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Theorem recmulnq 10379
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 6662 . . . 4 (*Q𝐴) ∈ V
21a1i 11 . . 3 (𝐴Q → (*Q𝐴) ∈ V)
3 eleq1 2880 . . 3 ((*Q𝐴) = 𝐵 → ((*Q𝐴) ∈ V ↔ 𝐵 ∈ V))
42, 3syl5ibcom 248 . 2 (𝐴Q → ((*Q𝐴) = 𝐵𝐵 ∈ V))
5 id 22 . . . . . 6 ((𝐴 ·Q 𝐵) = 1Q → (𝐴 ·Q 𝐵) = 1Q)
6 1nq 10343 . . . . . 6 1QQ
75, 6eqeltrdi 2901 . . . . 5 ((𝐴 ·Q 𝐵) = 1Q → (𝐴 ·Q 𝐵) ∈ Q)
8 mulnqf 10364 . . . . . . 7 ·Q :(Q × Q)⟶Q
98fdmi 6502 . . . . . 6 dom ·Q = (Q × Q)
10 0nnq 10339 . . . . . 6 ¬ ∅ ∈ Q
119, 10ndmovrcl 7318 . . . . 5 ((𝐴 ·Q 𝐵) ∈ Q → (𝐴Q𝐵Q))
127, 11syl 17 . . . 4 ((𝐴 ·Q 𝐵) = 1Q → (𝐴Q𝐵Q))
13 elex 3462 . . . 4 (𝐵Q𝐵 ∈ V)
1412, 13simpl2im 507 . . 3 ((𝐴 ·Q 𝐵) = 1Q𝐵 ∈ V)
1514a1i 11 . 2 (𝐴Q → ((𝐴 ·Q 𝐵) = 1Q𝐵 ∈ V))
16 oveq1 7146 . . . . 5 (𝑥 = 𝐴 → (𝑥 ·Q 𝑦) = (𝐴 ·Q 𝑦))
1716eqeq1d 2803 . . . 4 (𝑥 = 𝐴 → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝑦) = 1Q))
18 oveq2 7147 . . . . 5 (𝑦 = 𝐵 → (𝐴 ·Q 𝑦) = (𝐴 ·Q 𝐵))
1918eqeq1d 2803 . . . 4 (𝑦 = 𝐵 → ((𝐴 ·Q 𝑦) = 1Q ↔ (𝐴 ·Q 𝐵) = 1Q))
20 nqerid 10348 . . . . . . . . . 10 (𝑥Q → ([Q]‘𝑥) = 𝑥)
21 relxp 5541 . . . . . . . . . . . 12 Rel (N × N)
22 elpqn 10340 . . . . . . . . . . . 12 (𝑥Q𝑥 ∈ (N × N))
23 1st2nd 7724 . . . . . . . . . . . 12 ((Rel (N × N) ∧ 𝑥 ∈ (N × N)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2421, 22, 23sylancr 590 . . . . . . . . . . 11 (𝑥Q𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2524fveq2d 6653 . . . . . . . . . 10 (𝑥Q → ([Q]‘𝑥) = ([Q]‘⟨(1st𝑥), (2nd𝑥)⟩))
2620, 25eqtr3d 2838 . . . . . . . . 9 (𝑥Q𝑥 = ([Q]‘⟨(1st𝑥), (2nd𝑥)⟩))
2726oveq1d 7154 . . . . . . . 8 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = (([Q]‘⟨(1st𝑥), (2nd𝑥)⟩) ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)))
28 mulerpq 10372 . . . . . . . 8 (([Q]‘⟨(1st𝑥), (2nd𝑥)⟩) ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩))
2927, 28eqtrdi 2852 . . . . . . 7 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩)))
30 xp1st 7707 . . . . . . . . . . 11 (𝑥 ∈ (N × N) → (1st𝑥) ∈ N)
3122, 30syl 17 . . . . . . . . . 10 (𝑥Q → (1st𝑥) ∈ N)
32 xp2nd 7708 . . . . . . . . . . 11 (𝑥 ∈ (N × N) → (2nd𝑥) ∈ N)
3322, 32syl 17 . . . . . . . . . 10 (𝑥Q → (2nd𝑥) ∈ N)
34 mulpipq 10355 . . . . . . . . . 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 837 . . . . . . . . 9 (𝑥Q → (⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩) = ⟨((1st𝑥) ·N (2nd𝑥)), ((2nd𝑥) ·N (1st𝑥))⟩)
36 mulcompi 10311 . . . . . . . . . 10 ((2nd𝑥) ·N (1st𝑥)) = ((1st𝑥) ·N (2nd𝑥))
3736opeq2i 4772 . . . . . . . . 9 ⟨((1st𝑥) ·N (2nd𝑥)), ((2nd𝑥) ·N (1st𝑥))⟩ = ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩
3835, 37eqtrdi 2852 . . . . . . . 8 (𝑥Q → (⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩) = ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)
3938fveq2d 6653 . . . . . . 7 (𝑥Q → ([Q]‘(⟨(1st𝑥), (2nd𝑥)⟩ ·pQ ⟨(2nd𝑥), (1st𝑥)⟩)) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩))
40 nqerid 10348 . . . . . . . . 9 (1QQ → ([Q]‘1Q) = 1Q)
416, 40ax-mp 5 . . . . . . . 8 ([Q]‘1Q) = 1Q
42 mulclpi 10308 . . . . . . . . . . 11 (((1st𝑥) ∈ N ∧ (2nd𝑥) ∈ N) → ((1st𝑥) ·N (2nd𝑥)) ∈ N)
4331, 33, 42syl2anc 587 . . . . . . . . . 10 (𝑥Q → ((1st𝑥) ·N (2nd𝑥)) ∈ N)
44 1nqenq 10377 . . . . . . . . . 10 (((1st𝑥) ·N (2nd𝑥)) ∈ N → 1Q ~Q ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)
4543, 44syl 17 . . . . . . . . 9 (𝑥Q → 1Q ~Q ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)
46 elpqn 10340 . . . . . . . . . . 11 (1QQ → 1Q ∈ (N × N))
476, 46ax-mp 5 . . . . . . . . . 10 1Q ∈ (N × N)
4843, 43opelxpd 5561 . . . . . . . . . 10 (𝑥Q → ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ∈ (N × N))
49 nqereq 10350 . . . . . . . . . 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))⟩)))
5047, 48, 49sylancr 590 . . . . . . . . 9 (𝑥Q → (1Q ~Q ⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩ ↔ ([Q]‘1Q) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩)))
5145, 50mpbid 235 . . . . . . . 8 (𝑥Q → ([Q]‘1Q) = ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩))
5241, 51syl5reqr 2851 . . . . . . 7 (𝑥Q → ([Q]‘⟨((1st𝑥) ·N (2nd𝑥)), ((1st𝑥) ·N (2nd𝑥))⟩) = 1Q)
5329, 39, 523eqtrd 2840 . . . . . 6 (𝑥Q → (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q)
54 fvex 6662 . . . . . . 7 ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) ∈ V
55 oveq2 7147 . . . . . . . 8 (𝑦 = ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) → (𝑥 ·Q 𝑦) = (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)))
5655eqeq1d 2803 . . . . . . 7 (𝑦 = ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩) → ((𝑥 ·Q 𝑦) = 1Q ↔ (𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q))
5754, 56spcev 3558 . . . . . 6 ((𝑥 ·Q ([Q]‘⟨(2nd𝑥), (1st𝑥)⟩)) = 1Q → ∃𝑦(𝑥 ·Q 𝑦) = 1Q)
5853, 57syl 17 . . . . 5 (𝑥Q → ∃𝑦(𝑥 ·Q 𝑦) = 1Q)
59 mulcomnq 10368 . . . . . 6 (𝑟 ·Q 𝑠) = (𝑠 ·Q 𝑟)
60 mulassnq 10374 . . . . . 6 ((𝑟 ·Q 𝑠) ·Q 𝑡) = (𝑟 ·Q (𝑠 ·Q 𝑡))
61 mulidnq 10378 . . . . . 6 (𝑟Q → (𝑟 ·Q 1Q) = 𝑟)
626, 9, 10, 59, 60, 61caovmo 7369 . . . . 5 ∃*𝑦(𝑥 ·Q 𝑦) = 1Q
63 df-eu 2632 . . . . 5 (∃!𝑦(𝑥 ·Q 𝑦) = 1Q ↔ (∃𝑦(𝑥 ·Q 𝑦) = 1Q ∧ ∃*𝑦(𝑥 ·Q 𝑦) = 1Q))
6458, 62, 63sylanblrc 593 . . . 4 (𝑥Q → ∃!𝑦(𝑥 ·Q 𝑦) = 1Q)
65 cnvimass 5920 . . . . . . . 8 ( ·Q “ {1Q}) ⊆ dom ·Q
66 df-rq 10332 . . . . . . . 8 *Q = ( ·Q “ {1Q})
679eqcomi 2810 . . . . . . . 8 (Q × Q) = dom ·Q
6865, 66, 673sstr4i 3961 . . . . . . 7 *Q ⊆ (Q × Q)
69 relxp 5541 . . . . . . 7 Rel (Q × Q)
70 relss 5624 . . . . . . 7 (*Q ⊆ (Q × Q) → (Rel (Q × Q) → Rel *Q))
7168, 69, 70mp2 9 . . . . . 6 Rel *Q
7266eleq2i 2884 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ *Q ↔ ⟨𝑥, 𝑦⟩ ∈ ( ·Q “ {1Q}))
73 ffn 6491 . . . . . . . . 9 ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q))
74 fniniseg 6811 . . . . . . . . 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 464 . . . . . . . . 9 ((⟨𝑥, 𝑦⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q) ↔ (( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)))
77 ancom 464 . . . . . . . . . 10 ((𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q) ↔ ((𝑥 ·Q 𝑦) = 1Q𝑥Q))
78 eleq1 2880 . . . . . . . . . . . . . . 15 ((𝑥 ·Q 𝑦) = 1Q → ((𝑥 ·Q 𝑦) ∈ Q ↔ 1QQ))
796, 78mpbiri 261 . . . . . . . . . . . . . 14 ((𝑥 ·Q 𝑦) = 1Q → (𝑥 ·Q 𝑦) ∈ Q)
809, 10ndmovrcl 7318 . . . . . . . . . . . . . 14 ((𝑥 ·Q 𝑦) ∈ Q → (𝑥Q𝑦Q))
8179, 80syl 17 . . . . . . . . . . . . 13 ((𝑥 ·Q 𝑦) = 1Q → (𝑥Q𝑦Q))
82 opelxpi 5560 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q) → ⟨𝑥, 𝑦⟩ ∈ (Q × Q))
8381, 82syl 17 . . . . . . . . . . . 12 ((𝑥 ·Q 𝑦) = 1Q → ⟨𝑥, 𝑦⟩ ∈ (Q × Q))
8481simpld 498 . . . . . . . . . . . 12 ((𝑥 ·Q 𝑦) = 1Q𝑥Q)
8583, 842thd 268 . . . . . . . . . . 11 ((𝑥 ·Q 𝑦) = 1Q → (⟨𝑥, 𝑦⟩ ∈ (Q × Q) ↔ 𝑥Q))
8685pm5.32i 578 . . . . . . . . . 10 (((𝑥 ·Q 𝑦) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)) ↔ ((𝑥 ·Q 𝑦) = 1Q𝑥Q))
87 df-ov 7142 . . . . . . . . . . . 12 (𝑥 ·Q 𝑦) = ( ·Q ‘⟨𝑥, 𝑦⟩)
8887eqeq1i 2806 . . . . . . . . . . 11 ((𝑥 ·Q 𝑦) = 1Q ↔ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q)
8988anbi1i 626 . . . . . . . . . 10 (((𝑥 ·Q 𝑦) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)) ↔ (( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)))
9077, 86, 893bitr2ri 303 . . . . . . . . 9 ((( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q ∧ ⟨𝑥, 𝑦⟩ ∈ (Q × Q)) ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9176, 90bitri 278 . . . . . . . 8 ((⟨𝑥, 𝑦⟩ ∈ (Q × Q) ∧ ( ·Q ‘⟨𝑥, 𝑦⟩) = 1Q) ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9272, 75, 913bitri 300 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ *Q ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q))
9392a1i 11 . . . . . 6 (⊤ → (⟨𝑥, 𝑦⟩ ∈ *Q ↔ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)))
9471, 93opabbi2dv 5688 . . . . 5 (⊤ → *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)})
9594mptru 1545 . . . 4 *Q = {⟨𝑥, 𝑦⟩ ∣ (𝑥Q ∧ (𝑥 ·Q 𝑦) = 1Q)}
9617, 19, 64, 95fvopab3g 6744 . . 3 ((𝐴Q𝐵 ∈ V) → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
9796ex 416 . 2 (𝐴Q → (𝐵 ∈ V → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q)))
984, 15, 97pm5.21ndd 384 1 (𝐴Q → ((*Q𝐴) = 𝐵 ↔ (𝐴 ·Q 𝐵) = 1Q))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wtru 1539  wex 1781  wcel 2112  ∃*wmo 2599  ∃!weu 2631  Vcvv 3444  wss 3884  {csn 4528  cop 4534   class class class wbr 5033  {copab 5095   × cxp 5521  ccnv 5522  dom cdm 5523  cima 5526  Rel wrel 5528   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  Ncnpi 10259   ·N cmi 10261   ·pQ cmpq 10264   ~Q ceq 10266  Qcnq 10267  1Qc1q 10268  [Q]cerq 10269   ·Q cmq 10271  *Qcrq 10272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-omul 8094  df-er 8276  df-ni 10287  df-mi 10289  df-lti 10290  df-mpq 10324  df-enq 10326  df-nq 10327  df-erq 10328  df-mq 10330  df-1nq 10331  df-rq 10332
This theorem is referenced by:  recidnq  10380  recrecnq  10382  reclem3pr  10464
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