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Theorem ltrnqg 6403
 Description: Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 6404. (Contributed by Jim Kingdon, 29-Dec-2019.)
Assertion
Ref Expression
ltrnqg ((A Q B Q) → (A <Q B ↔ (*QB) <Q (*QA)))

Proof of Theorem ltrnqg
StepHypRef Expression
1 recclnq 6376 . . . 4 (A Q → (*QA) Q)
2 recclnq 6376 . . . 4 (B Q → (*QB) Q)
3 mulclnq 6360 . . . 4 (((*QA) Q (*QB) Q) → ((*QA) ·Q (*QB)) Q)
41, 2, 3syl2an 273 . . 3 ((A Q B Q) → ((*QA) ·Q (*QB)) Q)
5 ltmnqg 6385 . . 3 ((A Q B Q ((*QA) ·Q (*QB)) Q) → (A <Q B ↔ (((*QA) ·Q (*QB)) ·Q A) <Q (((*QA) ·Q (*QB)) ·Q B)))
64, 5mpd3an3 1232 . 2 ((A Q B Q) → (A <Q B ↔ (((*QA) ·Q (*QB)) ·Q A) <Q (((*QA) ·Q (*QB)) ·Q B)))
7 simpl 102 . . . . . 6 ((A Q B Q) → A Q)
8 mulcomnqg 6367 . . . . . 6 ((((*QA) ·Q (*QB)) Q A Q) → (((*QA) ·Q (*QB)) ·Q A) = (A ·Q ((*QA) ·Q (*QB))))
94, 7, 8syl2anc 391 . . . . 5 ((A Q B Q) → (((*QA) ·Q (*QB)) ·Q A) = (A ·Q ((*QA) ·Q (*QB))))
101adantr 261 . . . . . 6 ((A Q B Q) → (*QA) Q)
112adantl 262 . . . . . 6 ((A Q B Q) → (*QB) Q)
12 mulassnqg 6368 . . . . . 6 ((A Q (*QA) Q (*QB) Q) → ((A ·Q (*QA)) ·Q (*QB)) = (A ·Q ((*QA) ·Q (*QB))))
137, 10, 11, 12syl3anc 1134 . . . . 5 ((A Q B Q) → ((A ·Q (*QA)) ·Q (*QB)) = (A ·Q ((*QA) ·Q (*QB))))
14 mulclnq 6360 . . . . . . 7 ((A Q (*QA) Q) → (A ·Q (*QA)) Q)
157, 10, 14syl2anc 391 . . . . . 6 ((A Q B Q) → (A ·Q (*QA)) Q)
16 mulcomnqg 6367 . . . . . 6 (((A ·Q (*QA)) Q (*QB) Q) → ((A ·Q (*QA)) ·Q (*QB)) = ((*QB) ·Q (A ·Q (*QA))))
1715, 11, 16syl2anc 391 . . . . 5 ((A Q B Q) → ((A ·Q (*QA)) ·Q (*QB)) = ((*QB) ·Q (A ·Q (*QA))))
189, 13, 173eqtr2d 2075 . . . 4 ((A Q B Q) → (((*QA) ·Q (*QB)) ·Q A) = ((*QB) ·Q (A ·Q (*QA))))
19 recidnq 6377 . . . . . 6 (A Q → (A ·Q (*QA)) = 1Q)
2019oveq2d 5471 . . . . 5 (A Q → ((*QB) ·Q (A ·Q (*QA))) = ((*QB) ·Q 1Q))
21 mulidnq 6373 . . . . . 6 ((*QB) Q → ((*QB) ·Q 1Q) = (*QB))
222, 21syl 14 . . . . 5 (B Q → ((*QB) ·Q 1Q) = (*QB))
2320, 22sylan9eq 2089 . . . 4 ((A Q B Q) → ((*QB) ·Q (A ·Q (*QA))) = (*QB))
2418, 23eqtrd 2069 . . 3 ((A Q B Q) → (((*QA) ·Q (*QB)) ·Q A) = (*QB))
25 simpr 103 . . . . 5 ((A Q B Q) → B Q)
26 mulassnqg 6368 . . . . 5 (((*QA) Q (*QB) Q B Q) → (((*QA) ·Q (*QB)) ·Q B) = ((*QA) ·Q ((*QB) ·Q B)))
2710, 11, 25, 26syl3anc 1134 . . . 4 ((A Q B Q) → (((*QA) ·Q (*QB)) ·Q B) = ((*QA) ·Q ((*QB) ·Q B)))
28 mulcomnqg 6367 . . . . . 6 (((*QB) Q B Q) → ((*QB) ·Q B) = (B ·Q (*QB)))
2911, 25, 28syl2anc 391 . . . . 5 ((A Q B Q) → ((*QB) ·Q B) = (B ·Q (*QB)))
3029oveq2d 5471 . . . 4 ((A Q B Q) → ((*QA) ·Q ((*QB) ·Q B)) = ((*QA) ·Q (B ·Q (*QB))))
31 recidnq 6377 . . . . . 6 (B Q → (B ·Q (*QB)) = 1Q)
3231oveq2d 5471 . . . . 5 (B Q → ((*QA) ·Q (B ·Q (*QB))) = ((*QA) ·Q 1Q))
33 mulidnq 6373 . . . . . 6 ((*QA) Q → ((*QA) ·Q 1Q) = (*QA))
341, 33syl 14 . . . . 5 (A Q → ((*QA) ·Q 1Q) = (*QA))
3532, 34sylan9eqr 2091 . . . 4 ((A Q B Q) → ((*QA) ·Q (B ·Q (*QB))) = (*QA))
3627, 30, 353eqtrd 2073 . . 3 ((A Q B Q) → (((*QA) ·Q (*QB)) ·Q B) = (*QA))
3724, 36breq12d 3768 . 2 ((A Q B Q) → ((((*QA) ·Q (*QB)) ·Q A) <Q (((*QA) ·Q (*QB)) ·Q B) ↔ (*QB) <Q (*QA)))
386, 37bitrd 177 1 ((A Q B Q) → (A <Q B ↔ (*QB) <Q (*QA)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390   class class class wbr 3755  ‘cfv 4845  (class class class)co 5455  Qcnq 6264  1Qc1q 6265   ·Q cmq 6267  *Qcrq 6268
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