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Theorem ltexprlemfl 7544
Description: Lemma for ltexpri 7548. 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 7440 . . . . . 6 <P ⊆ (P × P)
21brel 4653 . . . . 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 7543 . . . 4 (𝐴<P 𝐵𝐶P)
6 df-iplp 7403 . . . . 5 +P = (𝑧P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑧) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑧) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
7 addclnq 7310 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
86, 7genpelvl 7447 . . . 4 ((𝐴P𝐶P) → (𝑧 ∈ (1st ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (1st𝐴)∃𝑢 ∈ (1st𝐶)𝑧 = (𝑤 +Q 𝑢)))
93, 5, 8syl2anc 409 . . 3 (𝐴<P 𝐵 → (𝑧 ∈ (1st ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (1st𝐴)∃𝑢 ∈ (1st𝐶)𝑧 = (𝑤 +Q 𝑢)))
10 simprr 522 . . . . . 6 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 = (𝑤 +Q 𝑢))
114ltexprlemell 7533 . . . . . . . . . . 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 481 . . . . . . . . 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 7410 . . . . . . . . . . . . . 14 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
173, 16syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
18 prltlu 7422 . . . . . . . . . . . . 13 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑤 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
1917, 18syl3an1 1260 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑤 ∈ (1st𝐴) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
20193adant2r 1222 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
21203adant2r 1222 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ 𝑦 ∈ (2nd𝐴)) → 𝑤 <Q 𝑦)
22213adant3r 1224 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → 𝑤 <Q 𝑦)
23 ltanqg 7335 . . . . . . . . . . . 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 7300 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
2625brel 4653 . . . . . . . . . . . . 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 7410 . . . . . . . . . . . . . . . 16 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
315, 30syl 14 . . . . . . . . . . . . . . 15 (𝐴<P 𝐵 → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
32 elprnql 7416 . . . . . . . . . . . . . . 15 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑢 ∈ (1st𝐶)) → 𝑢Q)
3331, 32sylan 281 . . . . . . . . . . . . . 14 ((𝐴<P 𝐵𝑢 ∈ (1st𝐶)) → 𝑢Q)
3433adantrl 470 . . . . . . . . . . . . 13 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶))) → 𝑢Q)
3534adantrr 471 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑢Q)
36353adant3 1006 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → 𝑢Q)
37 addcomnqg 7316 . . . . . . . . . . . 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 6005 . . . . . . . . . 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 7410 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
4240, 41syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
43 prcdnql 7419 . . . . . . . . . . . . 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 470 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → ((𝑤 +Q 𝑢) <Q (𝑦 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (1st𝐵)))
46453adant2 1005 . . . . . . . . . 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 1192 . . . . . . 7 (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) ∧ (𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑢) ∈ (1st𝐵))) → (𝑤 +Q 𝑢) ∈ (1st𝐵))
5015, 49exlimddv 1885 . . . . . 6 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → (𝑤 +Q 𝑢) ∈ (1st𝐵))
5110, 50eqeltrd 2241 . . . . 5 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 ∈ (1st𝐵))
5251expr 373 . . . 4 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (1st𝐴) ∧ 𝑢 ∈ (1st𝐶))) → (𝑧 = (𝑤 +Q 𝑢) → 𝑧 ∈ (1st𝐵)))
5352rexlimdvva 2589 . . 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 3146 1 (𝐴<P 𝐵 → (1st ‘(𝐴 +P 𝐶)) ⊆ (1st𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 967   = wceq 1342  wex 1479  wcel 2135  wrex 2443  {crab 2446  wss 3114  cop 3576   class class class wbr 3979  cfv 5185  (class class class)co 5839  1st c1st 6101  2nd c2nd 6102  Qcnq 7215   +Q cplq 7217   <Q cltq 7220  Pcnp 7226   +P cpp 7228  <P cltp 7230
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4094  ax-sep 4097  ax-nul 4105  ax-pow 4150  ax-pr 4184  ax-un 4408  ax-setind 4511  ax-iinf 4562
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2726  df-sbc 2950  df-csb 3044  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3408  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-int 3822  df-iun 3865  df-br 3980  df-opab 4041  df-mpt 4042  df-tr 4078  df-eprel 4264  df-id 4268  df-po 4271  df-iso 4272  df-iord 4341  df-on 4343  df-suc 4346  df-iom 4565  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-rn 4612  df-res 4613  df-ima 4614  df-iota 5150  df-fun 5187  df-fn 5188  df-f 5189  df-f1 5190  df-fo 5191  df-f1o 5192  df-fv 5193  df-ov 5842  df-oprab 5843  df-mpo 5844  df-1st 6103  df-2nd 6104  df-recs 6267  df-irdg 6332  df-1o 6378  df-2o 6379  df-oadd 6382  df-omul 6383  df-er 6495  df-ec 6497  df-qs 6501  df-ni 7239  df-pli 7240  df-mi 7241  df-lti 7242  df-plpq 7279  df-mpq 7280  df-enq 7282  df-nqqs 7283  df-plqqs 7284  df-mqqs 7285  df-1nqqs 7286  df-rq 7287  df-ltnqqs 7288  df-enq0 7359  df-nq0 7360  df-0nq0 7361  df-plq0 7362  df-mq0 7363  df-inp 7401  df-iplp 7403  df-iltp 7405
This theorem is referenced by:  ltexpri  7548
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