Theorem ltrnq 11094

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 11041	. . 3 ⊢ <Q ⊆ (Q × Q)
2	1	brel 5721	. 2 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
3	1	brel 5721	. . 3 ⊢ ((*Q‘𝐵) <Q (*Q‘𝐴) → ((*Q‘𝐵) ∈ Q ∧ (*Q‘𝐴) ∈ Q))
4		dmrecnq 11083	. . . . 5 ⊢ dom *Q = Q
5		0nnq 11039	. . . . 5 ⊢ ¬ ∅ ∈ Q
6	4, 5	ndmfvrcl 6944	. . . 4 ⊢ ((*Q‘𝐵) ∈ Q → 𝐵 ∈ Q)
7	4, 5	ndmfvrcl 6944	. . . 4 ⊢ ((*Q‘𝐴) ∈ Q → 𝐴 ∈ Q)
8	6, 7	anim12ci 626	. . 3 ⊢ (((*Q‘𝐵) ∈ Q ∧ (*Q‘𝐴) ∈ Q) → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
9	3, 8	syl 18	. 2 ⊢ ((*Q‘𝐵) <Q (*Q‘𝐴) → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
10		breq1 5109	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 <Q 𝑦 ↔ 𝐴 <Q 𝑦))
11		fveq2 6911	. . . . 5 ⊢ (𝑥 = 𝐴 → (*Q‘𝑥) = (*Q‘𝐴))
12	11	breq2d 5118	. . . 4 ⊢ (𝑥 = 𝐴 → ((*Q‘𝑦) <Q (*Q‘𝑥) ↔ (*Q‘𝑦) <Q (*Q‘𝐴)))
13	10, 12	bibi12d 348	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 <Q 𝑦 ↔ (*Q‘𝑦) <Q (*Q‘𝑥)) ↔ (𝐴 <Q 𝑦 ↔ (*Q‘𝑦) <Q (*Q‘𝐴))))
14		breq2 5110	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 <Q 𝑦 ↔ 𝐴 <Q 𝐵))
15		fveq2 6911	. . . . 5 ⊢ (𝑦 = 𝐵 → (*Q‘𝑦) = (*Q‘𝐵))
16	15	breq1d 5116	. . . 4 ⊢ (𝑦 = 𝐵 → ((*Q‘𝑦) <Q (*Q‘𝐴) ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))
17	14, 16	bibi12d 348	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 <Q 𝑦 ↔ (*Q‘𝑦) <Q (*Q‘𝐴)) ↔ (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴))))
18		recclnq 11081	. . . . . 6 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q)
19		recclnq 11081	. . . . . 6 ⊢ (𝑦 ∈ Q → (*Q‘𝑦) ∈ Q)
20		mulclnq 11062	. . . . . 6 ⊢ (((*Q‘𝑥) ∈ Q ∧ (*Q‘𝑦) ∈ Q) → ((*Q‘𝑥) ·Q (*Q‘𝑦)) ∈ Q)
21	18, 19, 20	syl2an 608	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑥) ·Q (*Q‘𝑦)) ∈ Q)
22		ltmnq 11087	. . . . 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 18	. . . 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 11068	. . . . . . 7 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = (𝑥 ·Q ((*Q‘𝑥) ·Q (*Q‘𝑦)))
25		mulassnq 11074	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘𝑥)) ·Q (*Q‘𝑦)) = (𝑥 ·Q ((*Q‘𝑥) ·Q (*Q‘𝑦)))
26		mulcomnq 11068	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘𝑥)) ·Q (*Q‘𝑦)) = ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥)))
27	24, 25, 26	3eqtr2i 2792	. . . . . 6 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥)))
28		recidnq 11080	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q)
29	28	oveq2d 7462	. . . . . . 7 ⊢ (𝑥 ∈ Q → ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥))) = ((*Q‘𝑦) ·Q 1Q))
30		mulidnq 11078	. . . . . . . 8 ⊢ ((*Q‘𝑦) ∈ Q → ((*Q‘𝑦) ·Q 1Q) = (*Q‘𝑦))
31	19, 30	syl 18	. . . . . . 7 ⊢ (𝑦 ∈ Q → ((*Q‘𝑦) ·Q 1Q) = (*Q‘𝑦))
32	29, 31	sylan9eq 2818	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥))) = (*Q‘𝑦))
33	27, 32	eqtrid 2810	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = (*Q‘𝑦))
34		mulassnq 11074	. . . . . . 7 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = ((*Q‘𝑥) ·Q ((*Q‘𝑦) ·Q 𝑦))
35		mulcomnq 11068	. . . . . . . 8 ⊢ ((*Q‘𝑦) ·Q 𝑦) = (𝑦 ·Q (*Q‘𝑦))
36	35	oveq2i 7457	. . . . . . 7 ⊢ ((*Q‘𝑥) ·Q ((*Q‘𝑦) ·Q 𝑦)) = ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦)))
37	34, 36	eqtri 2786	. . . . . 6 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦)))
38		recidnq 11080	. . . . . . . 8 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
39	38	oveq2d 7462	. . . . . . 7 ⊢ (𝑦 ∈ Q → ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦))) = ((*Q‘𝑥) ·Q 1Q))
40		mulidnq 11078	. . . . . . . 8 ⊢ ((*Q‘𝑥) ∈ Q → ((*Q‘𝑥) ·Q 1Q) = (*Q‘𝑥))
41	18, 40	syl 18	. . . . . . 7 ⊢ (𝑥 ∈ Q → ((*Q‘𝑥) ·Q 1Q) = (*Q‘𝑥))
42	39, 41	sylan9eqr 2820	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦))) = (*Q‘𝑥))
43	37, 42	eqtrid 2810	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = (*Q‘𝑥))
44	33, 43	breq12d 5119	. . . 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 282	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q 𝑦 ↔ (*Q‘𝑦) <Q (*Q‘𝑥)))
46	13, 17, 45	vtocl2ga 3541	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))
47	2, 9, 46	pm5.21nii 381	1 ⊢ (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   class class class wbr 5106  ‘cfv 6561  (class class class)co 7446  Qcnq 10967  1_Qc1q 10968   ·_Q cmq 10971  *_Qcrq 10972   <_Q cltq 10973

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-oadd 8511  df-omul 8512  df-er 8748  df-ni 10987  df-mi 10989  df-lti 10990  df-mpq 11024  df-ltpq 11025  df-enq 11026  df-nq 11027  df-erq 11028  df-mq 11030  df-1nq 11031  df-rq 11032  df-ltnq 11033

This theorem is used by:  addclprlem1  11131  reclem2pr  11163  reclem3pr  11164