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Theorem addlocprlemgt 7154
Description: Lemma for addlocpr 7156. 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 7152 . . . . . 6 (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
1312adantr 271 . . . . 5 ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
14 prop 7095 . . . . . . . . . . . 12 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
151, 14syl 14 . . . . . . . . . . 11 (𝜑 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
16 elprnql 7101 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐷 ∈ (1st𝐴)) → 𝐷Q)
1715, 6, 16syl2anc 404 . . . . . . . . . 10 (𝜑𝐷Q)
18 prop 7095 . . . . . . . . . . . 12 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
192, 18syl 14 . . . . . . . . . . 11 (𝜑 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
20 elprnql 7101 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐸 ∈ (1st𝐵)) → 𝐸Q)
2119, 9, 20syl2anc 404 . . . . . . . . . 10 (𝜑𝐸Q)
22 addclnq 6995 . . . . . . . . . 10 ((𝐷Q𝐸Q) → (𝐷 +Q 𝐸) ∈ Q)
2317, 21, 22syl2anc 404 . . . . . . . . 9 (𝜑 → (𝐷 +Q 𝐸) ∈ Q)
24 ltrelnq 6985 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
2524brel 4503 . . . . . . . . . . 11 (𝑄 <Q 𝑅 → (𝑄Q𝑅Q))
263, 25syl 14 . . . . . . . . . 10 (𝜑 → (𝑄Q𝑅Q))
2726simpld 111 . . . . . . . . 9 (𝜑𝑄Q)
28 addclnq 6995 . . . . . . . . . 10 ((𝑃Q𝑃Q) → (𝑃 +Q 𝑃) ∈ Q)
294, 4, 28syl2anc 404 . . . . . . . . 9 (𝜑 → (𝑃 +Q 𝑃) ∈ Q)
30 ltanqg 7020 . . . . . . . . 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 1175 . . . . . . . 8 (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 ↔ ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) <Q ((𝑃 +Q 𝑃) +Q 𝑄)))
32 addcomnqg 7001 . . . . . . . . . 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 404 . . . . . . . . 9 (𝜑 → ((𝑃 +Q 𝑃) +Q (𝐷 +Q 𝐸)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
34 addcomnqg 7001 . . . . . . . . . 10 (((𝑃 +Q 𝑃) ∈ Q𝑄Q) → ((𝑃 +Q 𝑃) +Q 𝑄) = (𝑄 +Q (𝑃 +Q 𝑃)))
3529, 27, 34syl2anc 404 . . . . . . . . 9 (𝜑 → ((𝑃 +Q 𝑃) +Q 𝑄) = (𝑄 +Q (𝑃 +Q 𝑃)))
3633, 35breq12d 3864 . . . . . . . 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 291 . . . . . 6 ((𝜑 ∧ (𝐷 +Q 𝐸) <Q 𝑄) → ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)) <Q (𝑄 +Q (𝑃 +Q 𝑃)))
395breq2d 3863 . . . . . . 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 271 . . . . . 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 301 . . . 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 7018 . . . . 5 <Q Or Q
4443, 24sotri 4840 . . . 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 301 . . . . 5 (𝜑 → (𝐴P𝑈 ∈ (2nd𝐴)))
472, 10jca 301 . . . . 5 (𝜑 → (𝐵P𝑇 ∈ (2nd𝐵)))
4826simprd 113 . . . . 5 (𝜑𝑅Q)
49 addnqpru 7150 . . . . 5 ((((𝐴P𝑈 ∈ (2nd𝐴)) ∧ (𝐵P𝑇 ∈ (2nd𝐵))) ∧ 𝑅Q) → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
5046, 47, 48, 49syl21anc 1174 . . . 4 (𝜑 → ((𝑈 +Q 𝑇) <Q 𝑅𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
5150adantr 271 . . 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 1290  wcel 1439  cop 3453   class class class wbr 3851  cfv 5028  (class class class)co 5666  1st c1st 5923  2nd c2nd 5924  Qcnq 6900   +Q cplq 6902   <Q cltq 6905  Pcnp 6911   +P cpp 6913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-pli 6925  df-mi 6926  df-lti 6927  df-plpq 6964  df-mpq 6965  df-enq 6967  df-nqqs 6968  df-plqqs 6969  df-mqqs 6970  df-1nqqs 6971  df-rq 6972  df-ltnqqs 6973  df-inp 7086  df-iplp 7088
This theorem is referenced by:  addlocprlem  7155
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