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Theorem addnqprlemfl 7118
Description: Lemma for addnqpr 7120. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
Assertion
Ref Expression
addnqprlemfl ((𝐴Q𝐵Q) → (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
Distinct variable groups:   𝐴,𝑙,𝑢   𝐵,𝑙,𝑢

Proof of Theorem addnqprlemfl
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 addnqprlemru 7117 . . . . . 6 ((𝐴Q𝐵Q) → (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
2 ltsonq 6957 . . . . . . . . 9 <Q Or Q
3 addclnq 6934 . . . . . . . . 9 ((𝐴Q𝐵Q) → (𝐴 +Q 𝐵) ∈ Q)
4 sonr 4144 . . . . . . . . 9 (( <Q Or Q ∧ (𝐴 +Q 𝐵) ∈ Q) → ¬ (𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵))
52, 3, 4sylancr 405 . . . . . . . 8 ((𝐴Q𝐵Q) → ¬ (𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵))
6 ltrelnq 6924 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
76brel 4490 . . . . . . . . . . 11 ((𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵) → ((𝐴 +Q 𝐵) ∈ Q ∧ (𝐴 +Q 𝐵) ∈ Q))
87simpld 110 . . . . . . . . . 10 ((𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵) → (𝐴 +Q 𝐵) ∈ Q)
9 elex 2630 . . . . . . . . . 10 ((𝐴 +Q 𝐵) ∈ Q → (𝐴 +Q 𝐵) ∈ V)
108, 9syl 14 . . . . . . . . 9 ((𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵) → (𝐴 +Q 𝐵) ∈ V)
11 breq2 3849 . . . . . . . . 9 (𝑢 = (𝐴 +Q 𝐵) → ((𝐴 +Q 𝐵) <Q 𝑢 ↔ (𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵)))
1210, 11elab3 2767 . . . . . . . 8 ((𝐴 +Q 𝐵) ∈ {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢} ↔ (𝐴 +Q 𝐵) <Q (𝐴 +Q 𝐵))
135, 12sylnibr 637 . . . . . . 7 ((𝐴Q𝐵Q) → ¬ (𝐴 +Q 𝐵) ∈ {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢})
14 ltnqex 7108 . . . . . . . . 9 {𝑙𝑙 <Q (𝐴 +Q 𝐵)} ∈ V
15 gtnqex 7109 . . . . . . . . 9 {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢} ∈ V
1614, 15op2nd 5918 . . . . . . . 8 (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) = {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}
1716eleq2i 2154 . . . . . . 7 ((𝐴 +Q 𝐵) ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ↔ (𝐴 +Q 𝐵) ∈ {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢})
1813, 17sylnibr 637 . . . . . 6 ((𝐴Q𝐵Q) → ¬ (𝐴 +Q 𝐵) ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
191, 18ssneldd 3028 . . . . 5 ((𝐴Q𝐵Q) → ¬ (𝐴 +Q 𝐵) ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
2019adantr 270 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)) → ¬ (𝐴 +Q 𝐵) ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
21 nqprlu 7106 . . . . . . 7 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
22 nqprlu 7106 . . . . . . 7 (𝐵Q → ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P)
23 addclpr 7096 . . . . . . 7 ((⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ∈ P)
2421, 22, 23syl2an 283 . . . . . 6 ((𝐴Q𝐵Q) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ∈ P)
25 prop 7034 . . . . . 6 ((⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ∈ P → ⟨(1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)), (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))⟩ ∈ P)
2624, 25syl 14 . . . . 5 ((𝐴Q𝐵Q) → ⟨(1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)), (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))⟩ ∈ P)
27 vex 2622 . . . . . . 7 𝑟 ∈ V
28 breq1 3848 . . . . . . 7 (𝑙 = 𝑟 → (𝑙 <Q (𝐴 +Q 𝐵) ↔ 𝑟 <Q (𝐴 +Q 𝐵)))
2914, 15op1st 5917 . . . . . . 7 (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) = {𝑙𝑙 <Q (𝐴 +Q 𝐵)}
3027, 28, 29elab2 2763 . . . . . 6 (𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ↔ 𝑟 <Q (𝐴 +Q 𝐵))
3130biimpi 118 . . . . 5 (𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) → 𝑟 <Q (𝐴 +Q 𝐵))
32 prloc 7050 . . . . 5 ((⟨(1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)), (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))⟩ ∈ P𝑟 <Q (𝐴 +Q 𝐵)) → (𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ∨ (𝐴 +Q 𝐵) ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3326, 31, 32syl2an 283 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)) → (𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ∨ (𝐴 +Q 𝐵) ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3420, 33ecased 1285 . . 3 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)) → 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
3534ex 113 . 2 ((𝐴Q𝐵Q) → (𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) → 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3635ssrdv 3031 1 ((𝐴Q𝐵Q) → (1st ‘⟨{𝑙𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 664  wcel 1438  {cab 2074  Vcvv 2619  wss 2999  cop 3449   class class class wbr 3845   Or wor 4122  cfv 5015  (class class class)co 5652  1st c1st 5909  2nd c2nd 5910  Qcnq 6839   +Q cplq 6841   <Q cltq 6844  Pcnp 6850   +P cpp 6852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912  df-enq0 6983  df-nq0 6984  df-0nq0 6985  df-plq0 6986  df-mq0 6987  df-inp 7025  df-iplp 7027
This theorem is referenced by:  addnqpr  7120
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