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Theorem ltexprlemfu 7640
Description: Lemma for ltexpri 7642. One direction of our result for upper 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
ltexprlemfu (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem ltexprlemfu
Dummy variables 𝑧 𝑤 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 7534 . . . . . 6 <P ⊆ (P × P)
21brel 4696 . . . . 5 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simpld 112 . . . 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 7637 . . . 4 (𝐴<P 𝐵𝐶P)
6 df-iplp 7497 . . . . 5 +P = (𝑧P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑧) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑧) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
7 addclnq 7404 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
86, 7genpelvu 7542 . . . 4 ((𝐴P𝐶P) → (𝑧 ∈ (2nd ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (2nd𝐴)∃𝑢 ∈ (2nd𝐶)𝑧 = (𝑤 +Q 𝑢)))
93, 5, 8syl2anc 411 . . 3 (𝐴<P 𝐵 → (𝑧 ∈ (2nd ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (2nd𝐴)∃𝑢 ∈ (2nd𝐶)𝑧 = (𝑤 +Q 𝑢)))
10 simprr 531 . . . . . 6 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 = (𝑤 +Q 𝑢))
114ltexprlemelu 7628 . . . . . . . . . . 11 (𝑢 ∈ (2nd𝐶) ↔ (𝑢Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))))
1211biimpi 120 . . . . . . . . . 10 (𝑢 ∈ (2nd𝐶) → (𝑢Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))))
1312ad2antlr 489 . . . . . . . . 9 (((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) → (𝑢Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))))
1413simprd 114 . . . . . . . 8 (((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵)))
1514adantl 277 . . . . . . 7 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵)))
16 prop 7504 . . . . . . . . . . . . . . 15 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
173, 16syl 14 . . . . . . . . . . . . . 14 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
18 prltlu 7516 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴) ∧ 𝑤 ∈ (2nd𝐴)) → 𝑦 <Q 𝑤)
1917, 18syl3an1 1282 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴) ∧ 𝑤 ∈ (2nd𝐴)) → 𝑦 <Q 𝑤)
20193com23 1211 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 <Q 𝑤)
21203adant2r 1235 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 <Q 𝑤)
22213adant2r 1235 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 <Q 𝑤)
23223adant3r 1237 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → 𝑦 <Q 𝑤)
24 ltanqg 7429 . . . . . . . . . . . 12 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
2524adantl 277 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
26 elprnql 7510 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
2717, 26sylan 283 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴)) → 𝑦Q)
2827adantrr 479 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → 𝑦Q)
29283adant2 1018 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → 𝑦Q)
30 elprnqu 7511 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑤 ∈ (2nd𝐴)) → 𝑤Q)
3117, 30sylan 283 . . . . . . . . . . . . . 14 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐴)) → 𝑤Q)
3231adantrr 479 . . . . . . . . . . . . 13 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶))) → 𝑤Q)
3332adantrr 479 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑤Q)
34333adant3 1019 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → 𝑤Q)
35 prop 7504 . . . . . . . . . . . . . . . 16 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
365, 35syl 14 . . . . . . . . . . . . . . 15 (𝐴<P 𝐵 → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
37 elprnqu 7511 . . . . . . . . . . . . . . 15 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑢 ∈ (2nd𝐶)) → 𝑢Q)
3836, 37sylan 283 . . . . . . . . . . . . . 14 ((𝐴<P 𝐵𝑢 ∈ (2nd𝐶)) → 𝑢Q)
3938adantrl 478 . . . . . . . . . . . . 13 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶))) → 𝑢Q)
4039adantrr 479 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑢Q)
41403adant3 1019 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → 𝑢Q)
42 addcomnqg 7410 . . . . . . . . . . . 12 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4342adantl 277 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4425, 29, 34, 41, 43caovord2d 6066 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → (𝑦 <Q 𝑤 ↔ (𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢)))
452simprd 114 . . . . . . . . . . . . . 14 (𝐴<P 𝐵𝐵P)
46 prop 7504 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
4745, 46syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
48 prcunqu 7514 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵)) → ((𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (2nd𝐵)))
4947, 48sylan 283 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵)) → ((𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (2nd𝐵)))
5049adantrl 478 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → ((𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (2nd𝐵)))
51503adant2 1018 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → ((𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (2nd𝐵)))
5244, 51sylbid 150 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → (𝑦 <Q 𝑤 → (𝑤 +Q 𝑢) ∈ (2nd𝐵)))
5323, 52mpd 13 . . . . . . . 8 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → (𝑤 +Q 𝑢) ∈ (2nd𝐵))
54533expa 1205 . . . . . . 7 (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → (𝑤 +Q 𝑢) ∈ (2nd𝐵))
5515, 54exlimddv 1910 . . . . . 6 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → (𝑤 +Q 𝑢) ∈ (2nd𝐵))
5610, 55eqeltrd 2266 . . . . 5 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 ∈ (2nd𝐵))
5756expr 375 . . . 4 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶))) → (𝑧 = (𝑤 +Q 𝑢) → 𝑧 ∈ (2nd𝐵)))
5857rexlimdvva 2615 . . 3 (𝐴<P 𝐵 → (∃𝑤 ∈ (2nd𝐴)∃𝑢 ∈ (2nd𝐶)𝑧 = (𝑤 +Q 𝑢) → 𝑧 ∈ (2nd𝐵)))
599, 58sylbid 150 . 2 (𝐴<P 𝐵 → (𝑧 ∈ (2nd ‘(𝐴 +P 𝐶)) → 𝑧 ∈ (2nd𝐵)))
6059ssrdv 3176 1 (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980   = wceq 1364  wex 1503  wcel 2160  wrex 2469  {crab 2472  wss 3144  cop 3610   class class class wbr 4018  cfv 5235  (class class class)co 5896  1st c1st 6163  2nd c2nd 6164  Qcnq 7309   +Q cplq 7311   <Q cltq 7314  Pcnp 7320   +P cpp 7322  <P cltp 7324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-1o 6441  df-2o 6442  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-pli 7334  df-mi 7335  df-lti 7336  df-plpq 7373  df-mpq 7374  df-enq 7376  df-nqqs 7377  df-plqqs 7378  df-mqqs 7379  df-1nqqs 7380  df-rq 7381  df-ltnqqs 7382  df-enq0 7453  df-nq0 7454  df-0nq0 7455  df-plq0 7456  df-mq0 7457  df-inp 7495  df-iplp 7497  df-iltp 7499
This theorem is referenced by:  ltexpri  7642
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