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Theorem mullocprlem 7560
Description: Calculations for mullocpr 7561. (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 112 . . . . . . . 8 (𝜑𝐸Q)
4 mullocprlem.duq . . . . . . . . 9 (𝜑 → (𝐷Q𝑈Q))
54simpld 112 . . . . . . . 8 (𝜑𝐷Q)
64simprd 114 . . . . . . . 8 (𝜑𝑈Q)
7 mulcomnqg 7373 . . . . . . . . 9 ((𝑥Q𝑦Q) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
87adantl 277 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑦Q)) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
9 mulassnqg 7374 . . . . . . . . 9 ((𝑥Q𝑦Q𝑧Q) → ((𝑥 ·Q 𝑦) ·Q 𝑧) = (𝑥 ·Q (𝑦 ·Q 𝑧)))
109adantl 277 . . . . . . . 8 ((𝜑 ∧ (𝑥Q𝑦Q𝑧Q)) → ((𝑥 ·Q 𝑦) ·Q 𝑧) = (𝑥 ·Q (𝑦 ·Q 𝑧)))
113, 5, 6, 8, 10caov13d 6052 . . . . . . 7 (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) = (𝑈 ·Q (𝐷 ·Q 𝐸)))
121, 11breqtrd 4026 . . . . . 6 (𝜑 → (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸)))
13 mullocprlem.qr . . . . . . . 8 (𝜑 → (𝑄Q𝑅Q))
1413simpld 112 . . . . . . 7 (𝜑𝑄Q)
15 mulclnq 7366 . . . . . . . 8 ((𝐷Q𝐸Q) → (𝐷 ·Q 𝐸) ∈ Q)
165, 3, 15syl2anc 411 . . . . . . 7 (𝜑 → (𝐷 ·Q 𝐸) ∈ Q)
17 ltmnqg 7391 . . . . . . 7 ((𝑄Q ∧ (𝐷 ·Q 𝐸) ∈ Q𝑈Q) → (𝑄 <Q (𝐷 ·Q 𝐸) ↔ (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸))))
1814, 16, 6, 17syl3anc 1238 . . . . . 6 (𝜑 → (𝑄 <Q (𝐷 ·Q 𝐸) ↔ (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸))))
1912, 18mpbird 167 . . . . 5 (𝜑𝑄 <Q (𝐷 ·Q 𝐸))
2019adantr 276 . . . 4 ((𝜑𝐸 ∈ (1st𝐵)) → 𝑄 <Q (𝐷 ·Q 𝐸))
21 mullocprlem.ab . . . . . . . 8 (𝜑 → (𝐴P𝐵P))
2221simpld 112 . . . . . . 7 (𝜑𝐴P)
23 mullocprlem.du . . . . . . . 8 (𝜑 → (𝐷 ∈ (1st𝐴) ∧ 𝑈 ∈ (2nd𝐴)))
2423simpld 112 . . . . . . 7 (𝜑𝐷 ∈ (1st𝐴))
2522, 24jca 306 . . . . . 6 (𝜑 → (𝐴P𝐷 ∈ (1st𝐴)))
2625adantr 276 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → (𝐴P𝐷 ∈ (1st𝐴)))
2721simprd 114 . . . . . 6 (𝜑𝐵P)
2827anim1i 340 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → (𝐵P𝐸 ∈ (1st𝐵)))
2914adantr 276 . . . . 5 ((𝜑𝐸 ∈ (1st𝐵)) → 𝑄Q)
30 mulnqprl 7558 . . . . 5 ((((𝐴P𝐷 ∈ (1st𝐴)) ∧ (𝐵P𝐸 ∈ (1st𝐵))) ∧ 𝑄Q) → (𝑄 <Q (𝐷 ·Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 ·P 𝐵))))
3126, 28, 29, 30syl21anc 1237 . . . 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 733 . 2 ((𝜑𝐸 ∈ (1st𝐵)) → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
342simprd 114 . . . . . . 7 (𝜑𝑇Q)
35 mulcomnqg 7373 . . . . . . 7 ((𝑇Q𝑈Q) → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
3634, 6, 35syl2anc 411 . . . . . 6 (𝜑 → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
37 mullocprlem.tdudr . . . . . . 7 (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))
38 mulclnq 7366 . . . . . . . . . 10 ((𝑇Q𝑈Q) → (𝑇 ·Q 𝑈) ∈ Q)
3934, 6, 38syl2anc 411 . . . . . . . . 9 (𝜑 → (𝑇 ·Q 𝑈) ∈ Q)
4013simprd 114 . . . . . . . . 9 (𝜑𝑅Q)
41 ltmnqg 7391 . . . . . . . . 9 (((𝑇 ·Q 𝑈) ∈ Q𝑅Q𝐷Q) → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4239, 40, 5, 41syl3anc 1238 . . . . . . . 8 (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4334, 5, 6, 8, 10caov12d 6050 . . . . . . . . 9 (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) = (𝐷 ·Q (𝑇 ·Q 𝑈)))
4443breq1d 4010 . . . . . . . 8 (𝜑 → ((𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅) ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4542, 44bitr4d 191 . . . . . . 7 (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
4637, 45mpbird 167 . . . . . 6 (𝜑 → (𝑇 ·Q 𝑈) <Q 𝑅)
4736, 46eqbrtrrd 4024 . . . . 5 (𝜑 → (𝑈 ·Q 𝑇) <Q 𝑅)
4847adantr 276 . . . 4 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝑈 ·Q 𝑇) <Q 𝑅)
4923simprd 114 . . . . . . 7 (𝜑𝑈 ∈ (2nd𝐴))
5022, 49jca 306 . . . . . 6 (𝜑 → (𝐴P𝑈 ∈ (2nd𝐴)))
5150adantr 276 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝐴P𝑈 ∈ (2nd𝐴)))
5227anim1i 340 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → (𝐵P𝑇 ∈ (2nd𝐵)))
5340adantr 276 . . . . 5 ((𝜑𝑇 ∈ (2nd𝐵)) → 𝑅Q)
54 mulnqpru 7559 . . . . 5 ((((𝐴P𝑈 ∈ (2nd𝐴)) ∧ (𝐵P𝑇 ∈ (2nd𝐵))) ∧ 𝑅Q) → ((𝑈 ·Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
5551, 52, 53, 54syl21anc 1237 . . . 4 ((𝜑𝑇 ∈ (2nd𝐵)) → ((𝑈 ·Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
5648, 55mpd 13 . . 3 ((𝜑𝑇 ∈ (2nd𝐵)) → 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵)))
5756olcd 734 . 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 7366 . . . . . . 7 ((𝐷Q𝑈Q) → (𝐷 ·Q 𝑈) ∈ Q)
604, 59syl 14 . . . . . 6 (𝜑 → (𝐷 ·Q 𝑈) ∈ Q)
61 ltmnqg 7391 . . . . . 6 ((𝐸Q𝑇Q ∧ (𝐷 ·Q 𝑈) ∈ Q) → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
623, 34, 60, 61syl3anc 1238 . . . . 5 (𝜑 → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
63 mulcomnqg 7373 . . . . . . 7 (((𝐷 ·Q 𝑈) ∈ Q𝐸Q) → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
6460, 3, 63syl2anc 411 . . . . . 6 (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
65 mulcomnqg 7373 . . . . . . 7 (((𝐷 ·Q 𝑈) ∈ Q𝑇Q) → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
6660, 34, 65syl2anc 411 . . . . . 6 (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
6764, 66breq12d 4013 . . . . 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 188 . . . 4 (𝜑 → (𝐸 <Q 𝑇 ↔ (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈))))
6958, 68mpbird 167 . . 3 (𝜑𝐸 <Q 𝑇)
70 prop 7465 . . . 4 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
71 prloc 7481 . . . 4 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐸 <Q 𝑇) → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7270, 71sylan 283 . . 3 ((𝐵P𝐸 <Q 𝑇) → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7327, 69, 72syl2anc 411 . 2 (𝜑 → (𝐸 ∈ (1st𝐵) ∨ 𝑇 ∈ (2nd𝐵)))
7433, 57, 73mpjaodan 798 1 (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708  w3a 978   = wceq 1353  wcel 2148  cop 3594   class class class wbr 4000  cfv 5212  (class class class)co 5869  1st c1st 6133  2nd c2nd 6134  Qcnq 7270   ·Q cmq 7273   <Q cltq 7275  Pcnp 7281   ·P cmp 7284
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-mi 7296  df-lti 7297  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-inp 7456  df-imp 7459
This theorem is referenced by:  mullocpr  7561
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