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Theorem recexprlemss1l 7450
Description: The lower cut of 𝐴 ·P 𝐵 is a subset of the lower cut of one. Lemma for recexpr 7453. (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 7447 . . . . 5 (𝐴P𝐵P)
3 df-imp 7284 . . . . . 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 7191 . . . . . 6 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
53, 4genpelvl 7327 . . . . 5 ((𝐴P𝐵P) → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑞 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑞)))
62, 5mpdan 417 . . . 4 (𝐴P → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑞 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑞)))
71recexprlemell 7437 . . . . . . . 8 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
8 ltrelnq 7180 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
98brel 4591 . . . . . . . . . . . . 13 (𝑞 <Q 𝑦 → (𝑞Q𝑦Q))
109simprd 113 . . . . . . . . . . . 12 (𝑞 <Q 𝑦𝑦Q)
11 prop 7290 . . . . . . . . . . . . . . . . . 18 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 elprnql 7296 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
1311, 12sylan 281 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (1st𝐴)) → 𝑧Q)
14 ltmnqi 7218 . . . . . . . . . . . . . . . . . 18 ((𝑞 <Q 𝑦𝑧Q) → (𝑧 ·Q 𝑞) <Q (𝑧 ·Q 𝑦))
1514expcom 115 . . . . . . . . . . . . . . . . 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 274 . . . . . . . . . . . . . . 15 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → (𝑞 <Q 𝑦 → (𝑧 ·Q 𝑞) <Q (𝑧 ·Q 𝑦)))
18 prltlu 7302 . . . . . . . . . . . . . . . . . . 19 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑧 <Q (*Q𝑦))
1911, 18syl3an1 1249 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝑧 ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑧 <Q (*Q𝑦))
20193expia 1183 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (1st𝐴)) → ((*Q𝑦) ∈ (2nd𝐴) → 𝑧 <Q (*Q𝑦)))
2120adantr 274 . . . . . . . . . . . . . . . 16 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → ((*Q𝑦) ∈ (2nd𝐴) → 𝑧 <Q (*Q𝑦)))
22 ltmnqi 7218 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 <Q (*Q𝑦) ∧ 𝑦Q) → (𝑦 ·Q 𝑧) <Q (𝑦 ·Q (*Q𝑦)))
2322expcom 115 . . . . . . . . . . . . . . . . . . . 20 (𝑦Q → (𝑧 <Q (*Q𝑦) → (𝑦 ·Q 𝑧) <Q (𝑦 ·Q (*Q𝑦))))
2423adantr 274 . . . . . . . . . . . . . . . . . . 19 ((𝑦Q𝑧Q) → (𝑧 <Q (*Q𝑦) → (𝑦 ·Q 𝑧) <Q (𝑦 ·Q (*Q𝑦))))
25 mulcomnqg 7198 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
26 recidnq 7208 . . . . . . . . . . . . . . . . . . . . 21 (𝑦Q → (𝑦 ·Q (*Q𝑦)) = 1Q)
2726adantr 274 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q (*Q𝑦)) = 1Q)
2825, 27breq12d 3942 . . . . . . . . . . . . . . . . . . 19 ((𝑦Q𝑧Q) → ((𝑦 ·Q 𝑧) <Q (𝑦 ·Q (*Q𝑦)) ↔ (𝑧 ·Q 𝑦) <Q 1Q))
2924, 28sylibd 148 . . . . . . . . . . . . . . . . . 18 ((𝑦Q𝑧Q) → (𝑧 <Q (*Q𝑦) → (𝑧 ·Q 𝑦) <Q 1Q))
3029ancoms 266 . . . . . . . . . . . . . . . . 17 ((𝑧Q𝑦Q) → (𝑧 <Q (*Q𝑦) → (𝑧 ·Q 𝑦) <Q 1Q))
3113, 30sylan 281 . . . . . . . . . . . . . . . 16 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → (𝑧 <Q (*Q𝑦) → (𝑧 ·Q 𝑦) <Q 1Q))
3221, 31syld 45 . . . . . . . . . . . . . . 15 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → ((*Q𝑦) ∈ (2nd𝐴) → (𝑧 ·Q 𝑦) <Q 1Q))
3317, 32anim12d 333 . . . . . . . . . . . . . 14 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → ((𝑧 ·Q 𝑞) <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q 1Q)))
34 ltsonq 7213 . . . . . . . . . . . . . . 15 <Q Or Q
3534, 8sotri 4934 . . . . . . . . . . . . . 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 364 . . . . . . . . . . . 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 50 . . . . . . . . . 10 ((𝐴P𝑧 ∈ (1st𝐴)) → (𝑞 <Q 𝑦 → ((*Q𝑦) ∈ (2nd𝐴) → (𝑧 ·Q 𝑞) <Q 1Q)))
4039impd 252 . . . . . . . . 9 ((𝐴P𝑧 ∈ (1st𝐴)) → ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → (𝑧 ·Q 𝑞) <Q 1Q))
4140exlimdv 1791 . . . . . . . 8 ((𝐴P𝑧 ∈ (1st𝐴)) → (∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → (𝑧 ·Q 𝑞) <Q 1Q))
427, 41syl5bi 151 . . . . . . 7 ((𝐴P𝑧 ∈ (1st𝐴)) → (𝑞 ∈ (1st𝐵) → (𝑧 ·Q 𝑞) <Q 1Q))
43 breq1 3932 . . . . . . . 8 (𝑤 = (𝑧 ·Q 𝑞) → (𝑤 <Q 1Q ↔ (𝑧 ·Q 𝑞) <Q 1Q))
4443biimprcd 159 . . . . . . 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 360 . . . . 5 (𝐴P → ((𝑧 ∈ (1st𝐴) ∧ 𝑞 ∈ (1st𝐵)) → (𝑤 = (𝑧 ·Q 𝑞) → 𝑤 <Q 1Q)))
4746rexlimdvv 2556 . . . 4 (𝐴P → (∃𝑧 ∈ (1st𝐴)∃𝑞 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑞) → 𝑤 <Q 1Q))
486, 47sylbid 149 . . 3 (𝐴P → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) → 𝑤 <Q 1Q))
49 1prl 7370 . . . 4 (1st ‘1P) = {𝑤𝑤 <Q 1Q}
5049abeq2i 2250 . . 3 (𝑤 ∈ (1st ‘1P) ↔ 𝑤 <Q 1Q)
5148, 50syl6ibr 161 . 2 (𝐴P → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) → 𝑤 ∈ (1st ‘1P)))
5251ssrdv 3103 1 (𝐴P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wex 1468  wcel 1480  {cab 2125  wrex 2417  wss 3071  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7095  1Qc1q 7096   ·Q cmq 7098  *Qcrq 7099   <Q cltq 7100  Pcnp 7106  1Pc1p 7107   ·P cmp 7109
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-inp 7281  df-i1p 7282  df-imp 7284
This theorem is referenced by:  recexprlemex  7452
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