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Theorem ltexprlemfl 7385
Description: Lemma for ltexpri 7389. 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 7281 . . . . . 6 <P ⊆ (P × P)
21brel 4561 . . . . 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 7384 . . . 4 (𝐴<P 𝐵𝐶P)
6 df-iplp 7244 . . . . 5 +P = (𝑧P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑧) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑧) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
7 addclnq 7151 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
86, 7genpelvl 7288 . . . 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 506 . . . . . 6 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 = (𝑤 +Q 𝑢))
114ltexprlemell 7374 . . . . . . . . . . 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 7251 . . . . . . . . . . . . . 14 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
173, 16syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
18 prltlu 7263 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑤 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
1917, 18syl3an1 1234 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑤 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
20193adant2r 1196 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
21203adant2r 1196 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
22213adant3r 1198 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → 𝑤 <Q 𝑦)
23 ltanqg 7176 . . . . . . . . . . . 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 7141 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
2625brel 4561 . . . . . . . . . . . . 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 7251 . . . . . . . . . . . . . . . 16 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
315, 30syl 14 . . . . . . . . . . . . . . 15 (𝐴<P 𝐵 → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
32 elprnql 7257 . . . . . . . . . . . . . . 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 986 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → 𝑢Q)
37 addcomnqg 7157 . . . . . . . . . . . 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 5908 . . . . . . . . . 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 7251 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
4240, 41syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
43 prcdnql 7260 . . . . . . . . . . . . 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 985 . . . . . . . . . 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 1166 . . . . . . 7 (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → (𝑤 +Q 𝑢) ∈ (1st𝐵))
5015, 49exlimddv 1854 . . . . . 6 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → (𝑤 +Q 𝑢) ∈ (1st𝐵))
5110, 50eqeltrd 2194 . . . . 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 2534 . . 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 3073 1 (𝐴<P 𝐵 → (1st ‘(𝐴 +P 𝐶)) ⊆ (1st𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 947   = wceq 1316  wex 1453  wcel 1465  wrex 2394  {crab 2397  wss 3041  cop 3500   class class class wbr 3899  cfv 5093  (class class class)co 5742  1st c1st 6004  2nd c2nd 6005  Qcnq 7056   +Q cplq 7058   <Q cltq 7061  Pcnp 7067   +P cpp 7069  <P cltp 7071
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244  df-iltp 7246
This theorem is referenced by:  ltexpri  7389
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