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Theorem recexprlemss1u 7585
Description: The upper cut of 𝐴 ·P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 7587. (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 7581 . . . . 5 (𝐴P𝐵P)
3 df-imp 7418 . . . . . 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 7325 . . . . . 6 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
53, 4genpelvu 7462 . . . . 5 ((𝐴P𝐵P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑞 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑞)))
62, 5mpdan 419 . . . 4 (𝐴P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd𝐴)∃𝑞 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑞)))
71recexprlemelu 7572 . . . . . . . 8 (𝑞 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)))
8 ltrelnq 7314 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
98brel 4661 . . . . . . . . . . . . 13 (𝑦 <Q 𝑞 → (𝑦Q𝑞Q))
109simpld 111 . . . . . . . . . . . 12 (𝑦 <Q 𝑞𝑦Q)
11 prop 7424 . . . . . . . . . . . . . . . . . 18 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 elprnqu 7431 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
1311, 12sylan 281 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
14 ltmnqi 7352 . . . . . . . . . . . . . . . . . 18 ((𝑦 <Q 𝑞𝑧Q) → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞))
1514expcom 115 . . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . 15 (((𝐴P𝑧 ∈ (2nd𝐴)) ∧ 𝑦Q) → (𝑦 <Q 𝑞 → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)))
18 prltlu 7436 . . . . . . . . . . . . . . . . . . . 20 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑦) ∈ (1st𝐴) ∧ 𝑧 ∈ (2nd𝐴)) → (*Q𝑦) <Q 𝑧)
1911, 18syl3an1 1266 . . . . . . . . . . . . . . . . . . 19 ((𝐴P ∧ (*Q𝑦) ∈ (1st𝐴) ∧ 𝑧 ∈ (2nd𝐴)) → (*Q𝑦) <Q 𝑧)
20193com23 1204 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝑧 ∈ (2nd𝐴) ∧ (*Q𝑦) ∈ (1st𝐴)) → (*Q𝑦) <Q 𝑧)
21203expia 1200 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (2nd𝐴)) → ((*Q𝑦) ∈ (1st𝐴) → (*Q𝑦) <Q 𝑧))
2221adantr 274 . . . . . . . . . . . . . . . 16 (((𝐴P𝑧 ∈ (2nd𝐴)) ∧ 𝑦Q) → ((*Q𝑦) ∈ (1st𝐴) → (*Q𝑦) <Q 𝑧))
23 ltmnqi 7352 . . . . . . . . . . . . . . . . . . . . 21 (((*Q𝑦) <Q 𝑧𝑦Q) → (𝑦 ·Q (*Q𝑦)) <Q (𝑦 ·Q 𝑧))
2423expcom 115 . . . . . . . . . . . . . . . . . . . 20 (𝑦Q → ((*Q𝑦) <Q 𝑧 → (𝑦 ·Q (*Q𝑦)) <Q (𝑦 ·Q 𝑧)))
2524adantr 274 . . . . . . . . . . . . . . . . . . 19 ((𝑦Q𝑧Q) → ((*Q𝑦) <Q 𝑧 → (𝑦 ·Q (*Q𝑦)) <Q (𝑦 ·Q 𝑧)))
26 recidnq 7342 . . . . . . . . . . . . . . . . . . . . 21 (𝑦Q → (𝑦 ·Q (*Q𝑦)) = 1Q)
2726adantr 274 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q (*Q𝑦)) = 1Q)
28 mulcomnqg 7332 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
2927, 28breq12d 4000 . . . . . . . . . . . . . . . . . . 19 ((𝑦Q𝑧Q) → ((𝑦 ·Q (*Q𝑦)) <Q (𝑦 ·Q 𝑧) ↔ 1Q <Q (𝑧 ·Q 𝑦)))
3025, 29sylibd 148 . . . . . . . . . . . . . . . . . 18 ((𝑦Q𝑧Q) → ((*Q𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
3130ancoms 266 . . . . . . . . . . . . . . . . 17 ((𝑧Q𝑦Q) → ((*Q𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
3213, 31sylan 281 . . . . . . . . . . . . . . . 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 333 . . . . . . . . . . . . . 14 (((𝐴P𝑧 ∈ (2nd𝐴)) ∧ 𝑦Q) → ((𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) → ((𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞) ∧ 1Q <Q (𝑧 ·Q 𝑦))))
35 ltsonq 7347 . . . . . . . . . . . . . . . 16 <Q Or Q
3635, 8sotri 5004 . . . . . . . . . . . . . . 15 ((1Q <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)) → 1Q <Q (𝑧 ·Q 𝑞))
3736ancoms 266 . . . . . . . . . . . . . 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 365 . . . . . . . . . . . 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 252 . . . . . . . . 9 ((𝐴P𝑧 ∈ (2nd𝐴)) → ((𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
4342exlimdv 1812 . . . . . . . 8 ((𝐴P𝑧 ∈ (2nd𝐴)) → (∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
447, 43syl5bi 151 . . . . . . 7 ((𝐴P𝑧 ∈ (2nd𝐴)) → (𝑞 ∈ (2nd𝐵) → 1Q <Q (𝑧 ·Q 𝑞)))
45 breq2 3991 . . . . . . . 8 (𝑤 = (𝑧 ·Q 𝑞) → (1Q <Q 𝑤 ↔ 1Q <Q (𝑧 ·Q 𝑞)))
4645biimprcd 159 . . . . . . 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 361 . . . . 5 (𝐴P → ((𝑧 ∈ (2nd𝐴) ∧ 𝑞 ∈ (2nd𝐵)) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤)))
4948rexlimdvv 2594 . . . 4 (𝐴P → (∃𝑧 ∈ (2nd𝐴)∃𝑞 ∈ (2nd𝐵)𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤))
506, 49sylbid 149 . . 3 (𝐴P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 1Q <Q 𝑤))
51 1pru 7505 . . . 4 (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
5251abeq2i 2281 . . 3 (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
5350, 52syl6ibr 161 . 2 (𝐴P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 𝑤 ∈ (2nd ‘1P)))
5453ssrdv 3153 1 (𝐴P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wex 1485  wcel 2141  {cab 2156  wrex 2449  wss 3121  cop 3584   class class class wbr 3987  cfv 5196  (class class class)co 5850  1st c1st 6114  2nd c2nd 6115  Qcnq 7229  1Qc1q 7230   ·Q cmq 7232  *Qcrq 7233   <Q cltq 7234  Pcnp 7240  1Pc1p 7241   ·P cmp 7243
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-1o 6392  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-mpq 7294  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-mqqs 7299  df-1nqqs 7300  df-rq 7301  df-ltnqqs 7302  df-inp 7415  df-i1p 7416  df-imp 7418
This theorem is referenced by:  recexprlemex  7586
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