Theorem ltrnqg 7418

Description: Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7419. (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 7390	. . . 4 ⊢ (𝐴 ∈ Q → (*Q‘𝐴) ∈ Q)
2		recclnq 7390	. . . 4 ⊢ (𝐵 ∈ Q → (*Q‘𝐵) ∈ Q)
3		mulclnq 7374	. . . 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 7399	. . 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 1338	. 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 7381	. . . . . 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 7382	. . . . . 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 1238	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((𝐴 ·Q (*Q‘𝐴)) ·Q (*Q‘𝐵)) = (𝐴 ·Q ((*Q‘𝐴) ·Q (*Q‘𝐵))))
14		mulclnq 7374	. . . . . . 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 7381	. . . . . 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 2216	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐴) = ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))))
19		recidnq 7391	. . . . . 6 ⊢ (𝐴 ∈ Q → (𝐴 ·Q (*Q‘𝐴)) = 1Q)
20	19	oveq2d 5890	. . . . 5 ⊢ (𝐴 ∈ Q → ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))) = ((*Q‘𝐵) ·Q 1Q))
21		mulidnq 7387	. . . . . 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 2230	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))) = (*Q‘𝐵))
24	18, 23	eqtrd 2210	. . 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 7382	. . . . 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 1238	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐵) = ((*Q‘𝐴) ·Q ((*Q‘𝐵) ·Q 𝐵)))
28		mulcomnqg 7381	. . . . . 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 5890	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) ·Q ((*Q‘𝐵) ·Q 𝐵)) = ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))))
31		recidnq 7391	. . . . . 6 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (*Q‘𝐵)) = 1Q)
32	31	oveq2d 5890	. . . . 5 ⊢ (𝐵 ∈ Q → ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))) = ((*Q‘𝐴) ·Q 1Q))
33		mulidnq 7387	. . . . . 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 2232	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))) = (*Q‘𝐴))
36	27, 30, 35	3eqtrd 2214	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐵) = (*Q‘𝐴))
37	24, 36	breq12d 4016	. 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 1353   ∈ wcel 2148   class class class wbr 4003  ‘cfv 5216  (class class class)co 5874  Qcnq 7278  1_Qc1q 7279   ·_Q cmq 7281  *_Qcrq 7282   <_Q cltq 7283

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-mi 7304  df-lti 7305  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351

This theorem is referenced by:  ltrnqi  7419  recexprlemloc  7629  archrecnq  7661