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Theorem addlocprlem 7355
Description: Lemma for addlocpr 7356. 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 7185 . . . . . 6 <Q ⊆ (Q × Q)
32brel 4591 . . . . 5 (𝑄 <Q 𝑅 → (𝑄Q𝑅Q))
43simpld 111 . . . 4 (𝑄 <Q 𝑅𝑄Q)
51, 4syl 14 . . 3 (𝜑𝑄Q)
6 addlocprlem.a . . . . . 6 (𝜑𝐴P)
7 prop 7295 . . . . . 6 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
86, 7syl 14 . . . . 5 (𝜑 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
9 addlocprlem.dlo . . . . 5 (𝜑𝐷 ∈ (1st𝐴))
10 elprnql 7301 . . . . 5 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝐷 ∈ (1st𝐴)) → 𝐷Q)
118, 9, 10syl2anc 408 . . . 4 (𝜑𝐷Q)
12 addlocprlem.b . . . . . 6 (𝜑𝐵P)
13 prop 7295 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
1412, 13syl 14 . . . . 5 (𝜑 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
15 addlocprlem.elo . . . . 5 (𝜑𝐸 ∈ (1st𝐵))
16 elprnql 7301 . . . . 5 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝐸 ∈ (1st𝐵)) → 𝐸Q)
1714, 15, 16syl2anc 408 . . . 4 (𝜑𝐸Q)
18 addclnq 7195 . . . 4 ((𝐷Q𝐸Q) → (𝐷 +Q 𝐸) ∈ Q)
1911, 17, 18syl2anc 408 . . 3 (𝜑 → (𝐷 +Q 𝐸) ∈ Q)
20 nqtri3or 7216 . . 3 ((𝑄Q ∧ (𝐷 +Q 𝐸) ∈ Q) → (𝑄 <Q (𝐷 +Q 𝐸) ∨ 𝑄 = (𝐷 +Q 𝐸) ∨ (𝐷 +Q 𝐸) <Q 𝑄))
215, 19, 20syl2anc 408 . 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 7351 . . . 4 (𝜑 → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 +P 𝐵))))
29 orc 701 . . . 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 7353 . . . 4 (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
32 olc 700 . . . 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 7354 . . . 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 1282 . 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 697  w3o 961   = wceq 1331  wcel 1480  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7100   +Q cplq 7102   <Q cltq 7105  Pcnp 7111   +P cpp 7113
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-inp 7286  df-iplp 7288
This theorem is referenced by:  addlocpr  7356
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