Theorem ltrnq 10939

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 10886	. . 3 ⊢ <Q ⊆ (Q × Q)
2	1	brel 5706	. 2 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
3	1	brel 5706	. . 3 ⊢ ((*Q‘𝐵) <Q (*Q‘𝐴) → ((*Q‘𝐵) ∈ Q ∧ (*Q‘𝐴) ∈ Q))
4		dmrecnq 10928	. . . . 5 ⊢ dom *Q = Q
5		0nnq 10884	. . . . 5 ⊢ ¬ ∅ ∈ Q
6	4, 5	ndmfvrcl 6897	. . . 4 ⊢ ((*Q‘𝐵) ∈ Q → 𝐵 ∈ Q)
7	4, 5	ndmfvrcl 6897	. . . 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 5113	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 <Q 𝑦 ↔ 𝐴 <Q 𝑦))
11		fveq2 6861	. . . . 5 ⊢ (𝑥 = 𝐴 → (*Q‘𝑥) = (*Q‘𝐴))
12	11	breq2d 5122	. . . 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 5114	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 <Q 𝑦 ↔ 𝐴 <Q 𝐵))
15		fveq2 6861	. . . . 5 ⊢ (𝑦 = 𝐵 → (*Q‘𝑦) = (*Q‘𝐵))
16	15	breq1d 5120	. . . 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 10926	. . . . . 6 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q)
19		recclnq 10926	. . . . . 6 ⊢ (𝑦 ∈ Q → (*Q‘𝑦) ∈ Q)
20		mulclnq 10907	. . . . . 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 10932	. . . . 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 10913	. . . . . . 7 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = (𝑥 ·Q ((*Q‘𝑥) ·Q (*Q‘𝑦)))
25		mulassnq 10919	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘𝑥)) ·Q (*Q‘𝑦)) = (𝑥 ·Q ((*Q‘𝑥) ·Q (*Q‘𝑦)))
26		mulcomnq 10913	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘𝑥)) ·Q (*Q‘𝑦)) = ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥)))
27	24, 25, 26	3eqtr2i 2759	. . . . . 6 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥)))
28		recidnq 10925	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q)
29	28	oveq2d 7406	. . . . . . 7 ⊢ (𝑥 ∈ Q → ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥))) = ((*Q‘𝑦) ·Q 1Q))
30		mulidnq 10923	. . . . . . . 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 2785	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥))) = (*Q‘𝑦))
33	27, 32	eqtrid 2777	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = (*Q‘𝑦))
34		mulassnq 10919	. . . . . . 7 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = ((*Q‘𝑥) ·Q ((*Q‘𝑦) ·Q 𝑦))
35		mulcomnq 10913	. . . . . . . 8 ⊢ ((*Q‘𝑦) ·Q 𝑦) = (𝑦 ·Q (*Q‘𝑦))
36	35	oveq2i 7401	. . . . . . 7 ⊢ ((*Q‘𝑥) ·Q ((*Q‘𝑦) ·Q 𝑦)) = ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦)))
37	34, 36	eqtri 2753	. . . . . 6 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦)))
38		recidnq 10925	. . . . . . . 8 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
39	38	oveq2d 7406	. . . . . . 7 ⊢ (𝑦 ∈ Q → ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦))) = ((*Q‘𝑥) ·Q 1Q))
40		mulidnq 10923	. . . . . . . 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 2787	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦))) = (*Q‘𝑥))
43	37, 42	eqtrid 2777	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = (*Q‘𝑥))
44	33, 43	breq12d 5123	. . . 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 3547	. 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 1540   ∈ wcel 2109   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Qcnq 10812  1_Qc1q 10813   ·_Q cmq 10816  *_Qcrq 10817   <_Q cltq 10818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-omul 8442  df-er 8674  df-ni 10832  df-mi 10834  df-lti 10835  df-mpq 10869  df-ltpq 10870  df-enq 10871  df-nq 10872  df-erq 10873  df-mq 10875  df-1nq 10876  df-rq 10877  df-ltnq 10878

This theorem is referenced by:  addclprlem1  10976  reclem2pr  11008  reclem3pr  11009