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Theorem addlocprlem 6518
Description: Lemma for addlocpr 6519. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.)
Hypotheses
Ref Expression
addlocprlem.a (φA P)
addlocprlem.b (φB P)
addlocprlem.qr (φ𝑄 <Q 𝑅)
addlocprlem.p (φ𝑃 Q)
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 6349 . . . . . 6 <Q ⊆ (Q × Q)
32brel 4335 . . . . 5 (𝑄 <Q 𝑅 → (𝑄 Q 𝑅 Q))
43simpld 105 . . . 4 (𝑄 <Q 𝑅𝑄 Q)
51, 4syl 14 . . 3 (φ𝑄 Q)
6 addlocprlem.a . . . . . 6 (φA P)
7 prop 6457 . . . . . 6 (A P → ⟨(1stA), (2ndA)⟩ P)
86, 7syl 14 . . . . 5 (φ → ⟨(1stA), (2ndA)⟩ P)
9 addlocprlem.dlo . . . . 5 (φ𝐷 (1stA))
10 elprnql 6463 . . . . 5 ((⟨(1stA), (2ndA)⟩ P 𝐷 (1stA)) → 𝐷 Q)
118, 9, 10syl2anc 391 . . . 4 (φ𝐷 Q)
12 addlocprlem.b . . . . . 6 (φB P)
13 prop 6457 . . . . . 6 (B P → ⟨(1stB), (2ndB)⟩ P)
1412, 13syl 14 . . . . 5 (φ → ⟨(1stB), (2ndB)⟩ P)
15 addlocprlem.elo . . . . 5 (φ𝐸 (1stB))
16 elprnql 6463 . . . . 5 ((⟨(1stB), (2ndB)⟩ P 𝐸 (1stB)) → 𝐸 Q)
1714, 15, 16syl2anc 391 . . . 4 (φ𝐸 Q)
18 addclnq 6359 . . . 4 ((𝐷 Q 𝐸 Q) → (𝐷 +Q 𝐸) Q)
1911, 17, 18syl2anc 391 . . 3 (φ → (𝐷 +Q 𝐸) Q)
20 nqtri3or 6380 . . 3 ((𝑄 Q (𝐷 +Q 𝐸) Q) → (𝑄 <Q (𝐷 +Q 𝐸) 𝑄 = (𝐷 +Q 𝐸) (𝐷 +Q 𝐸) <Q 𝑄))
215, 19, 20syl2anc 391 . 2 (φ → (𝑄 <Q (𝐷 +Q 𝐸) 𝑄 = (𝐷 +Q 𝐸) (𝐷 +Q 𝐸) <Q 𝑄))
22 addlocprlem.p . . . . 5 (φ𝑃 Q)
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 6514 . . . 4 (φ → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 (1st ‘(A +P B))))
29 orc 632 . . . 4 (𝑄 (1st ‘(A +P B)) → (𝑄 (1st ‘(A +P B)) 𝑅 (2nd ‘(A +P B))))
3028, 29syl6 29 . . 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 6516 . . . 4 (φ → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 (2nd ‘(A +P B))))
32 olc 631 . . . 4 (𝑅 (2nd ‘(A +P B)) → (𝑄 (1st ‘(A +P B)) 𝑅 (2nd ‘(A +P B))))
3331, 32syl6 29 . . 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 6517 . . . 4 (φ → ((𝐷 +Q 𝐸) <Q 𝑄𝑅 (2nd ‘(A +P B))))
3534, 32syl6 29 . . 3 (φ → ((𝐷 +Q 𝐸) <Q 𝑄 → (𝑄 (1st ‘(A +P B)) 𝑅 (2nd ‘(A +P B)))))
3630, 33, 353jaod 1198 . 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 628   w3o 883   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   +Q cplq 6266   <Q cltq 6269  Pcnp 6275   +P cpp 6277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6448  df-iplp 6450
This theorem is referenced by:  addlocpr  6519
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