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Theorem recexprlemss1u 7666
Description: The upper cut of 𝐴 ·P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 7668. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
Assertion
Ref Expression
recexprlemss1u (𝐴P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem recexprlemss1u
Dummy variables 𝑞 𝑧 𝑤 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 recexpr.1 . . . . . 6 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
21recexprlempr 7662 . . . . 5 (𝐴P𝐵P)
3 df-imp 7499 . . . . . 6 ·P = (𝑦P, 𝑤P ↦ ⟨{𝑢Q ∣ ∃𝑓Q𝑔Q (𝑓 ∈ (1st𝑦) ∧ 𝑔 ∈ (1st𝑤) ∧ 𝑢 = (𝑓 ·Q 𝑔))}, {𝑢Q ∣ ∃𝑓Q𝑔Q (𝑓 ∈ (2nd𝑦) ∧ 𝑔 ∈ (2nd𝑤) ∧ 𝑢 = (𝑓 ·Q 𝑔))}⟩)
4 mulclnq 7406 . . . . . 6 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
53, 4genpelvu 7543 . . . . 5 ((𝐴P𝐵P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑞 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑞)))
62, 5mpdan 421 . . . 4 (𝐴P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑞 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑞)))
71recexprlemelu 7653 . . . . . . . 8 (𝑞 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)))
8 ltrelnq 7395 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
98brel 4696 . . . . . . . . . . . . 13 (𝑦 <Q 𝑞 → (𝑦Q𝑞Q))
109simpld 112 . . . . . . . . . . . 12 (𝑦 <Q 𝑞𝑦Q)
11 prop 7505 . . . . . . . . . . . . . . . . . 18 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 elprnqu 7512 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
1311, 12sylan 283 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
14 ltmnqi 7433 . . . . . . . . . . . . . . . . . 18 ((𝑦 <Q 𝑞𝑧Q) → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞))
1514expcom 116 . . . . . . . . . . . . . . . . 17 (𝑧Q → (𝑦 <Q 𝑞 → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)))
1613, 15syl 14 . . . . . . . . . . . . . . . 16 ((𝐴P𝑧 ∈ (2nd𝐴)) → (𝑦 <Q 𝑞 → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)))
1716adantr 276 . . . . . . . . . . . . . . 15 (((𝐴P𝑧 ∈ (2nd𝐴)) ∧ 𝑦Q) → (𝑦 <Q 𝑞 → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)))
18 prltlu 7517 . . . . . . . . . . . . . . . . . . . 20 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑦) ∈ (1st𝐴) ∧ 𝑧 ∈ (2nd𝐴)) → (*Q𝑦) <Q 𝑧)
1911, 18syl3an1 1282 . . . . . . . . . . . . . . . . . . 19 ((𝐴P ∧ (*Q𝑦) ∈ (1st𝐴) ∧ 𝑧 ∈ (2nd𝐴)) → (*Q𝑦) <Q 𝑧)
20193com23 1211 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝑧 ∈ (2nd𝐴) ∧ (*Q𝑦) ∈ (1st𝐴)) → (*Q𝑦) <Q 𝑧)
21203expia 1207 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (2nd𝐴)) → ((*Q𝑦) ∈ (1st𝐴) → (*Q𝑦) <Q 𝑧))
2221adantr 276 . . . . . . . . . . . . . . . 16 (((𝐴P𝑧 ∈ (2nd𝐴)) ∧ 𝑦Q) → ((*Q𝑦) ∈ (1st𝐴) → (*Q𝑦) <Q 𝑧))
23 ltmnqi 7433 . . . . . . . . . . . . . . . . . . . . 21 (((*Q𝑦) <Q 𝑧𝑦Q) → (𝑦 ·Q (*Q𝑦)) <Q (𝑦 ·Q 𝑧))
2423expcom 116 . . . . . . . . . . . . . . . . . . . 20 (𝑦Q → ((*Q𝑦) <Q 𝑧 → (𝑦 ·Q (*Q𝑦)) <Q (𝑦 ·Q 𝑧)))
2524adantr 276 . . . . . . . . . . . . . . . . . . 19 ((𝑦Q𝑧Q) → ((*Q𝑦) <Q 𝑧 → (𝑦 ·Q (*Q𝑦)) <Q (𝑦 ·Q 𝑧)))
26 recidnq 7423 . . . . . . . . . . . . . . . . . . . . 21 (𝑦Q → (𝑦 ·Q (*Q𝑦)) = 1Q)
2726adantr 276 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q (*Q𝑦)) = 1Q)
28 mulcomnqg 7413 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
2927, 28breq12d 4031 . . . . . . . . . . . . . . . . . . 19 ((𝑦Q𝑧Q) → ((𝑦 ·Q (*Q𝑦)) <Q (𝑦 ·Q 𝑧) ↔ 1Q <Q (𝑧 ·Q 𝑦)))
3025, 29sylibd 149 . . . . . . . . . . . . . . . . . 18 ((𝑦Q𝑧Q) → ((*Q𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
3130ancoms 268 . . . . . . . . . . . . . . . . 17 ((𝑧Q𝑦Q) → ((*Q𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
3213, 31sylan 283 . . . . . . . . . . . . . . . 16 (((𝐴P𝑧 ∈ (2nd𝐴)) ∧ 𝑦Q) → ((*Q𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
3322, 32syld 45 . . . . . . . . . . . . . . 15 (((𝐴P𝑧 ∈ (2nd𝐴)) ∧ 𝑦Q) → ((*Q𝑦) ∈ (1st𝐴) → 1Q <Q (𝑧 ·Q 𝑦)))
3417, 33anim12d 335 . . . . . . . . . . . . . 14 (((𝐴P𝑧 ∈ (2nd𝐴)) ∧ 𝑦Q) → ((𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) → ((𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞) ∧ 1Q <Q (𝑧 ·Q 𝑦))))
35 ltsonq 7428 . . . . . . . . . . . . . . . 16 <Q Or Q
3635, 8sotri 5042 . . . . . . . . . . . . . . 15 ((1Q <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)) → 1Q <Q (𝑧 ·Q 𝑞))
3736ancoms 268 . . . . . . . . . . . . . 14 (((𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞) ∧ 1Q <Q (𝑧 ·Q 𝑦)) → 1Q <Q (𝑧 ·Q 𝑞))
3834, 37syl6 33 . . . . . . . . . . . . 13 (((𝐴P𝑧 ∈ (2nd𝐴)) ∧ 𝑦Q) → ((𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
3938exp4b 367 . . . . . . . . . . . 12 ((𝐴P𝑧 ∈ (2nd𝐴)) → (𝑦Q → (𝑦 <Q 𝑞 → ((*Q𝑦) ∈ (1st𝐴) → 1Q <Q (𝑧 ·Q 𝑞)))))
4010, 39syl5 32 . . . . . . . . . . 11 ((𝐴P𝑧 ∈ (2nd𝐴)) → (𝑦 <Q 𝑞 → (𝑦 <Q 𝑞 → ((*Q𝑦) ∈ (1st𝐴) → 1Q <Q (𝑧 ·Q 𝑞)))))
4140pm2.43d 50 . . . . . . . . . 10 ((𝐴P𝑧 ∈ (2nd𝐴)) → (𝑦 <Q 𝑞 → ((*Q𝑦) ∈ (1st𝐴) → 1Q <Q (𝑧 ·Q 𝑞))))
4241impd 254 . . . . . . . . 9 ((𝐴P𝑧 ∈ (2nd𝐴)) → ((𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
4342exlimdv 1830 . . . . . . . 8 ((𝐴P𝑧 ∈ (2nd𝐴)) → (∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
447, 43biimtrid 152 . . . . . . 7 ((𝐴P𝑧 ∈ (2nd𝐴)) → (𝑞 ∈ (2nd𝐵) → 1Q <Q (𝑧 ·Q 𝑞)))
45 breq2 4022 . . . . . . . 8 (𝑤 = (𝑧 ·Q 𝑞) → (1Q <Q 𝑤 ↔ 1Q <Q (𝑧 ·Q 𝑞)))
4645biimprcd 160 . . . . . . 7 (1Q <Q (𝑧 ·Q 𝑞) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤))
4744, 46syl6 33 . . . . . 6 ((𝐴P𝑧 ∈ (2nd𝐴)) → (𝑞 ∈ (2nd𝐵) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤)))
4847expimpd 363 . . . . 5 (𝐴P → ((𝑧 ∈ (2nd𝐴) ∧ 𝑞 ∈ (2nd𝐵)) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤)))
4948rexlimdvv 2614 . . . 4 (𝐴P → (∃𝑧 ∈ (2nd𝐴)∃𝑞 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤))
506, 49sylbid 150 . . 3 (𝐴P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 1Q <Q 𝑤))
51 1pru 7586 . . . 4 (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
5251abeq2i 2300 . . 3 (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
5350, 52imbitrrdi 162 . 2 (𝐴P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 𝑤 ∈ (2nd ‘1P)))
5453ssrdv 3176 1 (𝐴P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  {cab 2175  wrex 2469  wss 3144  cop 3610   class class class wbr 4018  cfv 5235  (class class class)co 5897  1st c1st 6164  2nd c2nd 6165  Qcnq 7310  1Qc1q 7311   ·Q cmq 7313  *Qcrq 7314   <Q cltq 7315  Pcnp 7321  1Pc1p 7322   ·P cmp 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 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383  df-inp 7496  df-i1p 7497  df-imp 7499
This theorem is referenced by:  recexprlemex  7667
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