Theorem ltrnq 10735

Description: Ordering property of reciprocal for positive fractions. Proposition 9-2.6(iv) of [Gleason] p. 120. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltrnq	⊢ (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴))

Proof of Theorem ltrnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 10682	. . 3 ⊢ <Q ⊆ (Q × Q)
2	1	brel 5652	. 2 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
3	1	brel 5652	. . 3 ⊢ ((*Q‘𝐵) <Q (*Q‘𝐴) → ((*Q‘𝐵) ∈ Q ∧ (*Q‘𝐴) ∈ Q))
4		dmrecnq 10724	. . . . 5 ⊢ dom *Q = Q
5		0nnq 10680	. . . . 5 ⊢ ¬ ∅ ∈ Q
6	4, 5	ndmfvrcl 6805	. . . 4 ⊢ ((*Q‘𝐵) ∈ Q → 𝐵 ∈ Q)
7	4, 5	ndmfvrcl 6805	. . . 4 ⊢ ((*Q‘𝐴) ∈ Q → 𝐴 ∈ Q)
8	6, 7	anim12ci 614	. . 3 ⊢ (((*Q‘𝐵) ∈ Q ∧ (*Q‘𝐴) ∈ Q) → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
9	3, 8	syl 17	. 2 ⊢ ((*Q‘𝐵) <Q (*Q‘𝐴) → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
10		breq1 5077	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 <Q 𝑦 ↔ 𝐴 <Q 𝑦))
11		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝐴 → (*Q‘𝑥) = (*Q‘𝐴))
12	11	breq2d 5086	. . . 4 ⊢ (𝑥 = 𝐴 → ((*Q‘𝑦) <Q (*Q‘𝑥) ↔ (*Q‘𝑦) <Q (*Q‘𝐴)))
13	10, 12	bibi12d 346	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 <Q 𝑦 ↔ (*Q‘𝑦) <Q (*Q‘𝑥)) ↔ (𝐴 <Q 𝑦 ↔ (*Q‘𝑦) <Q (*Q‘𝐴))))
14		breq2 5078	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 <Q 𝑦 ↔ 𝐴 <Q 𝐵))
15		fveq2 6774	. . . . 5 ⊢ (𝑦 = 𝐵 → (*Q‘𝑦) = (*Q‘𝐵))
16	15	breq1d 5084	. . . 4 ⊢ (𝑦 = 𝐵 → ((*Q‘𝑦) <Q (*Q‘𝐴) ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))
17	14, 16	bibi12d 346	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 <Q 𝑦 ↔ (*Q‘𝑦) <Q (*Q‘𝐴)) ↔ (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴))))
18		recclnq 10722	. . . . . 6 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q)
19		recclnq 10722	. . . . . 6 ⊢ (𝑦 ∈ Q → (*Q‘𝑦) ∈ Q)
20		mulclnq 10703	. . . . . 6 ⊢ (((*Q‘𝑥) ∈ Q ∧ (*Q‘𝑦) ∈ Q) → ((*Q‘𝑥) ·Q (*Q‘𝑦)) ∈ Q)
21	18, 19, 20	syl2an 596	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑥) ·Q (*Q‘𝑦)) ∈ Q)
22		ltmnq 10728	. . . . 5 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ∈ Q → (𝑥 <Q 𝑦 ↔ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) <Q (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦)))
23	21, 22	syl 17	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q 𝑦 ↔ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) <Q (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦)))
24		mulcomnq 10709	. . . . . . 7 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = (𝑥 ·Q ((*Q‘𝑥) ·Q (*Q‘𝑦)))
25		mulassnq 10715	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘𝑥)) ·Q (*Q‘𝑦)) = (𝑥 ·Q ((*Q‘𝑥) ·Q (*Q‘𝑦)))
26		mulcomnq 10709	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘𝑥)) ·Q (*Q‘𝑦)) = ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥)))
27	24, 25, 26	3eqtr2i 2772	. . . . . 6 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥)))
28		recidnq 10721	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q)
29	28	oveq2d 7291	. . . . . . 7 ⊢ (𝑥 ∈ Q → ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥))) = ((*Q‘𝑦) ·Q 1Q))
30		mulidnq 10719	. . . . . . . 8 ⊢ ((*Q‘𝑦) ∈ Q → ((*Q‘𝑦) ·Q 1Q) = (*Q‘𝑦))
31	19, 30	syl 17	. . . . . . 7 ⊢ (𝑦 ∈ Q → ((*Q‘𝑦) ·Q 1Q) = (*Q‘𝑦))
32	29, 31	sylan9eq 2798	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥))) = (*Q‘𝑦))
33	27, 32	eqtrid 2790	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = (*Q‘𝑦))
34		mulassnq 10715	. . . . . . 7 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = ((*Q‘𝑥) ·Q ((*Q‘𝑦) ·Q 𝑦))
35		mulcomnq 10709	. . . . . . . 8 ⊢ ((*Q‘𝑦) ·Q 𝑦) = (𝑦 ·Q (*Q‘𝑦))
36	35	oveq2i 7286	. . . . . . 7 ⊢ ((*Q‘𝑥) ·Q ((*Q‘𝑦) ·Q 𝑦)) = ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦)))
37	34, 36	eqtri 2766	. . . . . 6 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦)))
38		recidnq 10721	. . . . . . . 8 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
39	38	oveq2d 7291	. . . . . . 7 ⊢ (𝑦 ∈ Q → ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦))) = ((*Q‘𝑥) ·Q 1Q))
40		mulidnq 10719	. . . . . . . 8 ⊢ ((*Q‘𝑥) ∈ Q → ((*Q‘𝑥) ·Q 1Q) = (*Q‘𝑥))
41	18, 40	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ Q → ((*Q‘𝑥) ·Q 1Q) = (*Q‘𝑥))
42	39, 41	sylan9eqr 2800	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦))) = (*Q‘𝑥))
43	37, 42	eqtrid 2790	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = (*Q‘𝑥))
44	33, 43	breq12d 5087	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) <Q (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) ↔ (*Q‘𝑦) <Q (*Q‘𝑥)))
45	23, 44	bitrd 278	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q 𝑦 ↔ (*Q‘𝑦) <Q (*Q‘𝑥)))
46	13, 17, 45	vtocl2ga 3514	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))
47	2, 9, 46	pm5.21nii 380	1 ⊢ (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  Qcnq 10608  1_Qc1q 10609   ·_Q cmq 10612  *_Qcrq 10613   <_Q cltq 10614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-omul 8302  df-er 8498  df-ni 10628  df-mi 10630  df-lti 10631  df-mpq 10665  df-ltpq 10666  df-enq 10667  df-nq 10668  df-erq 10669  df-mq 10671  df-1nq 10672  df-rq 10673  df-ltnq 10674

This theorem is referenced by:  addclprlem1  10772  reclem2pr  10804  reclem3pr  10805