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Theorem lterpq 10906
Description: Compatibility of ordering on equivalent fractions. (Contributed by Mario Carneiro, 9-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
lterpq (𝐴 <pQ 𝐵 ↔ ([Q]‘𝐴) <Q ([Q]‘𝐵))

Proof of Theorem lterpq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltpq 10846 . . . 4 <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))}
2 opabssxp 5724 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))} ⊆ ((N × N) × (N × N))
31, 2eqsstri 3978 . . 3 <pQ ⊆ ((N × N) × (N × N))
43brel 5697 . 2 (𝐴 <pQ 𝐵 → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
5 ltrelnq 10862 . . . 4 <Q ⊆ (Q × Q)
65brel 5697 . . 3 (([Q]‘𝐴) <Q ([Q]‘𝐵) → (([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q))
7 elpqn 10861 . . . 4 (([Q]‘𝐴) ∈ Q → ([Q]‘𝐴) ∈ (N × N))
8 elpqn 10861 . . . 4 (([Q]‘𝐵) ∈ Q → ([Q]‘𝐵) ∈ (N × N))
9 nqerf 10866 . . . . . . 7 [Q]:(N × N)⟶Q
109fdmi 6680 . . . . . 6 dom [Q] = (N × N)
11 0nelxp 5667 . . . . . 6 ¬ ∅ ∈ (N × N)
1210, 11ndmfvrcl 6878 . . . . 5 (([Q]‘𝐴) ∈ (N × N) → 𝐴 ∈ (N × N))
1310, 11ndmfvrcl 6878 . . . . 5 (([Q]‘𝐵) ∈ (N × N) → 𝐵 ∈ (N × N))
1412, 13anim12i 613 . . . 4 ((([Q]‘𝐴) ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
157, 8, 14syl2an 596 . . 3 ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
166, 15syl 17 . 2 (([Q]‘𝐴) <Q ([Q]‘𝐵) → (𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)))
17 xp1st 7953 . . . . 5 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
18 xp2nd 7954 . . . . 5 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
19 mulclpi 10829 . . . . 5 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
2017, 18, 19syl2an 596 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
21 ltmpi 10840 . . . 4 (((1st𝐴) ·N (2nd𝐵)) ∈ N → (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) <N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))) ↔ (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) <N (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))))))
2220, 21syl 17 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) <N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))) ↔ (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) <N (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))))))
23 nqercl 10867 . . . 4 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q)
24 nqercl 10867 . . . 4 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q)
25 ordpinq 10879 . . . 4 ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q) → (([Q]‘𝐴) <Q ([Q]‘𝐵) ↔ ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) <N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))))
2623, 24, 25syl2an 596 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) <Q ([Q]‘𝐵) ↔ ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) <N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))))
27 1st2nd2 7960 . . . . . 6 (𝐴 ∈ (N × N) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
28 1st2nd2 7960 . . . . . 6 (𝐵 ∈ (N × N) → 𝐵 = ⟨(1st𝐵), (2nd𝐵)⟩)
2927, 28breqan12d 5121 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩))
30 ordpipq 10878 . . . . 5 (⟨(1st𝐴), (2nd𝐴)⟩ <pQ ⟨(1st𝐵), (2nd𝐵)⟩ ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)))
3129, 30bitrdi 286 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴))))
32 xp1st 7953 . . . . . . 7 (([Q]‘𝐴) ∈ (N × N) → (1st ‘([Q]‘𝐴)) ∈ N)
3323, 7, 323syl 18 . . . . . 6 (𝐴 ∈ (N × N) → (1st ‘([Q]‘𝐴)) ∈ N)
34 xp2nd 7954 . . . . . . 7 (([Q]‘𝐵) ∈ (N × N) → (2nd ‘([Q]‘𝐵)) ∈ N)
3524, 8, 343syl 18 . . . . . 6 (𝐵 ∈ (N × N) → (2nd ‘([Q]‘𝐵)) ∈ N)
36 mulclpi 10829 . . . . . 6 (((1st ‘([Q]‘𝐴)) ∈ N ∧ (2nd ‘([Q]‘𝐵)) ∈ N) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N)
3733, 35, 36syl2an 596 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N)
38 ltmpi 10840 . . . . 5 (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ∈ N → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐵) ·N (2nd𝐴)))))
3937, 38syl 17 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st𝐴) ·N (2nd𝐵)) <N ((1st𝐵) ·N (2nd𝐴)) ↔ (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐵) ·N (2nd𝐴)))))
40 mulcompi 10832 . . . . . 6 (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))))
4140a1i 11 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))))
42 nqerrel 10868 . . . . . . . . 9 (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴))
4323, 7syl 17 . . . . . . . . . 10 (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ (N × N))
44 enqbreq2 10856 . . . . . . . . . 10 ((𝐴 ∈ (N × N) ∧ ([Q]‘𝐴) ∈ (N × N)) → (𝐴 ~Q ([Q]‘𝐴) ↔ ((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd𝐴))))
4543, 44mpdan 685 . . . . . . . . 9 (𝐴 ∈ (N × N) → (𝐴 ~Q ([Q]‘𝐴) ↔ ((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd𝐴))))
4642, 45mpbid 231 . . . . . . . 8 (𝐴 ∈ (N × N) → ((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) = ((1st ‘([Q]‘𝐴)) ·N (2nd𝐴)))
4746eqcomd 2742 . . . . . . 7 (𝐴 ∈ (N × N) → ((1st ‘([Q]‘𝐴)) ·N (2nd𝐴)) = ((1st𝐴) ·N (2nd ‘([Q]‘𝐴))))
48 nqerrel 10868 . . . . . . . 8 (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵))
4924, 8syl 17 . . . . . . . . 9 (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ (N × N))
50 enqbreq2 10856 . . . . . . . . 9 ((𝐵 ∈ (N × N) ∧ ([Q]‘𝐵) ∈ (N × N)) → (𝐵 ~Q ([Q]‘𝐵) ↔ ((1st𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵))))
5149, 50mpdan 685 . . . . . . . 8 (𝐵 ∈ (N × N) → (𝐵 ~Q ([Q]‘𝐵) ↔ ((1st𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵))))
5248, 51mpbid 231 . . . . . . 7 (𝐵 ∈ (N × N) → ((1st𝐵) ·N (2nd ‘([Q]‘𝐵))) = ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵)))
5347, 52oveqan12d 7376 . . . . . 6 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘([Q]‘𝐴)) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd ‘([Q]‘𝐵)))) = (((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵))))
54 mulcompi 10832 . . . . . . 7 (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐵) ·N (2nd𝐴)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))))
55 fvex 6855 . . . . . . . 8 (1st𝐵) ∈ V
56 fvex 6855 . . . . . . . 8 (2nd𝐴) ∈ V
57 fvex 6855 . . . . . . . 8 (1st ‘([Q]‘𝐴)) ∈ V
58 mulcompi 10832 . . . . . . . 8 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
59 mulasspi 10833 . . . . . . . 8 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
60 fvex 6855 . . . . . . . 8 (2nd ‘([Q]‘𝐵)) ∈ V
6155, 56, 57, 58, 59, 60caov411 7586 . . . . . . 7 (((1st𝐵) ·N (2nd𝐴)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) = (((1st ‘([Q]‘𝐴)) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd ‘([Q]‘𝐵))))
6254, 61eqtri 2764 . . . . . 6 (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st ‘([Q]‘𝐴)) ·N (2nd𝐴)) ·N ((1st𝐵) ·N (2nd ‘([Q]‘𝐵))))
63 mulcompi 10832 . . . . . . 7 (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))) = (((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st𝐴) ·N (2nd𝐵)))
64 fvex 6855 . . . . . . . 8 (1st ‘([Q]‘𝐵)) ∈ V
65 fvex 6855 . . . . . . . 8 (2nd ‘([Q]‘𝐴)) ∈ V
66 fvex 6855 . . . . . . . 8 (1st𝐴) ∈ V
67 fvex 6855 . . . . . . . 8 (2nd𝐵) ∈ V
6864, 65, 66, 58, 59, 67caov411 7586 . . . . . . 7 (((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵)))
6963, 68eqtri 2764 . . . . . 6 (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))) = (((1st𝐴) ·N (2nd ‘([Q]‘𝐴))) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd𝐵)))
7053, 62, 693eqtr4g 2801 . . . . 5 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴)))))
7141, 70breq12d 5118 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → ((((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐴) ·N (2nd𝐵))) <N (((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵))) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) <N (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))))))
7231, 39, 713bitrd 304 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐴)) ·N (2nd ‘([Q]‘𝐵)))) <N (((1st𝐴) ·N (2nd𝐵)) ·N ((1st ‘([Q]‘𝐵)) ·N (2nd ‘([Q]‘𝐴))))))
7322, 26, 723bitr4rd 311 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 <pQ 𝐵 ↔ ([Q]‘𝐴) <Q ([Q]‘𝐵)))
744, 16, 73pm5.21nii 379 1 (𝐴 <pQ 𝐵 ↔ ([Q]‘𝐴) <Q ([Q]‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  cop 4592   class class class wbr 5105  {copab 5167   × cxp 5631  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Ncnpi 10780   ·N cmi 10782   <N clti 10783   <pQ cltpq 10786   ~Q ceq 10787  Qcnq 10788  [Q]cerq 10790   <Q cltq 10794
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-er 8648  df-ni 10808  df-mi 10810  df-lti 10811  df-ltpq 10846  df-enq 10847  df-nq 10848  df-erq 10849  df-1nq 10852  df-ltnq 10854
This theorem is referenced by:  ltanq  10907  ltmnq  10908  1lt2nq  10909
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