Theorem ltrnqg 7422

Description: Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 7423. (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 7394	. . . 4 ⊢ (𝐴 ∈ Q → (*Q‘𝐴) ∈ Q)
2		recclnq 7394	. . . 4 ⊢ (𝐵 ∈ Q → (*Q‘𝐵) ∈ Q)
3		mulclnq 7378	. . . 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 7403	. . 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 7385	. . . . . 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 7386	. . . . . 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 7378	. . . . . . 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 7385	. . . . . 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 7395	. . . . . 6 ⊢ (𝐴 ∈ Q → (𝐴 ·Q (*Q‘𝐴)) = 1Q)
20	19	oveq2d 5894	. . . . 5 ⊢ (𝐴 ∈ Q → ((*Q‘𝐵) ·Q (𝐴 ·Q (*Q‘𝐴))) = ((*Q‘𝐵) ·Q 1Q))
21		mulidnq 7391	. . . . . 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 7386	. . . . 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 7385	. . . . . 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 5894	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((*Q‘𝐴) ·Q ((*Q‘𝐵) ·Q 𝐵)) = ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))))
31		recidnq 7395	. . . . . 6 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (*Q‘𝐵)) = 1Q)
32	31	oveq2d 5894	. . . . 5 ⊢ (𝐵 ∈ Q → ((*Q‘𝐴) ·Q (𝐵 ·Q (*Q‘𝐵))) = ((*Q‘𝐴) ·Q 1Q))
33		mulidnq 7391	. . . . . 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 4018	. 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 4005  ‘cfv 5218  (class class class)co 5878  Qcnq 7282  1_Qc1q 7283   ·_Q cmq 7285  *_Qcrq 7286   <_Q cltq 7287

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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-1o 6420  df-oadd 6424  df-omul 6425  df-er 6538  df-ec 6540  df-qs 6544  df-ni 7306  df-mi 7308  df-lti 7309  df-mpq 7347  df-enq 7349  df-nqqs 7350  df-mqqs 7352  df-1nqqs 7353  df-rq 7354  df-ltnqqs 7355

This theorem is referenced by:  ltrnqi  7423  recexprlemloc  7633  archrecnq  7665