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Theorem recexprlemss1l 6791
Description: The lower cut of 𝐴 ·P 𝐵 is a subset of the lower cut of one. Lemma for recexpr 6794. (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
recexprlemss1l (𝐴P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem recexprlemss1l
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 6788 . . . . 5 (𝐴P𝐵P)
3 df-imp 6625 . . . . . 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 6532 . . . . . 6 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
53, 4genpelvl 6668 . . . . 5 ((𝐴P𝐵P) → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑞 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑞)))
62, 5mpdan 406 . . . 4 (𝐴P → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑞 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑞)))
71recexprlemell 6778 . . . . . . . 8 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
8 ltrelnq 6521 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
98brel 4420 . . . . . . . . . . . . 13 (𝑞 <Q 𝑦 → (𝑞Q𝑦Q))
109simprd 111 . . . . . . . . . . . 12 (𝑞 <Q 𝑦𝑦Q)
11 prop 6631 . . . . . . . . . . . . . . . . . 18 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 elprnql 6637 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
1311, 12sylan 271 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (1st𝐴)) → 𝑧Q)
14 ltmnqi 6559 . . . . . . . . . . . . . . . . . 18 ((𝑞 <Q 𝑦𝑧Q) → (𝑧 ·Q 𝑞) <Q (𝑧 ·Q 𝑦))
1514expcom 113 . . . . . . . . . . . . . . . . 17 (𝑧Q → (𝑞 <Q 𝑦 → (𝑧 ·Q 𝑞) <Q (𝑧 ·Q 𝑦)))
1613, 15syl 14 . . . . . . . . . . . . . . . 16 ((𝐴P𝑧 ∈ (1st𝐴)) → (𝑞 <Q 𝑦 → (𝑧 ·Q 𝑞) <Q (𝑧 ·Q 𝑦)))
1716adantr 265 . . . . . . . . . . . . . . 15 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → (𝑞 <Q 𝑦 → (𝑧 ·Q 𝑞) <Q (𝑧 ·Q 𝑦)))
18 prltlu 6643 . . . . . . . . . . . . . . . . . . 19 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑧 <Q (*Q𝑦))
1911, 18syl3an1 1179 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝑧 ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑧 <Q (*Q𝑦))
20193expia 1117 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (1st𝐴)) → ((*Q𝑦) ∈ (2nd𝐴) → 𝑧 <Q (*Q𝑦)))
2120adantr 265 . . . . . . . . . . . . . . . 16 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → ((*Q𝑦) ∈ (2nd𝐴) → 𝑧 <Q (*Q𝑦)))
22 ltmnqi 6559 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 <Q (*Q𝑦) ∧ 𝑦Q) → (𝑦 ·Q 𝑧) <Q (𝑦 ·Q (*Q𝑦)))
2322expcom 113 . . . . . . . . . . . . . . . . . . . 20 (𝑦Q → (𝑧 <Q (*Q𝑦) → (𝑦 ·Q 𝑧) <Q (𝑦 ·Q (*Q𝑦))))
2423adantr 265 . . . . . . . . . . . . . . . . . . 19 ((𝑦Q𝑧Q) → (𝑧 <Q (*Q𝑦) → (𝑦 ·Q 𝑧) <Q (𝑦 ·Q (*Q𝑦))))
25 mulcomnqg 6539 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
26 recidnq 6549 . . . . . . . . . . . . . . . . . . . . 21 (𝑦Q → (𝑦 ·Q (*Q𝑦)) = 1Q)
2726adantr 265 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q (*Q𝑦)) = 1Q)
2825, 27breq12d 3805 . . . . . . . . . . . . . . . . . . 19 ((𝑦Q𝑧Q) → ((𝑦 ·Q 𝑧) <Q (𝑦 ·Q (*Q𝑦)) ↔ (𝑧 ·Q 𝑦) <Q 1Q))
2924, 28sylibd 142 . . . . . . . . . . . . . . . . . 18 ((𝑦Q𝑧Q) → (𝑧 <Q (*Q𝑦) → (𝑧 ·Q 𝑦) <Q 1Q))
3029ancoms 259 . . . . . . . . . . . . . . . . 17 ((𝑧Q𝑦Q) → (𝑧 <Q (*Q𝑦) → (𝑧 ·Q 𝑦) <Q 1Q))
3113, 30sylan 271 . . . . . . . . . . . . . . . 16 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → (𝑧 <Q (*Q𝑦) → (𝑧 ·Q 𝑦) <Q 1Q))
3221, 31syld 44 . . . . . . . . . . . . . . 15 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → ((*Q𝑦) ∈ (2nd𝐴) → (𝑧 ·Q 𝑦) <Q 1Q))
3317, 32anim12d 322 . . . . . . . . . . . . . 14 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → ((𝑧 ·Q 𝑞) <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q 1Q)))
34 ltsonq 6554 . . . . . . . . . . . . . . 15 <Q Or Q
3534, 8sotri 4748 . . . . . . . . . . . . . 14 (((𝑧 ·Q 𝑞) <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q 1Q) → (𝑧 ·Q 𝑞) <Q 1Q)
3633, 35syl6 33 . . . . . . . . . . . . 13 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → (𝑧 ·Q 𝑞) <Q 1Q))
3736exp4b 353 . . . . . . . . . . . 12 ((𝐴P𝑧 ∈ (1st𝐴)) → (𝑦Q → (𝑞 <Q 𝑦 → ((*Q𝑦) ∈ (2nd𝐴) → (𝑧 ·Q 𝑞) <Q 1Q))))
3810, 37syl5 32 . . . . . . . . . . 11 ((𝐴P𝑧 ∈ (1st𝐴)) → (𝑞 <Q 𝑦 → (𝑞 <Q 𝑦 → ((*Q𝑦) ∈ (2nd𝐴) → (𝑧 ·Q 𝑞) <Q 1Q))))
3938pm2.43d 48 . . . . . . . . . 10 ((𝐴P𝑧 ∈ (1st𝐴)) → (𝑞 <Q 𝑦 → ((*Q𝑦) ∈ (2nd𝐴) → (𝑧 ·Q 𝑞) <Q 1Q)))
4039impd 246 . . . . . . . . 9 ((𝐴P𝑧 ∈ (1st𝐴)) → ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → (𝑧 ·Q 𝑞) <Q 1Q))
4140exlimdv 1716 . . . . . . . 8 ((𝐴P𝑧 ∈ (1st𝐴)) → (∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → (𝑧 ·Q 𝑞) <Q 1Q))
427, 41syl5bi 145 . . . . . . 7 ((𝐴P𝑧 ∈ (1st𝐴)) → (𝑞 ∈ (1st𝐵) → (𝑧 ·Q 𝑞) <Q 1Q))
43 breq1 3795 . . . . . . . 8 (𝑤 = (𝑧 ·Q 𝑞) → (𝑤 <Q 1Q ↔ (𝑧 ·Q 𝑞) <Q 1Q))
4443biimprcd 153 . . . . . . 7 ((𝑧 ·Q 𝑞) <Q 1Q → (𝑤 = (𝑧 ·Q 𝑞) → 𝑤 <Q 1Q))
4542, 44syl6 33 . . . . . 6 ((𝐴P𝑧 ∈ (1st𝐴)) → (𝑞 ∈ (1st𝐵) → (𝑤 = (𝑧 ·Q 𝑞) → 𝑤 <Q 1Q)))
4645expimpd 349 . . . . 5 (𝐴P → ((𝑧 ∈ (1st𝐴) ∧ 𝑞 ∈ (1st𝐵)) → (𝑤 = (𝑧 ·Q 𝑞) → 𝑤 <Q 1Q)))
4746rexlimdvv 2456 . . . 4 (𝐴P → (∃𝑧 ∈ (1st𝐴)∃𝑞 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑞) → 𝑤 <Q 1Q))
486, 47sylbid 143 . . 3 (𝐴P → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) → 𝑤 <Q 1Q))
49 1prl 6711 . . . 4 (1st ‘1P) = {𝑤𝑤 <Q 1Q}
5049abeq2i 2164 . . 3 (𝑤 ∈ (1st ‘1P) ↔ 𝑤 <Q 1Q)
5148, 50syl6ibr 155 . 2 (𝐴P → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) → 𝑤 ∈ (1st ‘1P)))
5251ssrdv 2979 1 (𝐴P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wrex 2324  wss 2945  cop 3406   class class class wbr 3792  cfv 4930  (class class class)co 5540  1st c1st 5793  2nd c2nd 5794  Qcnq 6436  1Qc1q 6437   ·Q cmq 6439  *Qcrq 6440   <Q cltq 6441  Pcnp 6447  1Pc1p 6448   ·P cmp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-inp 6622  df-i1p 6623  df-imp 6625
This theorem is referenced by:  recexprlemex  6793
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