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Theorem ordpipq 10017
Description: Ordering of positive fractions in terms of positive integers. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ordpipq (⟨𝐴, 𝐵⟩ <pQ𝐶, 𝐷⟩ ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))

Proof of Theorem ordpipq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 5088 . . 3 𝐴, 𝐵⟩ ∈ V
2 opex 5088 . . 3 𝐶, 𝐷⟩ ∈ V
3 eleq1 2832 . . . . . 6 (𝑥 = ⟨𝐴, 𝐵⟩ → (𝑥 ∈ (N × N) ↔ ⟨𝐴, 𝐵⟩ ∈ (N × N)))
43anbi1d 623 . . . . 5 (𝑥 = ⟨𝐴, 𝐵⟩ → ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))))
54anbi1d 623 . . . 4 (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))))
6 fveq2 6375 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → (1st𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
7 opelxp 5313 . . . . . . . . . 10 (⟨𝐴, 𝐵⟩ ∈ (N × N) ↔ (𝐴N𝐵N))
8 op1stg 7378 . . . . . . . . . 10 ((𝐴N𝐵N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
97, 8sylbi 208 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
109adantr 472 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
116, 10sylan9eq 2819 . . . . . . 7 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (1st𝑥) = 𝐴)
1211oveq1d 6857 . . . . . 6 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st𝑥) ·N (2nd𝑦)) = (𝐴 ·N (2nd𝑦)))
13 fveq2 6375 . . . . . . . 8 (𝑥 = ⟨𝐴, 𝐵⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
14 op2ndg 7379 . . . . . . . . . 10 ((𝐴N𝐵N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
157, 14sylbi 208 . . . . . . . . 9 (⟨𝐴, 𝐵⟩ ∈ (N × N) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
1615adantr 472 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
1713, 16sylan9eq 2819 . . . . . . 7 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (2nd𝑥) = 𝐵)
1817oveq2d 6858 . . . . . 6 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → ((1st𝑦) ·N (2nd𝑥)) = ((1st𝑦) ·N 𝐵))
1912, 18breq12d 4822 . . . . 5 ((𝑥 = ⟨𝐴, 𝐵⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N))) → (((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)) ↔ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵)))
2019pm5.32da 574 . . . 4 (𝑥 = ⟨𝐴, 𝐵⟩ → (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵))))
215, 20bitrd 270 . . 3 (𝑥 = ⟨𝐴, 𝐵⟩ → (((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥))) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵))))
22 eleq1 2832 . . . . . 6 (𝑦 = ⟨𝐶, 𝐷⟩ → (𝑦 ∈ (N × N) ↔ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
2322anbi2d 622 . . . . 5 (𝑦 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ↔ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))))
2423anbi1d 623 . . . 4 (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵)) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵))))
25 fveq2 6375 . . . . . . . 8 (𝑦 = ⟨𝐶, 𝐷⟩ → (2nd𝑦) = (2nd ‘⟨𝐶, 𝐷⟩))
26 opelxp 5313 . . . . . . . . . 10 (⟨𝐶, 𝐷⟩ ∈ (N × N) ↔ (𝐶N𝐷N))
27 op2ndg 7379 . . . . . . . . . 10 ((𝐶N𝐷N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
2826, 27sylbi 208 . . . . . . . . 9 (⟨𝐶, 𝐷⟩ ∈ (N × N) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
2928adantl 473 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
3025, 29sylan9eq 2819 . . . . . . 7 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (2nd𝑦) = 𝐷)
3130oveq2d 6858 . . . . . 6 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (𝐴 ·N (2nd𝑦)) = (𝐴 ·N 𝐷))
32 fveq2 6375 . . . . . . . 8 (𝑦 = ⟨𝐶, 𝐷⟩ → (1st𝑦) = (1st ‘⟨𝐶, 𝐷⟩))
33 op1stg 7378 . . . . . . . . . 10 ((𝐶N𝐷N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
3426, 33sylbi 208 . . . . . . . . 9 (⟨𝐶, 𝐷⟩ ∈ (N × N) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
3534adantl 473 . . . . . . . 8 ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
3632, 35sylan9eq 2819 . . . . . . 7 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → (1st𝑦) = 𝐶)
3736oveq1d 6857 . . . . . 6 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → ((1st𝑦) ·N 𝐵) = (𝐶 ·N 𝐵))
3831, 37breq12d 4822 . . . . 5 ((𝑦 = ⟨𝐶, 𝐷⟩ ∧ (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N))) → ((𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵) ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
3938pm5.32da 574 . . . 4 (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵)) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))))
4024, 39bitrd 270 . . 3 (𝑦 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ (𝐴 ·N (2nd𝑦)) <N ((1st𝑦) ·N 𝐵)) ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))))
41 df-ltpq 9985 . . 3 <pQ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ((1st𝑥) ·N (2nd𝑦)) <N ((1st𝑦) ·N (2nd𝑥)))}
421, 2, 21, 40, 41brab 5159 . 2 (⟨𝐴, 𝐵⟩ <pQ𝐶, 𝐷⟩ ↔ ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
43 simpr 477 . . 3 (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)) → (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))
44 ltrelpi 9964 . . . . . 6 <N ⊆ (N × N)
4544brel 5336 . . . . 5 ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → ((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N))
46 dmmulpi 9966 . . . . . . 7 dom ·N = (N × N)
47 0npi 9957 . . . . . . 7 ¬ ∅ ∈ N
4846, 47ndmovrcl 7018 . . . . . 6 ((𝐴 ·N 𝐷) ∈ N → (𝐴N𝐷N))
4946, 47ndmovrcl 7018 . . . . . 6 ((𝐶 ·N 𝐵) ∈ N → (𝐶N𝐵N))
5048, 49anim12i 606 . . . . 5 (((𝐴 ·N 𝐷) ∈ N ∧ (𝐶 ·N 𝐵) ∈ N) → ((𝐴N𝐷N) ∧ (𝐶N𝐵N)))
51 opelxpi 5314 . . . . . . 7 ((𝐴N𝐵N) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
5251ad2ant2rl 755 . . . . . 6 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → ⟨𝐴, 𝐵⟩ ∈ (N × N))
53 simprl 787 . . . . . . 7 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → 𝐶N)
54 simplr 785 . . . . . . 7 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → 𝐷N)
55 opelxpi 5314 . . . . . . 7 ((𝐶N𝐷N) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
5653, 54, 55syl2anc 579 . . . . . 6 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → ⟨𝐶, 𝐷⟩ ∈ (N × N))
5752, 56jca 507 . . . . 5 (((𝐴N𝐷N) ∧ (𝐶N𝐵N)) → (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
5845, 50, 573syl 18 . . . 4 ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → (⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)))
5958ancri 545 . . 3 ((𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵) → ((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)))
6043, 59impbii 200 . 2 (((⟨𝐴, 𝐵⟩ ∈ (N × N) ∧ ⟨𝐶, 𝐷⟩ ∈ (N × N)) ∧ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵)) ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))
6142, 60bitri 266 1 (⟨𝐴, 𝐵⟩ <pQ𝐶, 𝐷⟩ ↔ (𝐴 ·N 𝐷) <N (𝐶 ·N 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155  cop 4340   class class class wbr 4809   × cxp 5275  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  Ncnpi 9919   ·N cmi 9921   <N clti 9922   <pQ cltpq 9925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-omul 7769  df-ni 9947  df-mi 9949  df-lti 9950  df-ltpq 9985
This theorem is referenced by:  ordpinq  10018  lterpq  10045  ltanq  10046  ltmnq  10047  1lt2nq  10048
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