Theorem ltrnqg 7221

Description: Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7222. (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 7193	. . . 4 ⊢ (𝐴 ∈ Q → (*Q‘𝐴) ∈ Q)
2		recclnq 7193	. . . 4 ⊢ (𝐵 ∈ Q → (*Q‘𝐵) ∈ Q)
3		mulclnq 7177	. . . 4 ⊢ (((*Q‘𝐴) ∈ Q ∧ (*Q‘𝐵) ∈ Q) → ((*Q‘𝐴) ·Q (*Q‘𝐵)) ∈ Q)
4	1, 2, 3	syl2an 287	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) ·Q (*Q‘𝐵)) ∈ Q)
5		ltmnqg 7202	. . 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 1316	. 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 108	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 ∈ Q)
8		mulcomnqg 7184	. . . . . 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 408	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐴) = (𝐴 ·Q ((*Q‘𝐴) ·Q (*Q‘𝐵))))
10	1	adantr 274	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (*Q‘𝐴) ∈ Q)
11	2	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (*Q‘𝐵) ∈ Q)
12		mulassnqg 7185	. . . . . 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 1216	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((𝐴 ·Q (*Q‘𝐴)) ·Q (*Q‘𝐵)) = (𝐴 ·Q ((*Q‘𝐴) ·Q (*Q‘𝐵))))
14		mulclnq 7177	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ (*Q‘𝐴) ∈ Q) → (𝐴 ·Q (*Q‘𝐴)) ∈ Q)
15	7, 10, 14	syl2anc 408	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q (*Q‘𝐴)) ∈ Q)
16		mulcomnqg 7184	. . . . . 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 408	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((𝐴 ·Q (*Q‘𝐴)) ·Q (*Q‘𝐵)) = ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))))
18	9, 13, 17	3eqtr2d 2176	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐴) = ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))))
19		recidnq 7194	. . . . . 6 ⊢ (𝐴 ∈ Q → (𝐴 ·Q (*Q‘𝐴)) = 1Q)
20	19	oveq2d 5783	. . . . 5 ⊢ (𝐴 ∈ Q → ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))) = ((*Q‘𝐵) ·Q 1Q))
21		mulidnq 7190	. . . . . 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 2190	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))) = (*Q‘𝐵))
24	18, 23	eqtrd 2170	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐴) = (*Q‘𝐵))
25		simpr 109	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐵 ∈ Q)
26		mulassnqg 7185	. . . . 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 1216	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐵) = ((*Q‘𝐴) ·Q ((*Q‘𝐵) ·Q 𝐵)))
28		mulcomnqg 7184	. . . . . 6 ⊢ (((*Q‘𝐵) ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐵) ·Q 𝐵) = (𝐵 ·Q (*Q‘𝐵)))
29	11, 25, 28	syl2anc 408	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐵) ·Q 𝐵) = (𝐵 ·Q (*Q‘𝐵)))
30	29	oveq2d 5783	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) ·Q ((*Q‘𝐵) ·Q 𝐵)) = ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))))
31		recidnq 7194	. . . . . 6 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (*Q‘𝐵)) = 1Q)
32	31	oveq2d 5783	. . . . 5 ⊢ (𝐵 ∈ Q → ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))) = ((*Q‘𝐴) ·Q 1Q))
33		mulidnq 7190	. . . . . 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 2192	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))) = (*Q‘𝐴))
36	27, 30, 35	3eqtrd 2174	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐵) = (*Q‘𝐴))
37	24, 36	breq12d 3937	. 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 187	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ (*Q‘𝐵) <Q (*Q‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480   class class class wbr 3924  ‘cfv 5118  (class class class)co 5767  Qcnq 7081  1_Qc1q 7082   ·_Q cmq 7084  *_Qcrq 7085   <_Q cltq 7086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-mi 7107  df-lti 7108  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154

This theorem is referenced by:  ltrnqi  7222  recexprlemloc  7432  archrecnq  7464