ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  addlocprlem Structured version   GIF version

Theorem addlocprlem 6511
Description: Lemma for addlocpr 6512. 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 6342 . . . . . 6 <Q ⊆ (Q × Q)
32brel 4334 . . . . 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 6450 . . . . . 6 (A P → ⟨(1stA), (2ndA)⟩ P)
86, 7syl 14 . . . . 5 (φ → ⟨(1stA), (2ndA)⟩ P)
9 addlocprlem.dlo . . . . 5 (φ𝐷 (1stA))
10 elprnql 6456 . . . . 5 ((⟨(1stA), (2ndA)⟩ P 𝐷 (1stA)) → 𝐷 Q)
118, 9, 10syl2anc 391 . . . 4 (φ𝐷 Q)
12 addlocprlem.b . . . . . 6 (φB P)
13 prop 6450 . . . . . 6 (B P → ⟨(1stB), (2ndB)⟩ P)
1412, 13syl 14 . . . . 5 (φ → ⟨(1stB), (2ndB)⟩ P)
15 addlocprlem.elo . . . . 5 (φ𝐸 (1stB))
16 elprnql 6456 . . . . 5 ((⟨(1stB), (2ndB)⟩ P 𝐸 (1stB)) → 𝐸 Q)
1714, 15, 16syl2anc 391 . . . 4 (φ𝐸 Q)
18 addclnq 6352 . . . 4 ((𝐷 Q 𝐸 Q) → (𝐷 +Q 𝐸) Q)
1911, 17, 18syl2anc 391 . . 3 (φ → (𝐷 +Q 𝐸) Q)
20 nqtri3or 6373 . . 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 6507 . . . 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 6509 . . . 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 6510 . . . 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 3369   class class class wbr 3754  cfv 4844  (class class class)co 5452  1st c1st 5704  2nd c2nd 5705  Qcnq 6257   +Q cplq 6259   <Q cltq 6262  Pcnp 6268   +P cpp 6270
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 3862  ax-sep 3865  ax-nul 3873  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219  ax-iinf 4253
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 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-int 3606  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-eprel 4016  df-id 4020  df-po 4023  df-iso 4024  df-iord 4068  df-on 4070  df-suc 4073  df-iom 4256  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-ov 5455  df-oprab 5456  df-mpt2 5457  df-1st 5706  df-2nd 5707  df-recs 5858  df-irdg 5894  df-1o 5933  df-oadd 5937  df-omul 5938  df-er 6035  df-ec 6037  df-qs 6041  df-ni 6281  df-pli 6282  df-mi 6283  df-lti 6284  df-plpq 6321  df-mpq 6322  df-enq 6324  df-nqqs 6325  df-plqqs 6326  df-mqqs 6327  df-1nqqs 6328  df-rq 6329  df-ltnqqs 6330  df-inp 6441  df-iplp 6443
This theorem is referenced by:  addlocpr  6512
  Copyright terms: Public domain W3C validator