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Theorem addlocprlem 6661
Description: Lemma for addlocpr 6662. The result, in deduction form. (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
addlocprlem (𝜑 → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

Proof of Theorem addlocprlem
StepHypRef Expression
1 addlocprlem.qr . . . 4 (𝜑𝑄 <Q 𝑅)
2 ltrelnq 6491 . . . . . 6 <Q ⊆ (Q × Q)
32brel 4417 . . . . 5 (𝑄 <Q 𝑅 → (𝑄Q𝑅Q))
43simpld 109 . . . 4 (𝑄 <Q 𝑅𝑄Q)
51, 4syl 14 . . 3 (𝜑𝑄Q)
6 addlocprlem.a . . . . . 6 (𝜑𝐴P)
7 prop 6601 . . . . . 6 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
86, 7syl 14 . . . . 5 (𝜑 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
9 addlocprlem.dlo . . . . 5 (𝜑𝐷 ∈ (1st𝐴))
10 elprnql 6607 . . . . 5 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐷 ∈ (1st𝐴)) → 𝐷Q)
118, 9, 10syl2anc 397 . . . 4 (𝜑𝐷Q)
12 addlocprlem.b . . . . . 6 (𝜑𝐵P)
13 prop 6601 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
1412, 13syl 14 . . . . 5 (𝜑 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
15 addlocprlem.elo . . . . 5 (𝜑𝐸 ∈ (1st𝐵))
16 elprnql 6607 . . . . 5 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐸 ∈ (1st𝐵)) → 𝐸Q)
1714, 15, 16syl2anc 397 . . . 4 (𝜑𝐸Q)
18 addclnq 6501 . . . 4 ((𝐷Q𝐸Q) → (𝐷 +Q 𝐸) ∈ Q)
1911, 17, 18syl2anc 397 . . 3 (𝜑 → (𝐷 +Q 𝐸) ∈ Q)
20 nqtri3or 6522 . . 3 ((𝑄Q ∧ (𝐷 +Q 𝐸) ∈ Q) → (𝑄 <Q (𝐷 +Q 𝐸) ∨ 𝑄 = (𝐷 +Q 𝐸) ∨ (𝐷 +Q 𝐸) <Q 𝑄))
215, 19, 20syl2anc 397 . 2 (𝜑 → (𝑄 <Q (𝐷 +Q 𝐸) ∨ 𝑄 = (𝐷 +Q 𝐸) ∨ (𝐷 +Q 𝐸) <Q 𝑄))
22 addlocprlem.p . . . . 5 (𝜑𝑃Q)
23 addlocprlem.qppr . . . . 5 (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)
24 addlocprlem.uup . . . . 5 (𝜑𝑈 ∈ (2nd𝐴))
25 addlocprlem.du . . . . 5 (𝜑𝑈 <Q (𝐷 +Q 𝑃))
26 addlocprlem.tup . . . . 5 (𝜑𝑇 ∈ (2nd𝐵))
27 addlocprlem.et . . . . 5 (𝜑𝑇 <Q (𝐸 +Q 𝑃))
286, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27addlocprlemlt 6657 . . . 4 (𝜑 → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 +P 𝐵))))
29 orc 641 . . . 4 (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
3028, 29syl6 33 . . 3 (𝜑 → (𝑄 <Q (𝐷 +Q 𝐸) → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))))
316, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27addlocprlemeq 6659 . . . 4 (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
32 olc 640 . . . 4 (𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)) → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
3331, 32syl6 33 . . 3 (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))))
346, 12, 1, 22, 23, 9, 24, 25, 15, 26, 27addlocprlemgt 6660 . . . 4 (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
3534, 32syl6 33 . . 3 (𝜑 → ((𝐷 +Q 𝐸) <Q 𝑄 → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))))
3630, 33, 353jaod 1208 . 2 (𝜑 → ((𝑄 <Q (𝐷 +Q 𝐸) ∨ 𝑄 = (𝐷 +Q 𝐸) ∨ (𝐷 +Q 𝐸) <Q 𝑄) → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))))
3721, 36mpd 13 1 (𝜑 → (𝑄 ∈ (1st ‘(𝐴 +P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 637  w3o 893   = wceq 1257  wcel 1407  cop 3403   class class class wbr 3789  cfv 4927  (class class class)co 5537  1st c1st 5790  2nd c2nd 5791  Qcnq 6406   +Q cplq 6408   <Q cltq 6411  Pcnp 6417   +P cpp 6419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-dc 752  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-eprel 4051  df-id 4055  df-po 4058  df-iso 4059  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-1o 6029  df-oadd 6033  df-omul 6034  df-er 6134  df-ec 6136  df-qs 6140  df-ni 6430  df-pli 6431  df-mi 6432  df-lti 6433  df-plpq 6470  df-mpq 6471  df-enq 6473  df-nqqs 6474  df-plqqs 6475  df-mqqs 6476  df-1nqqs 6477  df-rq 6478  df-ltnqqs 6479  df-inp 6592  df-iplp 6594
This theorem is referenced by:  addlocpr  6662
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