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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
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