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Theorem mullocprlem 7342
Description: Calculations for mullocpr 7343. (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 111 . . . . . . . 8 (𝜑𝐸Q)
4 mullocprlem.duq . . . . . . . . 9 (𝜑 → (𝐷Q𝑈Q))
54simpld 111 . . . . . . . 8 (𝜑𝐷Q)
64simprd 113 . . . . . . . 8 (𝜑𝑈Q)
7 mulcomnqg 7155 . . . . . . . . 9 ((𝑥Q𝑦Q) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
87adantl 273 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑦Q)) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
9 mulassnqg 7156 . . . . . . . . 9 ((𝑥Q𝑦Q𝑧Q) → ((𝑥 ·Q 𝑦) ·Q 𝑧) = (𝑥 ·Q (𝑦 ·Q 𝑧)))
109adantl 273 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑦Q𝑧Q)) → ((𝑥 ·Q 𝑦) ·Q 𝑧) = (𝑥 ·Q (𝑦 ·Q 𝑧)))
113, 5, 6, 8, 10caov13d 5920 . . . . . . 7 (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) = (𝑈 ·Q (𝐷 ·Q 𝐸)))
121, 11breqtrd 3922 . . . . . 6 (𝜑 → (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸)))
13 mullocprlem.qr . . . . . . . 8 (𝜑 → (𝑄Q𝑅Q))
1413simpld 111 . . . . . . 7 (𝜑𝑄Q)
15 mulclnq 7148 . . . . . . . 8 ((𝐷Q𝐸Q) → (𝐷 ·Q 𝐸) ∈ Q)
165, 3, 15syl2anc 406 . . . . . . 7 (𝜑 → (𝐷 ·Q 𝐸) ∈ Q)
17 ltmnqg 7173 . . . . . . 7 ((𝑄Q ∧ (𝐷 ·Q 𝐸) ∈ Q𝑈Q) → (𝑄 <Q (𝐷 ·Q 𝐸) ↔ (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸))))
1814, 16, 6, 17syl3anc 1199 . . . . . 6 (𝜑 → (𝑄 <Q (𝐷 ·Q 𝐸) ↔ (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸))))
1912, 18mpbird 166 . . . . 5 (𝜑𝑄 <Q (𝐷 ·Q 𝐸))
2019adantr 272 . . . 4 ((𝜑𝐸 ∈ (1st𝐵)) → 𝑄 <Q (𝐷 ·Q 𝐸))
21 mullocprlem.ab . . . . . . . 8 (𝜑 → (𝐴P𝐵P))
2221simpld 111 . . . . . . 7 (𝜑𝐴P)
23 mullocprlem.du . . . . . . . 8 (𝜑 → (𝐷 ∈ (1st𝐴) ∧ 𝑈 ∈ (2nd𝐴)))
2423simpld 111 . . . . . . 7 (𝜑𝐷 ∈ (1st𝐴))
2522, 24jca 302 . . . . . 6 (𝜑 → (𝐴P𝐷 ∈ (1st𝐴)))
2625adantr 272 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → (𝐴P𝐷 ∈ (1st𝐴)))
2721simprd 113 . . . . . 6 (𝜑𝐵P)
2827anim1i 336 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → (𝐵P𝐸 ∈ (1st𝐵)))
2914adantr 272 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → 𝑄Q)
30 mulnqprl 7340 . . . . 5 ((((𝐴P𝐷 ∈ (1st𝐴)) ∧ (𝐵P𝐸 ∈ (1st𝐵))) ∧ 𝑄Q) → (𝑄 <Q (𝐷 ·Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 ·P 𝐵))))
3126, 28, 29, 30syl21anc 1198 . . . 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 705 . 2 ((𝜑𝐸 ∈ (1st𝐵)) → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
342simprd 113 . . . . . . 7 (𝜑𝑇Q)
35 mulcomnqg 7155 . . . . . . 7 ((𝑇Q𝑈Q) → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
3634, 6, 35syl2anc 406 . . . . . 6 (𝜑 → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
37 mullocprlem.tdudr . . . . . . 7 (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))
38 mulclnq 7148 . . . . . . . . . 10 ((𝑇Q𝑈Q) → (𝑇 ·Q 𝑈) ∈ Q)
3934, 6, 38syl2anc 406 . . . . . . . . 9 (𝜑 → (𝑇 ·Q 𝑈) ∈ Q)
4013simprd 113 . . . . . . . . 9 (𝜑𝑅Q)
41 ltmnqg 7173 . . . . . . . . 9 (((𝑇 ·Q 𝑈) ∈ Q𝑅Q𝐷Q) → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4239, 40, 5, 41syl3anc 1199 . . . . . . . 8 (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4334, 5, 6, 8, 10caov12d 5918 . . . . . . . . 9 (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) = (𝐷 ·Q (𝑇 ·Q 𝑈)))
4443breq1d 3907 . . . . . . . 8 (𝜑 → ((𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅) ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4542, 44bitr4d 190 . . . . . . 7 (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4637, 45mpbird 166 . . . . . 6 (𝜑 → (𝑇 ·Q 𝑈) <Q 𝑅)
4736, 46eqbrtrrd 3920 . . . . 5 (𝜑 → (𝑈 ·Q 𝑇) <Q 𝑅)
4847adantr 272 . . . 4 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝑈 ·Q 𝑇) <Q 𝑅)
4923simprd 113 . . . . . . 7 (𝜑𝑈 ∈ (2nd𝐴))
5022, 49jca 302 . . . . . 6 (𝜑 → (𝐴P𝑈 ∈ (2nd𝐴)))
5150adantr 272 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝐴P𝑈 ∈ (2nd𝐴)))
5227anim1i 336 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝐵P𝑇 ∈ (2nd𝐵)))
5340adantr 272 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → 𝑅Q)
54 mulnqpru 7341 . . . . 5 ((((𝐴P𝑈 ∈ (2nd𝐴)) ∧ (𝐵P𝑇 ∈ (2nd𝐵))) ∧ 𝑅Q) → ((𝑈 ·Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
5551, 52, 53, 54syl21anc 1198 . . . 4 ((𝜑𝑇 ∈ (2nd𝐵)) → ((𝑈 ·Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
5648, 55mpd 13 . . 3 ((𝜑𝑇 ∈ (2nd𝐵)) → 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵)))
5756olcd 706 . 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 7148 . . . . . . 7 ((𝐷Q𝑈Q) → (𝐷 ·Q 𝑈) ∈ Q)
604, 59syl 14 . . . . . 6 (𝜑 → (𝐷 ·Q 𝑈) ∈ Q)
61 ltmnqg 7173 . . . . . 6 ((𝐸Q𝑇Q ∧ (𝐷 ·Q 𝑈) ∈ Q) → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
623, 34, 60, 61syl3anc 1199 . . . . 5 (𝜑 → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
63 mulcomnqg 7155 . . . . . . 7 (((𝐷 ·Q 𝑈) ∈ Q𝐸Q) → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
6460, 3, 63syl2anc 406 . . . . . 6 (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
65 mulcomnqg 7155 . . . . . . 7 (((𝐷 ·Q 𝑈) ∈ Q𝑇Q) → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
6660, 34, 65syl2anc 406 . . . . . 6 (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
6764, 66breq12d 3910 . . . . 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 187 . . . 4 (𝜑 → (𝐸 <Q 𝑇 ↔ (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈))))
6958, 68mpbird 166 . . 3 (𝜑𝐸 <Q 𝑇)
70 prop 7247 . . . 4 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
71 prloc 7263 . . . 4 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐸 <Q 𝑇) → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7270, 71sylan 279 . . 3 ((𝐵P𝐸 <Q 𝑇) → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7327, 69, 72syl2anc 406 . 2 (𝜑 → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7433, 57, 73mpjaodan 770 1 (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 680  w3a 945   = wceq 1314  wcel 1463  cop 3498   class class class wbr 3897  cfv 5091  (class class class)co 5740  1st c1st 6002  2nd c2nd 6003  Qcnq 7052   ·Q cmq 7055   <Q cltq 7057  Pcnp 7063   ·P cmp 7066
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-mi 7078  df-lti 7079  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-inp 7238  df-imp 7241
This theorem is referenced by:  mullocpr  7343
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