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Theorem mullocprlem 6668
Description: Calculations for mullocpr 6669. (Contributed by Jim Kingdon, 10-Dec-2019.)
Hypotheses
Ref Expression
mullocprlem.ab (𝜑 → (𝐴P𝐵P))
mullocprlem.uqedu (𝜑 → (𝑈 ·Q 𝑄) <Q (𝐸 ·Q (𝐷 ·Q 𝑈)))
mullocprlem.edutdu (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈)))
mullocprlem.tdudr (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))
mullocprlem.qr (𝜑 → (𝑄Q𝑅Q))
mullocprlem.duq (𝜑 → (𝐷Q𝑈Q))
mullocprlem.du (𝜑 → (𝐷 ∈ (1st𝐴) ∧ 𝑈 ∈ (2nd𝐴)))
mullocprlem.et (𝜑 → (𝐸Q𝑇Q))
Assertion
Ref Expression
mullocprlem (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Proof of Theorem mullocprlem
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mullocprlem.uqedu . . . . . . 7 (𝜑 → (𝑈 ·Q 𝑄) <Q (𝐸 ·Q (𝐷 ·Q 𝑈)))
2 mullocprlem.et . . . . . . . . 9 (𝜑 → (𝐸Q𝑇Q))
32simpld 105 . . . . . . . 8 (𝜑𝐸Q)
4 mullocprlem.duq . . . . . . . . 9 (𝜑 → (𝐷Q𝑈Q))
54simpld 105 . . . . . . . 8 (𝜑𝐷Q)
64simprd 107 . . . . . . . 8 (𝜑𝑈Q)
7 mulcomnqg 6481 . . . . . . . . 9 ((𝑥Q𝑦Q) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
87adantl 262 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑦Q)) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
9 mulassnqg 6482 . . . . . . . . 9 ((𝑥Q𝑦Q𝑧Q) → ((𝑥 ·Q 𝑦) ·Q 𝑧) = (𝑥 ·Q (𝑦 ·Q 𝑧)))
109adantl 262 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑦Q𝑧Q)) → ((𝑥 ·Q 𝑦) ·Q 𝑧) = (𝑥 ·Q (𝑦 ·Q 𝑧)))
113, 5, 6, 8, 10caov13d 5684 . . . . . . 7 (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) = (𝑈 ·Q (𝐷 ·Q 𝐸)))
121, 11breqtrd 3788 . . . . . 6 (𝜑 → (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸)))
13 mullocprlem.qr . . . . . . . 8 (𝜑 → (𝑄Q𝑅Q))
1413simpld 105 . . . . . . 7 (𝜑𝑄Q)
15 mulclnq 6474 . . . . . . . 8 ((𝐷Q𝐸Q) → (𝐷 ·Q 𝐸) ∈ Q)
165, 3, 15syl2anc 391 . . . . . . 7 (𝜑 → (𝐷 ·Q 𝐸) ∈ Q)
17 ltmnqg 6499 . . . . . . 7 ((𝑄Q ∧ (𝐷 ·Q 𝐸) ∈ Q𝑈Q) → (𝑄 <Q (𝐷 ·Q 𝐸) ↔ (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸))))
1814, 16, 6, 17syl3anc 1135 . . . . . 6 (𝜑 → (𝑄 <Q (𝐷 ·Q 𝐸) ↔ (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸))))
1912, 18mpbird 156 . . . . 5 (𝜑𝑄 <Q (𝐷 ·Q 𝐸))
2019adantr 261 . . . 4 ((𝜑𝐸 ∈ (1st𝐵)) → 𝑄 <Q (𝐷 ·Q 𝐸))
21 mullocprlem.ab . . . . . . . 8 (𝜑 → (𝐴P𝐵P))
2221simpld 105 . . . . . . 7 (𝜑𝐴P)
23 mullocprlem.du . . . . . . . 8 (𝜑 → (𝐷 ∈ (1st𝐴) ∧ 𝑈 ∈ (2nd𝐴)))
2423simpld 105 . . . . . . 7 (𝜑𝐷 ∈ (1st𝐴))
2522, 24jca 290 . . . . . 6 (𝜑 → (𝐴P𝐷 ∈ (1st𝐴)))
2625adantr 261 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → (𝐴P𝐷 ∈ (1st𝐴)))
2721simprd 107 . . . . . 6 (𝜑𝐵P)
2827anim1i 323 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → (𝐵P𝐸 ∈ (1st𝐵)))
2914adantr 261 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → 𝑄Q)
30 mulnqprl 6666 . . . . 5 ((((𝐴P𝐷 ∈ (1st𝐴)) ∧ (𝐵P𝐸 ∈ (1st𝐵))) ∧ 𝑄Q) → (𝑄 <Q (𝐷 ·Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 ·P 𝐵))))
3126, 28, 29, 30syl21anc 1134 . . . 4 ((𝜑𝐸 ∈ (1st𝐵)) → (𝑄 <Q (𝐷 ·Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 ·P 𝐵))))
3220, 31mpd 13 . . 3 ((𝜑𝐸 ∈ (1st𝐵)) → 𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)))
3332orcd 652 . 2 ((𝜑𝐸 ∈ (1st𝐵)) → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
342simprd 107 . . . . . . 7 (𝜑𝑇Q)
35 mulcomnqg 6481 . . . . . . 7 ((𝑇Q𝑈Q) → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
3634, 6, 35syl2anc 391 . . . . . 6 (𝜑 → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
37 mullocprlem.tdudr . . . . . . 7 (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))
38 mulclnq 6474 . . . . . . . . . 10 ((𝑇Q𝑈Q) → (𝑇 ·Q 𝑈) ∈ Q)
3934, 6, 38syl2anc 391 . . . . . . . . 9 (𝜑 → (𝑇 ·Q 𝑈) ∈ Q)
4013simprd 107 . . . . . . . . 9 (𝜑𝑅Q)
41 ltmnqg 6499 . . . . . . . . 9 (((𝑇 ·Q 𝑈) ∈ Q𝑅Q𝐷Q) → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4239, 40, 5, 41syl3anc 1135 . . . . . . . 8 (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4334, 5, 6, 8, 10caov12d 5682 . . . . . . . . 9 (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) = (𝐷 ·Q (𝑇 ·Q 𝑈)))
4443breq1d 3774 . . . . . . . 8 (𝜑 → ((𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅) ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4542, 44bitr4d 180 . . . . . . 7 (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4637, 45mpbird 156 . . . . . 6 (𝜑 → (𝑇 ·Q 𝑈) <Q 𝑅)
4736, 46eqbrtrrd 3786 . . . . 5 (𝜑 → (𝑈 ·Q 𝑇) <Q 𝑅)
4847adantr 261 . . . 4 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝑈 ·Q 𝑇) <Q 𝑅)
4923simprd 107 . . . . . . 7 (𝜑𝑈 ∈ (2nd𝐴))
5022, 49jca 290 . . . . . 6 (𝜑 → (𝐴P𝑈 ∈ (2nd𝐴)))
5150adantr 261 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝐴P𝑈 ∈ (2nd𝐴)))
5227anim1i 323 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝐵P𝑇 ∈ (2nd𝐵)))
5340adantr 261 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → 𝑅Q)
54 mulnqpru 6667 . . . . 5 ((((𝐴P𝑈 ∈ (2nd𝐴)) ∧ (𝐵P𝑇 ∈ (2nd𝐵))) ∧ 𝑅Q) → ((𝑈 ·Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
5551, 52, 53, 54syl21anc 1134 . . . 4 ((𝜑𝑇 ∈ (2nd𝐵)) → ((𝑈 ·Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
5648, 55mpd 13 . . 3 ((𝜑𝑇 ∈ (2nd𝐵)) → 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵)))
5756olcd 653 . 2 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
58 mullocprlem.edutdu . . . 4 (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈)))
59 mulclnq 6474 . . . . . . 7 ((𝐷Q𝑈Q) → (𝐷 ·Q 𝑈) ∈ Q)
604, 59syl 14 . . . . . 6 (𝜑 → (𝐷 ·Q 𝑈) ∈ Q)
61 ltmnqg 6499 . . . . . 6 ((𝐸Q𝑇Q ∧ (𝐷 ·Q 𝑈) ∈ Q) → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
623, 34, 60, 61syl3anc 1135 . . . . 5 (𝜑 → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
63 mulcomnqg 6481 . . . . . . 7 (((𝐷 ·Q 𝑈) ∈ Q𝐸Q) → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
6460, 3, 63syl2anc 391 . . . . . 6 (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
65 mulcomnqg 6481 . . . . . . 7 (((𝐷 ·Q 𝑈) ∈ Q𝑇Q) → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
6660, 34, 65syl2anc 391 . . . . . 6 (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
6764, 66breq12d 3777 . . . . 5 (𝜑 → (((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇) ↔ (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈))))
6862, 67bitrd 177 . . . 4 (𝜑 → (𝐸 <Q 𝑇 ↔ (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈))))
6958, 68mpbird 156 . . 3 (𝜑𝐸 <Q 𝑇)
70 prop 6573 . . . 4 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
71 prloc 6589 . . . 4 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐸 <Q 𝑇) → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7270, 71sylan 267 . . 3 ((𝐵P𝐸 <Q 𝑇) → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7327, 69, 72syl2anc 391 . 2 (𝜑 → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7433, 57, 73mpjaodan 711 1 (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  wo 629  w3a 885   = wceq 1243  wcel 1393  cop 3378   class class class wbr 3764  cfv 4902  (class class class)co 5512  1st c1st 5765  2nd c2nd 5766  Qcnq 6378   ·Q cmq 6381   <Q cltq 6383  Pcnp 6389   ·P cmp 6392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-lti 6405  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-inp 6564  df-imp 6567
This theorem is referenced by:  mullocpr  6669
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