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Theorem addnqpr1lemiu 6540
Description: Lemma for addnqpr1 6541. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 28-Apr-2020.)
Assertion
Ref Expression
addnqpr1lemiu (A Q → (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩) ⊆ (2nd ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)))
Distinct variable group:   A,𝑙,u

Proof of Theorem addnqpr1lemiu
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 addnqpr1lemrl 6537 . . . . . 6 (A Q → (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩))
2 ltsonq 6382 . . . . . . . . 9 <Q Or Q
3 1nq 6350 . . . . . . . . . 10 1Q Q
4 addclnq 6359 . . . . . . . . . 10 ((A Q 1Q Q) → (A +Q 1Q) Q)
53, 4mpan2 401 . . . . . . . . 9 (A Q → (A +Q 1Q) Q)
6 sonr 4045 . . . . . . . . 9 (( <Q Or Q (A +Q 1Q) Q) → ¬ (A +Q 1Q) <Q (A +Q 1Q))
72, 5, 6sylancr 393 . . . . . . . 8 (A Q → ¬ (A +Q 1Q) <Q (A +Q 1Q))
8 ltrelnq 6349 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
98brel 4335 . . . . . . . . . . 11 ((A +Q 1Q) <Q (A +Q 1Q) → ((A +Q 1Q) Q (A +Q 1Q) Q))
109simpld 105 . . . . . . . . . 10 ((A +Q 1Q) <Q (A +Q 1Q) → (A +Q 1Q) Q)
11 elex 2560 . . . . . . . . . 10 ((A +Q 1Q) Q → (A +Q 1Q) V)
1210, 11syl 14 . . . . . . . . 9 ((A +Q 1Q) <Q (A +Q 1Q) → (A +Q 1Q) V)
13 breq1 3758 . . . . . . . . 9 (𝑙 = (A +Q 1Q) → (𝑙 <Q (A +Q 1Q) ↔ (A +Q 1Q) <Q (A +Q 1Q)))
1412, 13elab3 2688 . . . . . . . 8 ((A +Q 1Q) {𝑙𝑙 <Q (A +Q 1Q)} ↔ (A +Q 1Q) <Q (A +Q 1Q))
157, 14sylnibr 601 . . . . . . 7 (A Q → ¬ (A +Q 1Q) {𝑙𝑙 <Q (A +Q 1Q)})
16 ltnqex 6531 . . . . . . . . 9 {𝑙𝑙 <Q (A +Q 1Q)} V
17 gtnqex 6532 . . . . . . . . 9 {u ∣ (A +Q 1Q) <Q u} V
1816, 17op1st 5715 . . . . . . . 8 (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩) = {𝑙𝑙 <Q (A +Q 1Q)}
1918eleq2i 2101 . . . . . . 7 ((A +Q 1Q) (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩) ↔ (A +Q 1Q) {𝑙𝑙 <Q (A +Q 1Q)})
2015, 19sylnibr 601 . . . . . 6 (A Q → ¬ (A +Q 1Q) (1st ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩))
211, 20ssneldd 2942 . . . . 5 (A Q → ¬ (A +Q 1Q) (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)))
2221adantr 261 . . . 4 ((A Q 𝑟 (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩)) → ¬ (A +Q 1Q) (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)))
23 nqprlu 6530 . . . . . . . . 9 (A Q → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ P)
2423adantr 261 . . . . . . . 8 ((A Q 𝑟 (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩)) → ⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ P)
25 1pr 6534 . . . . . . . 8 1P P
26 addclpr 6520 . . . . . . . 8 ((⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ P 1P P) → (⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P) P)
2724, 25, 26sylancl 392 . . . . . . 7 ((A Q 𝑟 (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩)) → (⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P) P)
28 prop 6457 . . . . . . 7 ((⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P) P → ⟨(1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)), (2nd ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))⟩ P)
2927, 28syl 14 . . . . . 6 ((A Q 𝑟 (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩)) → ⟨(1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)), (2nd ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))⟩ P)
30 vex 2554 . . . . . . . . 9 𝑟 V
31 breq2 3759 . . . . . . . . 9 (u = 𝑟 → ((A +Q 1Q) <Q u ↔ (A +Q 1Q) <Q 𝑟))
3216, 17op2nd 5716 . . . . . . . . 9 (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩) = {u ∣ (A +Q 1Q) <Q u}
3330, 31, 32elab2 2684 . . . . . . . 8 (𝑟 (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩) ↔ (A +Q 1Q) <Q 𝑟)
3433biimpi 113 . . . . . . 7 (𝑟 (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩) → (A +Q 1Q) <Q 𝑟)
3534adantl 262 . . . . . 6 ((A Q 𝑟 (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩)) → (A +Q 1Q) <Q 𝑟)
36 prloc 6473 . . . . . 6 ((⟨(1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)), (2nd ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))⟩ P (A +Q 1Q) <Q 𝑟) → ((A +Q 1Q) (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)) 𝑟 (2nd ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))))
3729, 35, 36syl2anc 391 . . . . 5 ((A Q 𝑟 (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩)) → ((A +Q 1Q) (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)) 𝑟 (2nd ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))))
3837orcomd 647 . . . 4 ((A Q 𝑟 (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩)) → (𝑟 (2nd ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)) (A +Q 1Q) (1st ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))))
3922, 38ecased 1238 . . 3 ((A Q 𝑟 (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩)) → 𝑟 (2nd ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)))
4039ex 108 . 2 (A Q → (𝑟 (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩) → 𝑟 (2nd ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P))))
4140ssrdv 2945 1 (A Q → (2nd ‘⟨{𝑙𝑙 <Q (A +Q 1Q)}, {u ∣ (A +Q 1Q) <Q u}⟩) ⊆ (2nd ‘(⟨{𝑙𝑙 <Q A}, {uA <Q u}⟩ +P 1P)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97   wo 628   wcel 1390  {cab 2023  Vcvv 2551  wss 2911  cop 3370   class class class wbr 3755   Or wor 4023  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  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-2o 5941  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-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450
This theorem is referenced by:  addnqpr1  6541
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