Theorem ltrnq 11067

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 11014	. . 3 ⊢ <Q ⊆ (Q × Q)
2	1	brel 5715	. 2 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
3	1	brel 5715	. . 3 ⊢ ((*Q‘𝐵) <Q (*Q‘𝐴) → ((*Q‘𝐵) ∈ Q ∧ (*Q‘𝐴) ∈ Q))
4		dmrecnq 11056	. . . . 5 ⊢ dom *Q = Q
5		0nnq 11012	. . . . 5 ⊢ ¬ ∅ ∈ Q
6	4, 5	ndmfvrcl 6918	. . . 4 ⊢ ((*Q‘𝐵) ∈ Q → 𝐵 ∈ Q)
7	4, 5	ndmfvrcl 6918	. . . 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 5105	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 <Q 𝑦 ↔ 𝐴 <Q 𝑦))
11		fveq2 6885	. . . . 5 ⊢ (𝑥 = 𝐴 → (*Q‘𝑥) = (*Q‘𝐴))
12	11	breq2d 5114	. . . 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 5106	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴 <Q 𝑦 ↔ 𝐴 <Q 𝐵))
15		fveq2 6885	. . . . 5 ⊢ (𝑦 = 𝐵 → (*Q‘𝑦) = (*Q‘𝐵))
16	15	breq1d 5112	. . . 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 11054	. . . . . 6 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q)
19		recclnq 11054	. . . . . 6 ⊢ (𝑦 ∈ Q → (*Q‘𝑦) ∈ Q)
20		mulclnq 11035	. . . . . 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 11060	. . . . 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 11041	. . . . . . 7 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = (𝑥 ·Q ((*Q‘𝑥) ·Q (*Q‘𝑦)))
25		mulassnq 11047	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘𝑥)) ·Q (*Q‘𝑦)) = (𝑥 ·Q ((*Q‘𝑥) ·Q (*Q‘𝑦)))
26		mulcomnq 11041	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘𝑥)) ·Q (*Q‘𝑦)) = ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥)))
27	24, 25, 26	3eqtr2i 2789	. . . . . 6 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥)))
28		recidnq 11053	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q)
29	28	oveq2d 7436	. . . . . . 7 ⊢ (𝑥 ∈ Q → ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥))) = ((*Q‘𝑦) ·Q 1Q))
30		mulidnq 11051	. . . . . . . 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 2815	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ·Q (𝑥 ·Q (*Q‘𝑥))) = (*Q‘𝑦))
33	27, 32	eqtrid 2807	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑥) = (*Q‘𝑦))
34		mulassnq 11047	. . . . . . 7 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = ((*Q‘𝑥) ·Q ((*Q‘𝑦) ·Q 𝑦))
35		mulcomnq 11041	. . . . . . . 8 ⊢ ((*Q‘𝑦) ·Q 𝑦) = (𝑦 ·Q (*Q‘𝑦))
36	35	oveq2i 7431	. . . . . . 7 ⊢ ((*Q‘𝑥) ·Q ((*Q‘𝑦) ·Q 𝑦)) = ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦)))
37	34, 36	eqtri 2783	. . . . . 6 ⊢ (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦)))
38		recidnq 11053	. . . . . . . 8 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
39	38	oveq2d 7436	. . . . . . 7 ⊢ (𝑦 ∈ Q → ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦))) = ((*Q‘𝑥) ·Q 1Q))
40		mulidnq 11051	. . . . . . . 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 2817	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑥) ·Q (𝑦 ·Q (*Q‘𝑦))) = (*Q‘𝑥))
43	37, 42	eqtrid 2807	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (((*Q‘𝑥) ·Q (*Q‘𝑦)) ·Q 𝑦) = (*Q‘𝑥))
44	33, 43	breq12d 5115	. . . 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 3537	. 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 1569   ∈ wcel 2144   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Qcnq 10940  1_Qc1q 10941   ·_Q cmq 10944  *_Qcrq 10945   <_Q cltq 10946

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-oadd 8483  df-omul 8484  df-er 8720  df-ni 10960  df-mi 10962  df-lti 10963  df-mpq 10997  df-ltpq 10998  df-enq 10999  df-nq 11000  df-erq 11001  df-mq 11003  df-1nq 11004  df-rq 11005  df-ltnq 11006

This theorem is used by:  addclprlem1  11104  reclem2pr  11136  reclem3pr  11137