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Theorem addlocprlem 6384
Description: Lemma for addlocpr 6385. 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 6218 . . . . . 6 <Q ⊆ (Q × Q)
32brel 4315 . . . . 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 6323 . . . . . 6 (A P → ⟨(1stA), (2ndA)⟩ P)
86, 7syl 14 . . . . 5 (φ → ⟨(1stA), (2ndA)⟩ P)
9 addlocprlem.dlo . . . . 5 (φ𝐷 (1stA))
10 elprnql 6329 . . . . 5 ((⟨(1stA), (2ndA)⟩ P 𝐷 (1stA)) → 𝐷 Q)
118, 9, 10syl2anc 393 . . . 4 (φ𝐷 Q)
12 addlocprlem.b . . . . . 6 (φB P)
13 prop 6323 . . . . . 6 (B P → ⟨(1stB), (2ndB)⟩ P)
1412, 13syl 14 . . . . 5 (φ → ⟨(1stB), (2ndB)⟩ P)
15 addlocprlem.elo . . . . 5 (φ𝐸 (1stB))
16 elprnql 6329 . . . . 5 ((⟨(1stB), (2ndB)⟩ P 𝐸 (1stB)) → 𝐸 Q)
1714, 15, 16syl2anc 393 . . . 4 (φ𝐸 Q)
18 addclnq 6228 . . . 4 ((𝐷 Q 𝐸 Q) → (𝐷 +Q 𝐸) Q)
1911, 17, 18syl2anc 393 . . 3 (φ → (𝐷 +Q 𝐸) Q)
20 nqtri3or 6249 . . 3 ((𝑄 Q (𝐷 +Q 𝐸) Q) → (𝑄 <Q (𝐷 +Q 𝐸) 𝑄 = (𝐷 +Q 𝐸) (𝐷 +Q 𝐸) <Q 𝑄))
215, 19, 20syl2anc 393 . 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 6380 . . . 4 (φ → (𝑄 <Q (𝐷 +Q 𝐸) → 𝑄 (1st ‘(A +P B))))
29 orc 620 . . . 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 6382 . . . 4 (φ → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 (2nd ‘(A +P B))))
32 olc 619 . . . 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 6383 . . . 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 1183 . 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 616   w3o 870   = wceq 1226   wcel 1370  cop 3349   class class class wbr 3734  cfv 4825  (class class class)co 5432  1st c1st 5684  2nd c2nd 5685  Qcnq 6134   +Q cplq 6136   <Q cltq 6139  Pcnp 6145   +P cpp 6147
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-dc 731  df-3or 872  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-eprel 3996  df-id 4000  df-po 4003  df-iso 4004  df-iord 4048  df-on 4050  df-suc 4053  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687  df-recs 5838  df-irdg 5874  df-1o 5912  df-oadd 5916  df-omul 5917  df-er 6013  df-ec 6015  df-qs 6019  df-ni 6158  df-pli 6159  df-mi 6160  df-lti 6161  df-plpq 6197  df-mpq 6198  df-enq 6200  df-nqqs 6201  df-plqqs 6202  df-mqqs 6203  df-1nqqs 6204  df-rq 6205  df-ltnqqs 6206  df-inp 6314  df-iplp 6316
This theorem is referenced by:  addlocpr  6385
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