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Theorem recexprlemss1u 7626
Description: The upper cut of 𝐴 ·P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 7628. (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 7622 . . . . 5 (𝐴P𝐵P)
3 df-imp 7459 . . . . . 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 7366 . . . . . 6 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
53, 4genpelvu 7503 . . . . 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 7613 . . . . . . . 8 (𝑞 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q𝑦) ∈ (1st𝐴)))
8 ltrelnq 7355 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
98brel 4675 . . . . . . . . . . . . 13 (𝑦 <Q 𝑞 → (𝑦Q𝑞Q))
109simpld 112 . . . . . . . . . . . 12 (𝑦 <Q 𝑞𝑦Q)
11 prop 7465 . . . . . . . . . . . . . . . . . 18 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 elprnqu 7472 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
1311, 12sylan 283 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (2nd𝐴)) → 𝑧Q)
14 ltmnqi 7393 . . . . . . . . . . . . . . . . . 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 7477 . . . . . . . . . . . . . . . . . . . 20 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑦) ∈ (1st𝐴) ∧ 𝑧 ∈ (2nd𝐴)) → (*Q𝑦) <Q 𝑧)
1911, 18syl3an1 1271 . . . . . . . . . . . . . . . . . . 19 ((𝐴P ∧ (*Q𝑦) ∈ (1st𝐴) ∧ 𝑧 ∈ (2nd𝐴)) → (*Q𝑦) <Q 𝑧)
20193com23 1209 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝑧 ∈ (2nd𝐴) ∧ (*Q𝑦) ∈ (1st𝐴)) → (*Q𝑦) <Q 𝑧)
21203expia 1205 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (2nd𝐴)) → ((*Q𝑦) ∈ (1st𝐴) → (*Q𝑦) <Q 𝑧))
2221adantr 276 . . . . . . . . . . . . . . . 16 (((𝐴P𝑧 ∈ (2nd𝐴)) ∧ 𝑦Q) → ((*Q𝑦) ∈ (1st𝐴) → (*Q𝑦) <Q 𝑧))
23 ltmnqi 7393 . . . . . . . . . . . . . . . . . . . . 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 7383 . . . . . . . . . . . . . . . . . . . . 21 (𝑦Q → (𝑦 ·Q (*Q𝑦)) = 1Q)
2726adantr 276 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q (*Q𝑦)) = 1Q)
28 mulcomnqg 7373 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
2927, 28breq12d 4013 . . . . . . . . . . . . . . . . . . 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 7388 . . . . . . . . . . . . . . . 16 <Q Or Q
3635, 8sotri 5020 . . . . . . . . . . . . . . 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 1819 . . . . . . . 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 4004 . . . . . . . 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 2601 . . . 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 7546 . . . 4 (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
5251abeq2i 2288 . . 3 (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
5350, 52syl6ibr 162 . 2 (𝐴P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 𝑤 ∈ (2nd ‘1P)))
5453ssrdv 3161 1 (𝐴P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wrex 2456  wss 3129  cop 3594   class class class wbr 4000  cfv 5212  (class class class)co 5869  1st c1st 6133  2nd c2nd 6134  Qcnq 7270  1Qc1q 7271   ·Q cmq 7273  *Qcrq 7274   <Q cltq 7275  Pcnp 7281  1Pc1p 7282   ·P cmp 7284
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-inp 7456  df-i1p 7457  df-imp 7459
This theorem is referenced by:  recexprlemex  7627
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