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Theorem addlocprlemgt 7365
Description: Lemma for addlocpr 7367. The (𝐷 +Q 𝐸) <Q 𝑄 case. (Contributed by Jim Kingdon, 6-Dec-2019.)
Hypotheses
Ref Expression
addlocprlem.a (𝜑𝐴P)
addlocprlem.b (𝜑𝐵P)
addlocprlem.qr (𝜑𝑄 <Q 𝑅)
addlocprlem.p (𝜑𝑃Q)
addlocprlem.qppr (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)
addlocprlem.dlo (𝜑𝐷 ∈ (1st𝐴))
addlocprlem.uup (𝜑𝑈 ∈ (2nd𝐴))
addlocprlem.du (𝜑𝑈 <Q (𝐷 +Q 𝑃))
addlocprlem.elo (𝜑𝐸 ∈ (1st𝐵))
addlocprlem.tup (𝜑𝑇 ∈ (2nd𝐵))
addlocprlem.et (𝜑𝑇 <Q (𝐸 +Q 𝑃))
Assertion
Ref Expression
addlocprlemgt (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

Proof of Theorem addlocprlemgt
StepHypRef Expression
1 addlocprlem.a . . . . . . 7 (𝜑𝐴P)
2 addlocprlem.b . . . . . . 7 (𝜑𝐵P)
3 addlocprlem.qr . . . . . . 7 (𝜑𝑄 <Q 𝑅)
4 addlocprlem.p . . . . . . 7 (𝜑𝑃Q)
5 addlocprlem.qppr . . . . . . 7 (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)
6 addlocprlem.dlo . . . . . . 7 (𝜑𝐷 ∈ (1st𝐴))
7 addlocprlem.uup . . . . . . 7 (𝜑𝑈 ∈ (2nd𝐴))
8 addlocprlem.du . . . . . . 7 (𝜑𝑈 <Q (𝐷 +Q 𝑃))
9 addlocprlem.elo . . . . . . 7 (𝜑𝐸 ∈ (1st𝐵))
10 addlocprlem.tup . . . . . . 7 (𝜑𝑇 ∈ (2nd𝐵))
11 addlocprlem.et . . . . . . 7 (𝜑𝑇 <Q (𝐸 +Q 𝑃))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11addlocprlemeqgt 7363 . . . . . 6 (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
1312adantr 274 . . . . 5 ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
14 prop 7306 . . . . . . . . . . . 12 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
151, 14syl 14 . . . . . . . . . . 11 (𝜑 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
16 elprnql 7312 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐷 ∈ (1st𝐴)) → 𝐷Q)
1715, 6, 16syl2anc 409 . . . . . . . . . 10 (𝜑𝐷Q)
18 prop 7306 . . . . . . . . . . . 12 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
192, 18syl 14 . . . . . . . . . . 11 (𝜑 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
20 elprnql 7312 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐸 ∈ (1st𝐵)) → 𝐸Q)
2119, 9, 20syl2anc 409 . . . . . . . . . 10 (𝜑𝐸Q)
22 addclnq 7206 . . . . . . . . . 10 ((𝐷Q𝐸Q) → (𝐷 +Q 𝐸) ∈ Q)
2317, 21, 22syl2anc 409 . . . . . . . . 9 (𝜑 → (𝐷 +Q 𝐸) ∈ Q)
24 ltrelnq 7196 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
2524brel 4598 . . . . . . . . . . 11 (𝑄 <Q 𝑅 → (𝑄Q𝑅Q))
263, 25syl 14 . . . . . . . . . 10 (𝜑 → (𝑄Q𝑅Q))
2726simpld 111 . . . . . . . . 9 (𝜑𝑄Q)
28 addclnq 7206 . . . . . . . . . 10 ((𝑃Q𝑃Q) → (𝑃 +Q 𝑃) ∈ Q)
294, 4, 28syl2anc 409 . . . . . . . . 9 (𝜑 → (𝑃 +Q 𝑃) ∈ Q)
30 ltanqg 7231 . . . . . . . . 9 (((𝐷 +Q 𝐸) ∈ Q𝑄Q ∧ (𝑃 +Q 𝑃) ∈ Q) → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) <Q ((𝑃 +Q 𝑃) +Q 𝑄)))
3123, 27, 29, 30syl3anc 1217 . . . . . . . 8 (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) <Q ((𝑃 +Q 𝑃) +Q 𝑄)))
32 addcomnqg 7212 . . . . . . . . . 10 (((𝑃 +Q 𝑃) ∈ Q ∧ (𝐷 +Q 𝐸) ∈ Q) → ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
3329, 23, 32syl2anc 409 . . . . . . . . 9 (𝜑 → ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
34 addcomnqg 7212 . . . . . . . . . 10 (((𝑃 +Q 𝑃) ∈ Q𝑄Q) → ((𝑃 +Q 𝑃) +Q 𝑄) = (𝑄 +Q (𝑃 +Q 𝑃)))
3529, 27, 34syl2anc 409 . . . . . . . . 9 (𝜑 → ((𝑃 +Q 𝑃) +Q 𝑄) = (𝑄 +Q (𝑃 +Q 𝑃)))
3633, 35breq12d 3949 . . . . . . . 8 (𝜑 → (((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) <Q ((𝑃 +Q 𝑃) +Q 𝑄) ↔ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃))))
3731, 36bitrd 187 . . . . . . 7 (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃))))
3837biimpa 294 . . . . . 6 ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃)))
395breq2d 3948 . . . . . . 7 (𝜑 → (((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃)) ↔ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅))
4039adantr 274 . . . . . 6 ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → (((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃)) ↔ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅))
4138, 40mpbid 146 . . . . 5 ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅)
4213, 41jca 304 . . . 4 ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) ∧ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅))
43 ltsonq 7229 . . . . 5 <Q Or Q
4443, 24sotri 4941 . . . 4 (((𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) ∧ ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q 𝑅) → (𝑈 +Q 𝑇) <Q 𝑅)
4542, 44syl 14 . . 3 ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → (𝑈 +Q 𝑇) <Q 𝑅)
461, 7jca 304 . . . . 5 (𝜑 → (𝐴P𝑈 ∈ (2nd𝐴)))
472, 10jca 304 . . . . 5 (𝜑 → (𝐵P𝑇 ∈ (2nd𝐵)))
4826simprd 113 . . . . 5 (𝜑𝑅Q)
49 addnqpru 7361 . . . . 5 ((((𝐴P𝑈 ∈ (2nd𝐴)) ∧ (𝐵P𝑇 ∈ (2nd𝐵))) ∧ 𝑅Q) → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
5046, 47, 48, 49syl21anc 1216 . . . 4 (𝜑 → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
5150adantr 274 . . 3 ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
5245, 51mpd 13 . 2 ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))
5352ex 114 1 (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  cop 3534   class class class wbr 3936  cfv 5130  (class class class)co 5781  1st c1st 6043  2nd c2nd 6044  Qcnq 7111   +Q cplq 7113   <Q cltq 7116  Pcnp 7122   +P cpp 7124
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-1o 6320  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-mpq 7176  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-mqqs 7181  df-1nqqs 7182  df-rq 7183  df-ltnqqs 7184  df-inp 7297  df-iplp 7299
This theorem is referenced by:  addlocprlem  7366
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