Theorem ltrnq 10867

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 10814	. . 3 ⊢ <Q ⊆ (Q × Q)
2	1	brel 5681	. 2 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
3	1	brel 5681	. . 3 ⊢ ((*Q‘𝐵) <Q (*Q‘𝐴) → ((*Q‘𝐵) ∈ Q ∧ (*Q‘𝐴) ∈ Q))
4		dmrecnq 10856	. . . . 5 ⊢ dom *Q = Q
5		0nnq 10812	. . . . 5 ⊢ ¬ ∅ ∈ Q
6	4, 5	ndmfvrcl 6855	. . . 4 ⊢ ((*Q‘𝐵) ∈ Q → 𝐵 ∈ Q)
7	4, 5	ndmfvrcl 6855	. . . 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 5094	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 <Q 𝑦 ↔ 𝐴 <Q 𝑦))
11		fveq2 6822	. . . . 5 ⊢ (𝑥 = 𝐴 → (*Q‘𝑥) = (*Q‘𝐴))
12	11	breq2d 5103	. . . 4 ⊢ (𝑥 = 𝐴 → ((*Q‘𝑦) <Q (*Q‘𝑥) ↔ (*Q‘𝑦) <Q (*Q‘𝐴)))
13	10, 12	bibi12d 345	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 <Q 𝑦 ↔ (*Q‘𝑦) <Q (*Q‘𝑥)) ↔ (𝐴 <Q 𝑦 ↔ (*Q‘𝑦) <Q (*Q‘𝐴))))
14		breq2 5095	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 <Q 𝑦 ↔ 𝐴 <Q 𝐵))
15		fveq2 6822	. . . . 5 ⊢ (𝑦 = 𝐵 → (*Q‘𝑦) = (*Q‘𝐵))
16	15	breq1d 5101	. . . 4 ⊢ (𝑦 = 𝐵 → ((*Q‘𝑦) <Q (*Q‘𝐴) ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))
17	14, 16	bibi12d 345	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 <Q 𝑦 ↔ (*Q‘𝑦) <Q (*Q‘𝐴)) ↔ (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴))))
18		recclnq 10854	. . . . . 6 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q)
19		recclnq 10854	. . . . . 6 ⊢ (𝑦 ∈ Q → (*Q‘𝑦) ∈ Q)
20		mulclnq 10835	. . . . . 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 10860	. . . . 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 10841	. . . . . . 7 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = (𝑥 ·Q ((*Q‘𝑥) ·Q (*Q‘𝑦)))
25		mulassnq 10847	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘𝑥)) ·Q (*Q‘𝑦)) = (𝑥 ·Q ((*Q‘𝑥) ·Q (*Q‘𝑦)))
26		mulcomnq 10841	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘𝑥)) ·Q (*Q‘𝑦)) = ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥)))
27	24, 25, 26	3eqtr2i 2760	. . . . . 6 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥)))
28		recidnq 10853	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q)
29	28	oveq2d 7362	. . . . . . 7 ⊢ (𝑥 ∈ Q → ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥))) = ((*Q‘𝑦) ·Q 1Q))
30		mulidnq 10851	. . . . . . . 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 2786	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥))) = (*Q‘𝑦))
33	27, 32	eqtrid 2778	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = (*Q‘𝑦))
34		mulassnq 10847	. . . . . . 7 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = ((*Q‘𝑥) ·Q ((*Q‘𝑦) ·Q 𝑦))
35		mulcomnq 10841	. . . . . . . 8 ⊢ ((*Q‘𝑦) ·Q 𝑦) = (𝑦 ·Q (*Q‘𝑦))
36	35	oveq2i 7357	. . . . . . 7 ⊢ ((*Q‘𝑥) ·Q ((*Q‘𝑦) ·Q 𝑦)) = ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦)))
37	34, 36	eqtri 2754	. . . . . 6 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦)))
38		recidnq 10853	. . . . . . . 8 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
39	38	oveq2d 7362	. . . . . . 7 ⊢ (𝑦 ∈ Q → ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦))) = ((*Q‘𝑥) ·Q 1Q))
40		mulidnq 10851	. . . . . . . 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 2788	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦))) = (*Q‘𝑥))
43	37, 42	eqtrid 2778	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = (*Q‘𝑥))
44	33, 43	breq12d 5104	. . . 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 279	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q 𝑦 ↔ (*Q‘𝑦) <Q (*Q‘𝑥)))
46	13, 17, 45	vtocl2ga 3533	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))
47	2, 9, 46	pm5.21nii 378	1 ⊢ (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346  Qcnq 10740  1_Qc1q 10741   ·_Q cmq 10744  *_Qcrq 10745   <_Q cltq 10746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-omul 8390  df-er 8622  df-ni 10760  df-mi 10762  df-lti 10763  df-mpq 10797  df-ltpq 10798  df-enq 10799  df-nq 10800  df-erq 10801  df-mq 10803  df-1nq 10804  df-rq 10805  df-ltnq 10806

This theorem is referenced by:  addclprlem1  10904  reclem2pr  10936  reclem3pr  10937