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Theorem ltexprlemfl 7071
Description: Lemma for ltexpri 7075. One directon of our result for lower cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
Assertion
Ref Expression
ltexprlemfl (𝐴<P 𝐵 → (1st ‘(𝐴 +P 𝐶)) ⊆ (1st𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem ltexprlemfl
Dummy variables 𝑧 𝑤 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 6967 . . . . . 6 <P ⊆ (P × P)
21brel 4448 . . . . 5 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simpld 110 . . . 4 (𝐴<P 𝐵𝐴P)
4 ltexprlem.1 . . . . 5 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
54ltexprlempr 7070 . . . 4 (𝐴<P 𝐵𝐶P)
6 df-iplp 6930 . . . . 5 +P = (𝑧P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑧) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑧) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
7 addclnq 6837 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
86, 7genpelvl 6974 . . . 4 ((𝐴P𝐶P) → (𝑧 ∈ (1st ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (1st𝐴)∃𝑢 ∈ (1st𝐶)𝑧 = (𝑤 +Q 𝑢)))
93, 5, 8syl2anc 403 . . 3 (𝐴<P 𝐵 → (𝑧 ∈ (1st ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (1st𝐴)∃𝑢 ∈ (1st𝐶)𝑧 = (𝑤 +Q 𝑢)))
10 simprr 499 . . . . . 6 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 = (𝑤 +Q 𝑢))
114ltexprlemell 7060 . . . . . . . . . . 11 (𝑢 ∈ (1st𝐶) ↔ (𝑢Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))))
1211biimpi 118 . . . . . . . . . 10 (𝑢 ∈ (1st𝐶) → (𝑢Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))))
1312ad2antlr 473 . . . . . . . . 9 (((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) → (𝑢Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))))
1413adantl 271 . . . . . . . 8 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → (𝑢Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))))
1514simprd 112 . . . . . . 7 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵)))
16 prop 6937 . . . . . . . . . . . . . 14 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
173, 16syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
18 prltlu 6949 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑤 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
1917, 18syl3an1 1203 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑤 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
20193adant2r 1165 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
21203adant2r 1165 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
22213adant3r 1167 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → 𝑤 <Q 𝑦)
23 ltanqg 6862 . . . . . . . . . . . 12 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
2423adantl 271 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
25 ltrelnq 6827 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
2625brel 4448 . . . . . . . . . . . . 13 (𝑤 <Q 𝑦 → (𝑤Q𝑦Q))
2722, 26syl 14 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → (𝑤Q𝑦Q))
2827simpld 110 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → 𝑤Q)
2927simprd 112 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → 𝑦Q)
30 prop 6937 . . . . . . . . . . . . . . . 16 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
315, 30syl 14 . . . . . . . . . . . . . . 15 (𝐴<P 𝐵 → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
32 elprnql 6943 . . . . . . . . . . . . . . 15 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑢 ∈ (1st𝐶)) → 𝑢Q)
3331, 32sylan 277 . . . . . . . . . . . . . 14 ((𝐴<P 𝐵𝑢 ∈ (1st𝐶)) → 𝑢Q)
3433adantrl 462 . . . . . . . . . . . . 13 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶))) → 𝑢Q)
3534adantrr 463 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑢Q)
36353adant3 959 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → 𝑢Q)
37 addcomnqg 6843 . . . . . . . . . . . 12 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3837adantl 271 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3924, 28, 29, 36, 38caovord2d 5749 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → (𝑤 <Q 𝑦 ↔ (𝑤 +Q 𝑢) <Q (𝑦 +Q 𝑢)))
402simprd 112 . . . . . . . . . . . . . 14 (𝐴<P 𝐵𝐵P)
41 prop 6937 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
4240, 41syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
43 prcdnql 6946 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵)) → ((𝑤 +Q 𝑢) <Q (𝑦 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (1st𝐵)))
4442, 43sylan 277 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵)) → ((𝑤 +Q 𝑢) <Q (𝑦 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (1st𝐵)))
4544adantrl 462 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → ((𝑤 +Q 𝑢) <Q (𝑦 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (1st𝐵)))
46453adant2 958 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → ((𝑤 +Q 𝑢) <Q (𝑦 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (1st𝐵)))
4739, 46sylbid 148 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → (𝑤 <Q 𝑦 → (𝑤 +Q 𝑢) ∈ (1st𝐵)))
4822, 47mpd 13 . . . . . . . 8 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → (𝑤 +Q 𝑢) ∈ (1st𝐵))
49483expa 1139 . . . . . . 7 (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → (𝑤 +Q 𝑢) ∈ (1st𝐵))
5015, 49exlimddv 1821 . . . . . 6 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → (𝑤 +Q 𝑢) ∈ (1st𝐵))
5110, 50eqeltrd 2159 . . . . 5 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 ∈ (1st𝐵))
5251expr 367 . . . 4 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶))) → (𝑧 = (𝑤 +Q 𝑢) → 𝑧 ∈ (1st𝐵)))
5352rexlimdvva 2490 . . 3 (𝐴<P 𝐵 → (∃𝑤 ∈ (1st𝐴)∃𝑢 ∈ (1st𝐶)𝑧 = (𝑤 +Q 𝑢) → 𝑧 ∈ (1st𝐵)))
549, 53sylbid 148 . 2 (𝐴<P 𝐵 → (𝑧 ∈ (1st ‘(𝐴 +P 𝐶)) → 𝑧 ∈ (1st𝐵)))
5554ssrdv 3016 1 (𝐴<P 𝐵 → (1st ‘(𝐴 +P 𝐶)) ⊆ (1st𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wex 1422  wcel 1434  wrex 2354  {crab 2357  wss 2984  cop 3425   class class class wbr 3811  cfv 4969  (class class class)co 5591  1st c1st 5844  2nd c2nd 5845  Qcnq 6742   +Q cplq 6744   <Q cltq 6747  Pcnp 6753   +P cpp 6755  <P cltp 6757
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-2o 6114  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-enq0 6886  df-nq0 6887  df-0nq0 6888  df-plq0 6889  df-mq0 6890  df-inp 6928  df-iplp 6930  df-iltp 6932
This theorem is referenced by:  ltexpri  7075
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