Theorem ltrnqg 7382

Description: Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7383. (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 7354	. . . 4 ⊢ (𝐴 ∈ Q → (*Q‘𝐴) ∈ Q)
2		recclnq 7354	. . . 4 ⊢ (𝐵 ∈ Q → (*Q‘𝐵) ∈ Q)
3		mulclnq 7338	. . . 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 7363	. . 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 1333	. 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 7345	. . . . . 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 409	. . . . 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 7346	. . . . . 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 1233	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((𝐴 ·Q (*Q‘𝐴)) ·Q (*Q‘𝐵)) = (𝐴 ·Q ((*Q‘𝐴) ·Q (*Q‘𝐵))))
14		mulclnq 7338	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ (*Q‘𝐴) ∈ Q) → (𝐴 ·Q (*Q‘𝐴)) ∈ Q)
15	7, 10, 14	syl2anc 409	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 ·Q (*Q‘𝐴)) ∈ Q)
16		mulcomnqg 7345	. . . . . 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 409	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((𝐴 ·Q (*Q‘𝐴)) ·Q (*Q‘𝐵)) = ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))))
18	9, 13, 17	3eqtr2d 2209	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐴) = ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))))
19		recidnq 7355	. . . . . 6 ⊢ (𝐴 ∈ Q → (𝐴 ·Q (*Q‘𝐴)) = 1Q)
20	19	oveq2d 5869	. . . . 5 ⊢ (𝐴 ∈ Q → ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))) = ((*Q‘𝐵) ·Q 1Q))
21		mulidnq 7351	. . . . . 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 2223	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))) = (*Q‘𝐵))
24	18, 23	eqtrd 2203	. . 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 7346	. . . . 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 1233	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐵) = ((*Q‘𝐴) ·Q ((*Q‘𝐵) ·Q 𝐵)))
28		mulcomnqg 7345	. . . . . 6 ⊢ (((*Q‘𝐵) ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐵) ·Q 𝐵) = (𝐵 ·Q (*Q‘𝐵)))
29	11, 25, 28	syl2anc 409	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐵) ·Q 𝐵) = (𝐵 ·Q (*Q‘𝐵)))
30	29	oveq2d 5869	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) ·Q ((*Q‘𝐵) ·Q 𝐵)) = ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))))
31		recidnq 7355	. . . . . 6 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (*Q‘𝐵)) = 1Q)
32	31	oveq2d 5869	. . . . 5 ⊢ (𝐵 ∈ Q → ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))) = ((*Q‘𝐴) ·Q 1Q))
33		mulidnq 7351	. . . . . 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 2225	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))) = (*Q‘𝐴))
36	27, 30, 35	3eqtrd 2207	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((*Q‘𝐴) ·Q (*Q‘𝐵)) ·Q 𝐵) = (*Q‘𝐴))
37	24, 36	breq12d 4002	. 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 1348   ∈ wcel 2141   class class class wbr 3989  ‘cfv 5198  (class class class)co 5853  Qcnq 7242  1_Qc1q 7243   ·_Q cmq 7245  *_Qcrq 7246   <_Q cltq 7247

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-mi 7268  df-lti 7269  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315

This theorem is referenced by:  ltrnqi  7383  recexprlemloc  7593  archrecnq  7625