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Theorem addnqpr1lemrl 6537
Description: Lemma for addnqpr1 6541. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 28-Apr-2020.)
Assertion
Ref Expression
addnqpr1lemrl (A Q → (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩))
Distinct variable group:   A,𝑙,u

Proof of Theorem addnqpr1lemrl
Dummy variables f g 𝑟 𝑠 𝑡 x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqprlu 6530 . . . . . 6 (A Q → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ P)
2 1pr 6534 . . . . . 6 1P P
3 df-iplp 6450 . . . . . . 7 +P = (x P, y P ↦ ⟨{f Qg Q Q (g (1stx) (1sty) f = (g +Q ))}, {f Qg Q Q (g (2ndx) (2ndy) f = (g +Q ))}⟩)
4 addclnq 6359 . . . . . . 7 ((g Q Q) → (g +Q ) Q)
53, 4genpelvl 6494 . . . . . 6 ((⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ P 1P P) → (𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)) ↔ 𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩)𝑡 (1st ‘1P)𝑟 = (𝑠 +Q 𝑡)))
61, 2, 5sylancl 392 . . . . 5 (A Q → (𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)) ↔ 𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩)𝑡 (1st ‘1P)𝑟 = (𝑠 +Q 𝑡)))
76biimpa 280 . . . 4 ((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) → 𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩)𝑡 (1st ‘1P)𝑟 = (𝑠 +Q 𝑡))
8 vex 2554 . . . . . . . . . . . . 13 𝑠 V
9 breq1 3758 . . . . . . . . . . . . 13 (𝑙 = 𝑠 → (𝑙 <Q A𝑠 <Q A))
10 ltnqex 6531 . . . . . . . . . . . . . 14 {𝑙𝑙 <Q A} V
11 gtnqex 6532 . . . . . . . . . . . . . 14 {uA <Q u} V
1210, 11op1st 5715 . . . . . . . . . . . . 13 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) = {𝑙𝑙 <Q A}
138, 9, 12elab2 2684 . . . . . . . . . . . 12 (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) ↔ 𝑠 <Q A)
1413biimpi 113 . . . . . . . . . . 11 (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) → 𝑠 <Q A)
1514ad2antrl 459 . . . . . . . . . 10 (((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) 𝑡 (1st ‘1P))) → 𝑠 <Q A)
1615adantr 261 . . . . . . . . 9 ((((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) 𝑡 (1st ‘1P))) 𝑟 = (𝑠 +Q 𝑡)) → 𝑠 <Q A)
17 vex 2554 . . . . . . . . . . . . 13 𝑡 V
18 breq1 3758 . . . . . . . . . . . . 13 (𝑙 = 𝑡 → (𝑙 <Q 1Q𝑡 <Q 1Q))
19 1prl 6535 . . . . . . . . . . . . 13 (1st ‘1P) = {𝑙𝑙 <Q 1Q}
2017, 18, 19elab2 2684 . . . . . . . . . . . 12 (𝑡 (1st ‘1P) ↔ 𝑡 <Q 1Q)
2120biimpi 113 . . . . . . . . . . 11 (𝑡 (1st ‘1P) → 𝑡 <Q 1Q)
2221ad2antll 460 . . . . . . . . . 10 (((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) 𝑡 (1st ‘1P))) → 𝑡 <Q 1Q)
2322adantr 261 . . . . . . . . 9 ((((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) 𝑡 (1st ‘1P))) 𝑟 = (𝑠 +Q 𝑡)) → 𝑡 <Q 1Q)
24 ltrelnq 6349 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
2524brel 4335 . . . . . . . . . . 11 (𝑠 <Q A → (𝑠 Q A Q))
2616, 25syl 14 . . . . . . . . . 10 ((((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) 𝑡 (1st ‘1P))) 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 Q A Q))
2724brel 4335 . . . . . . . . . . 11 (𝑡 <Q 1Q → (𝑡 Q 1Q Q))
2823, 27syl 14 . . . . . . . . . 10 ((((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) 𝑡 (1st ‘1P))) 𝑟 = (𝑠 +Q 𝑡)) → (𝑡 Q 1Q Q))
29 lt2addnq 6388 . . . . . . . . . 10 (((𝑠 Q A Q) (𝑡 Q 1Q Q)) → ((𝑠 <Q A 𝑡 <Q 1Q) → (𝑠 +Q 𝑡) <Q (A +Q 1Q)))
3026, 28, 29syl2anc 391 . . . . . . . . 9 ((((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) 𝑡 (1st ‘1P))) 𝑟 = (𝑠 +Q 𝑡)) → ((𝑠 <Q A 𝑡 <Q 1Q) → (𝑠 +Q 𝑡) <Q (A +Q 1Q)))
3116, 23, 30mp2and 409 . . . . . . . 8 ((((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) 𝑡 (1st ‘1P))) 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 +Q 𝑡) <Q (A +Q 1Q))
32 breq1 3758 . . . . . . . . 9 (𝑟 = (𝑠 +Q 𝑡) → (𝑟 <Q (A +Q 1Q) ↔ (𝑠 +Q 𝑡) <Q (A +Q 1Q)))
3332adantl 262 . . . . . . . 8 ((((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) 𝑡 (1st ‘1P))) 𝑟 = (𝑠 +Q 𝑡)) → (𝑟 <Q (A +Q 1Q) ↔ (𝑠 +Q 𝑡) <Q (A +Q 1Q)))
3431, 33mpbird 156 . . . . . . 7 ((((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) 𝑡 (1st ‘1P))) 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 <Q (A +Q 1Q))
35 vex 2554 . . . . . . . 8 𝑟 V
36 breq1 3758 . . . . . . . 8 (𝑙 = 𝑟 → (𝑙 <Q (A +Q 1Q) ↔ 𝑟 <Q (A +Q 1Q)))
37 ltnqex 6531 . . . . . . . . 9 {𝑙𝑙 <Q (A +Q 1Q)} V
38 gtnqex 6532 . . . . . . . . 9 {u ∣ (A +Q 1Q) <Q u} V
3937, 38op1st 5715 . . . . . . . 8 (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩) = {𝑙𝑙 <Q (A +Q 1Q)}
4035, 36, 39elab2 2684 . . . . . . 7 (𝑟 (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩) ↔ 𝑟 <Q (A +Q 1Q))
4134, 40sylibr 137 . . . . . 6 ((((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) 𝑡 (1st ‘1P))) 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩))
4241ex 108 . . . . 5 (((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩) 𝑡 (1st ‘1P))) → (𝑟 = (𝑠 +Q 𝑡) → 𝑟 (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩)))
4342rexlimdvva 2434 . . . 4 ((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) → (𝑠 (1st ‘⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩)𝑡 (1st ‘1P)𝑟 = (𝑠 +Q 𝑡) → 𝑟 (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩)))
447, 43mpd 13 . . 3 ((A Q 𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))) → 𝑟 (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩))
4544ex 108 . 2 (A Q → (𝑟 (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)) → 𝑟 (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩)))
4645ssrdv 2945 1 (A Q → (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  {cab 2023  wrex 2301  wss 2911  cop 3370   class class class wbr 3755  cfv 4845  (class class class)co 5455  1st c1st 5707  Qcnq 6264  1Qc1q 6265   +Q cplq 6266   <Q cltq 6269  Pcnp 6275  1Pc1p 6276   +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-i1p 6449  df-iplp 6450
This theorem is referenced by:  addnqpr1lemiu  6540  addnqpr1  6541
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