Theorem ltrnqg 7630

Description: Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7631. (Contributed by Jim Kingdon, 29-Dec-2019.)

Assertion

Ref	Expression
ltrnqg	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))

Proof of Theorem ltrnqg

Step	Hyp	Ref	Expression
1		recclnq 7602	. . . 4 ⊢ (𝐴 ∈ Q → (*Q‘𝐴) ∈ Q)
2		recclnq 7602	. . . 4 ⊢ (𝐵 ∈ Q → (*Q‘𝐵) ∈ Q)
3		mulclnq 7586	. . . 4 ⊢ (((*Q‘𝐴) ∈ Q ∧ (*Q‘𝐵) ∈ Q) → ((*Q‘𝐴) ·Q (*Q‘𝐵)) ∈ Q)
4	1, 2, 3	syl2an 289	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) ·Q (*Q‘𝐵)) ∈ Q)
5		ltmnqg 7611	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ ((*Q‘𝐴) ·Q (*Q‘𝐵)) ∈ Q) → (𝐴 <Q 𝐵 ↔ (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐴) <Q (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐵)))
6	4, 5	mpd3an3 1372	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐴) <Q (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐵)))
7		simpl 109	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 ∈ Q)
8		mulcomnqg 7593	. . . . . 6 ⊢ ((((*Q‘𝐴) ·Q (*Q‘𝐵)) ∈ Q ∧ 𝐴 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐴) = (𝐴 ·Q ((*Q‘𝐴) ·Q (*Q‘𝐵))))
9	4, 7, 8	syl2anc 411	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐴) = (𝐴 ·Q ((*Q‘𝐴) ·Q (*Q‘𝐵))))
10	1	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (*Q‘𝐴) ∈ Q)
11	2	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (*Q‘𝐵) ∈ Q)
12		mulassnqg 7594	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ (*Q‘𝐴) ∈ Q ∧ (*Q‘𝐵) ∈ Q) → ((𝐴 ·Q (*Q‘𝐴)) ·Q (*Q‘𝐵)) = (𝐴 ·Q ((*Q‘𝐴) ·Q (*Q‘𝐵))))
13	7, 10, 11, 12	syl3anc 1271	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((𝐴 ·Q (*Q‘𝐴)) ·Q (*Q‘𝐵)) = (𝐴 ·Q ((*Q‘𝐴) ·Q (*Q‘𝐵))))
14		mulclnq 7586	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ (*Q‘𝐴) ∈ Q) → (𝐴 ·Q (*Q‘𝐴)) ∈ Q)
15	7, 10, 14	syl2anc 411	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q (*Q‘𝐴)) ∈ Q)
16		mulcomnqg 7593	. . . . . 6 ⊢ (((𝐴 ·Q (*Q‘𝐴)) ∈ Q ∧ (*Q‘𝐵) ∈ Q) → ((𝐴 ·Q (*Q‘𝐴)) ·Q (*Q‘𝐵)) = ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))))
17	15, 11, 16	syl2anc 411	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((𝐴 ·Q (*Q‘𝐴)) ·Q (*Q‘𝐵)) = ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))))
18	9, 13, 17	3eqtr2d 2268	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐴) = ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))))
19		recidnq 7603	. . . . . 6 ⊢ (𝐴 ∈ Q → (𝐴 ·Q (*Q‘𝐴)) = 1Q)
20	19	oveq2d 6029	. . . . 5 ⊢ (𝐴 ∈ Q → ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))) = ((*Q‘𝐵) ·Q 1Q))
21		mulidnq 7599	. . . . . 6 ⊢ ((*Q‘𝐵) ∈ Q → ((*Q‘𝐵) ·Q 1Q) = (*Q‘𝐵))
22	2, 21	syl 14	. . . . 5 ⊢ (𝐵 ∈ Q → ((*Q‘𝐵) ·Q 1Q) = (*Q‘𝐵))
23	20, 22	sylan9eq 2282	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))) = (*Q‘𝐵))
24	18, 23	eqtrd 2262	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐴) = (*Q‘𝐵))
25		simpr 110	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐵 ∈ Q)
26		mulassnqg 7594	. . . . 5 ⊢ (((*Q‘𝐴) ∈ Q ∧ (*Q‘𝐵) ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐵) = ((*Q‘𝐴) ·Q ((*Q‘𝐵) ·Q 𝐵)))
27	10, 11, 25, 26	syl3anc 1271	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐵) = ((*Q‘𝐴) ·Q ((*Q‘𝐵) ·Q 𝐵)))
28		mulcomnqg 7593	. . . . . 6 ⊢ (((*Q‘𝐵) ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐵) ·Q 𝐵) = (𝐵 ·Q (*Q‘𝐵)))
29	11, 25, 28	syl2anc 411	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐵) ·Q 𝐵) = (𝐵 ·Q (*Q‘𝐵)))
30	29	oveq2d 6029	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) ·Q ((*Q‘𝐵) ·Q 𝐵)) = ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))))
31		recidnq 7603	. . . . . 6 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (*Q‘𝐵)) = 1Q)
32	31	oveq2d 6029	. . . . 5 ⊢ (𝐵 ∈ Q → ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))) = ((*Q‘𝐴) ·Q 1Q))
33		mulidnq 7599	. . . . . 6 ⊢ ((*Q‘𝐴) ∈ Q → ((*Q‘𝐴) ·Q 1Q) = (*Q‘𝐴))
34	1, 33	syl 14	. . . . 5 ⊢ (𝐴 ∈ Q → ((*Q‘𝐴) ·Q 1Q) = (*Q‘𝐴))
35	32, 34	sylan9eqr 2284	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))) = (*Q‘𝐴))
36	27, 30, 35	3eqtrd 2266	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐵) = (*Q‘𝐴))
37	24, 36	breq12d 4099	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐴) <Q (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐵) ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))
38	6, 37	bitrd 188	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200   class class class wbr 4086  ‘cfv 5324  (class class class)co 6013  Qcnq 7490  1_Qc1q 7491   ·_Q cmq 7493  *_Qcrq 7494   <_Q cltq 7495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-mi 7516  df-lti 7517  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-mqqs 7560  df-1nqqs 7561  df-rq 7562  df-ltnqqs 7563

This theorem is referenced by:  ltrnqi  7631  recexprlemloc  7841  archrecnq  7873