))
42, 3syl5ibcom 245 . 2 (𝐴Q → ((*Q𝐴) = 𝐵𝐵 ∈ V))
5 id 22 . . . . . 6 ((𝐴 ·Q 𝐵) = 1Q → (𝐴 ·Q 𝐵) = 1Q)
6 1nq 10968 . . . . . 6 1QQ
75, 6eqeltrdi 2849 . . . . 5 ((𝐴 ·Q 𝐵) = 1Q → (𝐴 ·Q 𝐵) ∈ Q)
8 mulnqf 10989 . . . . . . 7 ·Q :(Q × Q)⟶Q
98fdmi 6747 . . . . . 6 dom ·Q = (Q × Q)
10 0nnq 10964 . . . . . 6 ¬ ∅ ∈ Q
119, 10ndmovrcl 7619 . . . . 5 ((𝐴 ·Q 𝐵) ∈ Q → (𝐴Q𝐵Q))
127, 11syl 17 . . . 4 ((𝐴 ·Q 𝐵) = 1Q → (𝐴Q𝐵Q))
13 elex 3501 . . . 4 (𝐵Q𝐵 ∈ V)
1412, 13simpl2im 503 . . 3 ((𝐴 ·Q 𝐵) = 1Q𝐵 ∈ V)
1514a1i 11 . 2 (𝐴Q → ((𝐴 ·Q 𝐵) = 1Q𝐵 ∈ V))
16 oveq1 7438 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·Q 𝑦) = (𝐴 ·Q 𝑦))
1716eqeq1d 2739 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
18 oveq2 7439 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·Q 𝑦) = (𝐴 ·Q 𝐵))
1918eqeq1d 2739 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝐵) = 1Q))
20 nqerid 10973 . . . . . . . . . 10 (𝑥Q → ([Q]‘𝑥) = 𝑥)
21 relxp 5703 . . . . . . . . . . . 12 Rel (N × N)
22 elpqn 10965 . . . . . . . . . . . 12 (𝑥Q𝑥 ∈ (N × N))
23 1st2nd 8064 . . . . . . . . . . . 12 ((Rel (N × N) ∧ 𝑥 ∈ (N × N)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2421, 22, 23sylancr 587 . . . . . . . . . . 11 (𝑥Q𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2524fveq2d 6910 . . . . . . . . . 10 (𝑥Q → ([Q]‘𝑥) = ([Q]‘⟨(1st𝑥), (2nd𝑥)⟩))
2620, 25eqtr3d 2779 . . . . . . . . 9 (𝑥Q𝑥 = ([Q]‘⟨(1st𝑥), (2nd𝑥)⟩))
2726oveq1d 7446 . . . . . . . 8 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = (([Q]‘⟨(1st𝑥), (2nd𝑥)⟩) ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)))
28 mulerpq 10997 . . . . . . . 8 (([Q]‘⟨(1st𝑥), (2nd𝑥)⟩) ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩))
2927, 28eqtrdi 2793 . . . . . . 7 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩)))
30 xp1st 8046 . . . . . . . . . . 11 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
3122, 30syl 17 . . . . . . . . . 10 (𝑥Q → (1st𝑥) ∈ N)
32 xp2nd 8047 . . . . . . . . . . 11 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
3322, 32syl 17 . . . . . . . . . 10 (𝑥Q → (2nd𝑥) ∈ N)
34 mulpipq 10980 . . . . . . . . . 10 ((((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N) ∧ ((2nd𝑥) ∈ N ∧ (1st𝑥) ∈ N)) → (⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩) = ⟨((1st𝑥) ·N (2nd𝑥)), ((2nd𝑥) ·N (1st𝑥))⟩)
3531, 33, 33, 31, 34syl22anc 839 . . . . . . . . 9 (𝑥Q → (⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩) = ⟨((1st𝑥) ·N (2nd𝑥)), ((2nd𝑥) ·N (1st𝑥))⟩)
36 mulcompi 10936 . . . . . . . . . 10 ((2nd𝑥) ·N (1st𝑥)) = ((1st𝑥) ·N (2nd𝑥))
3736opeq2i 4877 . . . . . . . . 9 ⟨((1st𝑥) ·N (2nd𝑥)), ((2nd𝑥) ·N (1st𝑥))⟩ = ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩
3835, 37eqtrdi 2793 . . . . . . . 8 (𝑥Q → (⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩) = ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)
3938fveq2d 6910 . . . . . . 7 (𝑥Q → ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩))
40 mulclpi 10933 . . . . . . . . . . 11 (((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N) → ((1st𝑥) ·N (2nd𝑥)) ∈ N)
4131, 33, 40syl2anc 584 . . . . . . . . . 10 (𝑥Q → ((1st𝑥) ·N (2nd𝑥)) ∈ N)
42 1nqenq 11002 . . . . . . . . . 10 (((1st𝑥) ·N (2nd𝑥)) ∈ N → 1Q ~Q ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)
4341, 42syl 17 . . . . . . . . 9 (𝑥Q → 1Q ~Q ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)
44 elpqn 10965 . . . . . . . . . . 11 (1QQ → 1Q ∈ (N × N))
456, 44ax-mp 5 . . . . . . . . . 10 1Q ∈ (N × N)
4641, 41opelxpd 5724 . . . . . . . . . 10 (𝑥Q → ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ∈ (N × N))
47 nqereq 10975 . . . . . . . . . 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))⟩)))
4845, 46, 47sylancr 587 . . . . . . . . 9 (𝑥Q → (1Q ~Q ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ↔ ([Q]‘1Q) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)))
4943, 48mpbid 232 . . . . . . . 8 (𝑥Q → ([Q]‘1Q) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩))
50 nqerid 10973 . . . . . . . . 9 (1QQ → ([Q]‘1Q) = 1Q)
516, 50ax-mp 5 . . . . . . . 8 ([Q]‘1Q) = 1Q
5249, 51eqtr3di 2792 . . . . . . 7 (𝑥Q → ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩) = 1Q)
5329, 39, 523eqtrd 2781 . . . . . 6 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q)
54 fvex 6919 . . . . . . 7 ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) ∈ V
55 oveq2 7439 . . . . . . . 8 (𝑦 = ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) → (𝑥 ·Q 𝑦) = (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)))
5655eqeq1d 2739 . . . . . . 7 (𝑦 = ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q))
5754, 56spcev 3606 . . . . . 6 ((𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q → ∃𝑦(𝑥 ·Q 𝑦) = 1Q)
5853, 57syl 17 . . . . 5 (𝑥Q → ∃𝑦(𝑥 ·Q 𝑦) = 1Q)
59 mulcomnq 10993 . . . . . 6 (𝑟 ·Q 𝑠) = (𝑠 ·Q 𝑟)
60 mulassnq 10999 . . . . . 6 ((𝑟 ·Q 𝑠) ·Q 𝑡) = (𝑟 ·Q (𝑠 ·Q 𝑡))
61 mulidnq 11003 . . . . . 6 (𝑟Q → (𝑟 ·Q 1Q) = 𝑟)
626, 9, 10, 59, 60, 61caovmo 7670 . . . . 5 ∃*𝑦(𝑥 ·Q 𝑦) = 1Q
63 df-eu 2569 . . . . 5 (∃!𝑦(𝑥 ·Q 𝑦) = 1Q ↔ (∃𝑦(𝑥 ·Q 𝑦) = 1Q ∧ ∃*𝑦(𝑥 ·Q 𝑦) = 1Q))
6458, 62, 63sylanblrc 590 . . . 4 (𝑥Q → ∃!𝑦(𝑥 ·Q 𝑦) = 1Q)
65 cnvimass 6100 . . . . . . . 8 ( ·Q “ {1Q}) ⊆ dom ·Q
66 df-rq 10957 . . . . . . . 8 *Q = ( ·Q “ {1Q})
679eqcomi 2746 . . . . . . . 8 (Q × Q) = dom ·Q
6865, 66, 673sstr4i 4035 . . . . . . 7 *Q ⊆ (Q × Q)
69 relxp 5703 . . . . . . 7 Rel (Q × Q)
70 relss 5791 . . . . . . 7 (*Q ⊆ (Q × Q) → (Rel (Q × Q) → Rel *Q))
7168, 69, 70mp2 9 . . . . . 6 Rel *Q
7266eleq2i 2833 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ *Q ↔ ⟨𝑥, 𝑦⟩ ∈ ( ·Q “ {1Q}))
73 ffn 6736 . . . . . . . . 9 ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
74 fniniseg 7080 . . . . . . . . 9 ( ·Q Fn (Q × Q) → (⟨𝑥, 𝑦⟩ ∈ ( ·Q “ {1Q}) ↔ (⟨𝑥, 𝑦⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q)))
758, 73, 74mp2b 10 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ ( ·Q “ {1Q}) ↔ (⟨𝑥, 𝑦⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q))
76 ancom 460 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q) ↔ (( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)))
77 ancom 460 . . . . . . . . . 10 ((𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ ((𝑥 ·Q 𝑦) = 1Q𝑥Q))
78 eleq1 2829 . . . . . . . . . . . . . . 15 ((𝑥 ·Q 𝑦) = 1Q → ((𝑥 ·Q 𝑦) ∈ Q ↔ 1QQ))
796, 78mpbiri 258 . . . . . . . . . . . . . 14 ((𝑥 ·Q 𝑦) = 1Q → (𝑥 ·Q 𝑦) ∈ Q)
809, 10ndmovrcl 7619 . . . . . . . . . . . . . 14 ((𝑥 ·Q 𝑦) ∈ Q → (𝑥Q𝑦Q))
8179, 80syl 17 . . . . . . . . . . . . 13 ((𝑥 ·Q 𝑦) = 1Q → (𝑥Q𝑦Q))
82 opelxpi 5722 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q) → ⟨𝑥, 𝑦⟩ ∈ (Q × Q))
8381, 82syl 17 . . . . . . . . . . . 12 ((𝑥 ·Q 𝑦) = 1Q → ⟨𝑥, 𝑦⟩ ∈ (Q × Q))
8481simpld 494 . . . . . . . . . . . 12 ((𝑥 ·Q 𝑦) = 1Q𝑥Q)
8583, 842thd 265 . . . . . . . . . . 11 ((𝑥 ·Q 𝑦) = 1Q → (⟨𝑥, 𝑦⟩ ∈ (Q × Q) ↔ 𝑥Q))
8685pm5.32i 574 . . . . . . . . . 10 (((𝑥 ·Q 𝑦) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)) ↔ ((𝑥 ·Q 𝑦) = 1Q𝑥Q))
87 df-ov 7434 . . . . . . . . . . . 12 (𝑥 ·Q 𝑦) = ( ·Q ‘⟨𝑥, 𝑦⟩)
8887eqeq1i 2742 . . . . . . . . . . 11 ((𝑥 ·Q 𝑦) = 1Q ↔ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q)
8988anbi1i 624 . . . . . . . . . 10 (((𝑥 ·Q 𝑦) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)) ↔ (( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)))
9077, 86, 893bitr2ri 300 . . . . . . . . 9 ((( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)) ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9176, 90bitri 275 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q) ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9272, 75, 913bitri 297 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ *Q ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9392a1i 11 . . . . . 6 (⊤ → (⟨𝑥, 𝑦⟩ ∈ *Q ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)))
9471, 93opabbi2dv 5860 . . . . 5 (⊤ → *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)})
9594mptru 1547 . . . 4 *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)}
9617, 19, 64, 95fvopab3g 7011 . . 3 ((𝐴Q𝐵 ∈ V) → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
9796ex 412 . 2 (𝐴Q → (𝐵 ∈ V → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q)))
984, 15, 97pm5.21ndd 379 1 (𝐴Q → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wtru 1541  wex 1779  wcel 2108  ∃*wmo 2538  ∃!weu 2568  Vcvv 3480  wss 3951  {csn 4626  cop 4632   class class class wbr 5143  {copab 5205   × cxp 5683  ccnv 5684  dom cdm 5685  cima 5688  Rel wrel 5690   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Ncnpi 10884   ·N cmi 10886   ·pQ cmpq 10889   ~Q ceq 10891  Qcnq 10892  1Qc1q 10893  [Q]cerq 10894   ·Q cmq 10896  *Qcrq 10897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-er 8745  df-ni 10912  df-mi 10914  df-lti 10915  df-mpq 10949  df-enq 10951  df-nq 10952  df-erq 10953  df-mq 10955  df-1nq 10956  df-rq 10957
This theorem is referenced by:  recidnq  11005  recrecnq  11007  reclem3pr  11089
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