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Theorem mullocprlem 7032
Description: Calculations for mullocpr 7033. (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 110 . . . . . . . 8 (𝜑𝐸Q)
4 mullocprlem.duq . . . . . . . . 9 (𝜑 → (𝐷Q𝑈Q))
54simpld 110 . . . . . . . 8 (𝜑𝐷Q)
64simprd 112 . . . . . . . 8 (𝜑𝑈Q)
7 mulcomnqg 6845 . . . . . . . . 9 ((𝑥Q𝑦Q) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
87adantl 271 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑦Q)) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
9 mulassnqg 6846 . . . . . . . . 9 ((𝑥Q𝑦Q𝑧Q) → ((𝑥 ·Q 𝑦) ·Q 𝑧) = (𝑥 ·Q (𝑦 ·Q 𝑧)))
109adantl 271 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑦Q𝑧Q)) → ((𝑥 ·Q 𝑦) ·Q 𝑧) = (𝑥 ·Q (𝑦 ·Q 𝑧)))
113, 5, 6, 8, 10caov13d 5763 . . . . . . 7 (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) = (𝑈 ·Q (𝐷 ·Q 𝐸)))
121, 11breqtrd 3835 . . . . . 6 (𝜑 → (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸)))
13 mullocprlem.qr . . . . . . . 8 (𝜑 → (𝑄Q𝑅Q))
1413simpld 110 . . . . . . 7 (𝜑𝑄Q)
15 mulclnq 6838 . . . . . . . 8 ((𝐷Q𝐸Q) → (𝐷 ·Q 𝐸) ∈ Q)
165, 3, 15syl2anc 403 . . . . . . 7 (𝜑 → (𝐷 ·Q 𝐸) ∈ Q)
17 ltmnqg 6863 . . . . . . 7 ((𝑄Q ∧ (𝐷 ·Q 𝐸) ∈ Q𝑈Q) → (𝑄 <Q (𝐷 ·Q 𝐸) ↔ (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸))))
1814, 16, 6, 17syl3anc 1170 . . . . . 6 (𝜑 → (𝑄 <Q (𝐷 ·Q 𝐸) ↔ (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸))))
1912, 18mpbird 165 . . . . 5 (𝜑𝑄 <Q (𝐷 ·Q 𝐸))
2019adantr 270 . . . 4 ((𝜑𝐸 ∈ (1st𝐵)) → 𝑄 <Q (𝐷 ·Q 𝐸))
21 mullocprlem.ab . . . . . . . 8 (𝜑 → (𝐴P𝐵P))
2221simpld 110 . . . . . . 7 (𝜑𝐴P)
23 mullocprlem.du . . . . . . . 8 (𝜑 → (𝐷 ∈ (1st𝐴) ∧ 𝑈 ∈ (2nd𝐴)))
2423simpld 110 . . . . . . 7 (𝜑𝐷 ∈ (1st𝐴))
2522, 24jca 300 . . . . . 6 (𝜑 → (𝐴P𝐷 ∈ (1st𝐴)))
2625adantr 270 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → (𝐴P𝐷 ∈ (1st𝐴)))
2721simprd 112 . . . . . 6 (𝜑𝐵P)
2827anim1i 333 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → (𝐵P𝐸 ∈ (1st𝐵)))
2914adantr 270 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → 𝑄Q)
30 mulnqprl 7030 . . . . 5 ((((𝐴P𝐷 ∈ (1st𝐴)) ∧ (𝐵P𝐸 ∈ (1st𝐵))) ∧ 𝑄Q) → (𝑄 <Q (𝐷 ·Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 ·P 𝐵))))
3126, 28, 29, 30syl21anc 1169 . . . 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 685 . 2 ((𝜑𝐸 ∈ (1st𝐵)) → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
342simprd 112 . . . . . . 7 (𝜑𝑇Q)
35 mulcomnqg 6845 . . . . . . 7 ((𝑇Q𝑈Q) → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
3634, 6, 35syl2anc 403 . . . . . 6 (𝜑 → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
37 mullocprlem.tdudr . . . . . . 7 (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))
38 mulclnq 6838 . . . . . . . . . 10 ((𝑇Q𝑈Q) → (𝑇 ·Q 𝑈) ∈ Q)
3934, 6, 38syl2anc 403 . . . . . . . . 9 (𝜑 → (𝑇 ·Q 𝑈) ∈ Q)
4013simprd 112 . . . . . . . . 9 (𝜑𝑅Q)
41 ltmnqg 6863 . . . . . . . . 9 (((𝑇 ·Q 𝑈) ∈ Q𝑅Q𝐷Q) → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4239, 40, 5, 41syl3anc 1170 . . . . . . . 8 (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4334, 5, 6, 8, 10caov12d 5761 . . . . . . . . 9 (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) = (𝐷 ·Q (𝑇 ·Q 𝑈)))
4443breq1d 3821 . . . . . . . 8 (𝜑 → ((𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅) ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4542, 44bitr4d 189 . . . . . . 7 (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4637, 45mpbird 165 . . . . . 6 (𝜑 → (𝑇 ·Q 𝑈) <Q 𝑅)
4736, 46eqbrtrrd 3833 . . . . 5 (𝜑 → (𝑈 ·Q 𝑇) <Q 𝑅)
4847adantr 270 . . . 4 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝑈 ·Q 𝑇) <Q 𝑅)
4923simprd 112 . . . . . . 7 (𝜑𝑈 ∈ (2nd𝐴))
5022, 49jca 300 . . . . . 6 (𝜑 → (𝐴P𝑈 ∈ (2nd𝐴)))
5150adantr 270 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝐴P𝑈 ∈ (2nd𝐴)))
5227anim1i 333 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝐵P𝑇 ∈ (2nd𝐵)))
5340adantr 270 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → 𝑅Q)
54 mulnqpru 7031 . . . . 5 ((((𝐴P𝑈 ∈ (2nd𝐴)) ∧ (𝐵P𝑇 ∈ (2nd𝐵))) ∧ 𝑅Q) → ((𝑈 ·Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
5551, 52, 53, 54syl21anc 1169 . . . 4 ((𝜑𝑇 ∈ (2nd𝐵)) → ((𝑈 ·Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
5648, 55mpd 13 . . 3 ((𝜑𝑇 ∈ (2nd𝐵)) → 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵)))
5756olcd 686 . 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 6838 . . . . . . 7 ((𝐷Q𝑈Q) → (𝐷 ·Q 𝑈) ∈ Q)
604, 59syl 14 . . . . . 6 (𝜑 → (𝐷 ·Q 𝑈) ∈ Q)
61 ltmnqg 6863 . . . . . 6 ((𝐸Q𝑇Q ∧ (𝐷 ·Q 𝑈) ∈ Q) → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
623, 34, 60, 61syl3anc 1170 . . . . 5 (𝜑 → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
63 mulcomnqg 6845 . . . . . . 7 (((𝐷 ·Q 𝑈) ∈ Q𝐸Q) → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
6460, 3, 63syl2anc 403 . . . . . 6 (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
65 mulcomnqg 6845 . . . . . . 7 (((𝐷 ·Q 𝑈) ∈ Q𝑇Q) → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
6660, 34, 65syl2anc 403 . . . . . 6 (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
6764, 66breq12d 3824 . . . . 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 186 . . . 4 (𝜑 → (𝐸 <Q 𝑇 ↔ (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈))))
6958, 68mpbird 165 . . 3 (𝜑𝐸 <Q 𝑇)
70 prop 6937 . . . 4 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
71 prloc 6953 . . . 4 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐸 <Q 𝑇) → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7270, 71sylan 277 . . 3 ((𝐵P𝐸 <Q 𝑇) → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7327, 69, 72syl2anc 403 . 2 (𝜑 → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7433, 57, 73mpjaodan 745 1 (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  w3a 920   = wceq 1285  wcel 1434  cop 3425   class class class wbr 3811  cfv 4969  (class class class)co 5591  1st c1st 5844  2nd c2nd 5845  Qcnq 6742   ·Q cmq 6745   <Q cltq 6747  Pcnp 6753   ·P cmp 6756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-mi 6768  df-lti 6769  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-inp 6928  df-imp 6931
This theorem is referenced by:  mullocpr  7033
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