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Theorem ltexprlemfl 7424
Description: Lemma for ltexpri 7428. One direction 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 7320 . . . . . 6 <P ⊆ (P × P)
21brel 4591 . . . . 5 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simpld 111 . . . 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 7423 . . . 4 (𝐴<P 𝐵𝐶P)
6 df-iplp 7283 . . . . 5 +P = (𝑧P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑧) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑧) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
7 addclnq 7190 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
86, 7genpelvl 7327 . . . 4 ((𝐴P𝐶P) → (𝑧 ∈ (1st ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (1st𝐴)∃𝑢 ∈ (1st𝐶)𝑧 = (𝑤 +Q 𝑢)))
93, 5, 8syl2anc 408 . . 3 (𝐴<P 𝐵 → (𝑧 ∈ (1st ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (1st𝐴)∃𝑢 ∈ (1st𝐶)𝑧 = (𝑤 +Q 𝑢)))
10 simprr 521 . . . . . 6 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 = (𝑤 +Q 𝑢))
114ltexprlemell 7413 . . . . . . . . . . 11 (𝑢 ∈ (1st𝐶) ↔ (𝑢Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))))
1211biimpi 119 . . . . . . . . . 10 (𝑢 ∈ (1st𝐶) → (𝑢Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))))
1312ad2antlr 480 . . . . . . . . 9 (((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) → (𝑢Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))))
1413adantl 275 . . . . . . . 8 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → (𝑢Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))))
1514simprd 113 . . . . . . 7 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵)))
16 prop 7290 . . . . . . . . . . . . . 14 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
173, 16syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
18 prltlu 7302 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑤 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
1917, 18syl3an1 1249 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑤 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
20193adant2r 1211 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
21203adant2r 1211 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
22213adant3r 1213 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → 𝑤 <Q 𝑦)
23 ltanqg 7215 . . . . . . . . . . . 12 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
2423adantl 275 . . . . . . . . . . 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 7180 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
2625brel 4591 . . . . . . . . . . . . 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 111 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → 𝑤Q)
2927simprd 113 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → 𝑦Q)
30 prop 7290 . . . . . . . . . . . . . . . 16 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
315, 30syl 14 . . . . . . . . . . . . . . 15 (𝐴<P 𝐵 → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
32 elprnql 7296 . . . . . . . . . . . . . . 15 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑢 ∈ (1st𝐶)) → 𝑢Q)
3331, 32sylan 281 . . . . . . . . . . . . . 14 ((𝐴<P 𝐵𝑢 ∈ (1st𝐶)) → 𝑢Q)
3433adantrl 469 . . . . . . . . . . . . 13 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶))) → 𝑢Q)
3534adantrr 470 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑢Q)
36353adant3 1001 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → 𝑢Q)
37 addcomnqg 7196 . . . . . . . . . . . 12 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
3837adantl 275 . . . . . . . . . . 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 5940 . . . . . . . . . 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 113 . . . . . . . . . . . . . 14 (𝐴<P 𝐵𝐵P)
41 prop 7290 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
4240, 41syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
43 prcdnql 7299 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵)) → ((𝑤 +Q 𝑢) <Q (𝑦 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (1st𝐵)))
4442, 43sylan 281 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵)) → ((𝑤 +Q 𝑢) <Q (𝑦 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (1st𝐵)))
4544adantrl 469 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → ((𝑤 +Q 𝑢) <Q (𝑦 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (1st𝐵)))
46453adant2 1000 . . . . . . . . . 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 149 . . . . . . . . 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 1181 . . . . . . 7 (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → (𝑤 +Q 𝑢) ∈ (1st𝐵))
5015, 49exlimddv 1870 . . . . . 6 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → (𝑤 +Q 𝑢) ∈ (1st𝐵))
5110, 50eqeltrd 2216 . . . . 5 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 ∈ (1st𝐵))
5251expr 372 . . . 4 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶))) → (𝑧 = (𝑤 +Q 𝑢) → 𝑧 ∈ (1st𝐵)))
5352rexlimdvva 2557 . . 3 (𝐴<P 𝐵 → (∃𝑤 ∈ (1st𝐴)∃𝑢 ∈ (1st𝐶)𝑧 = (𝑤 +Q 𝑢) → 𝑧 ∈ (1st𝐵)))
549, 53sylbid 149 . 2 (𝐴<P 𝐵 → (𝑧 ∈ (1st ‘(𝐴 +P 𝐶)) → 𝑧 ∈ (1st𝐵)))
5554ssrdv 3103 1 (𝐴<P 𝐵 → (1st ‘(𝐴 +P 𝐶)) ⊆ (1st𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wex 1468  wcel 1480  wrex 2417  {crab 2420  wss 3071  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7095   +Q cplq 7097   <Q cltq 7100  Pcnp 7106   +P cpp 7108  <P cltp 7110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iplp 7283  df-iltp 7285
This theorem is referenced by:  ltexpri  7428
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