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Theorem recexprlemss1l 7291
Description: The lower cut of 𝐴 ·P 𝐵 is a subset of the lower cut of one. Lemma for recexpr 7294. (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 7288 . . . . 5 (𝐴P𝐵P)
3 df-imp 7125 . . . . . 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 7032 . . . . . 6 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
53, 4genpelvl 7168 . . . . 5 ((𝐴P𝐵P) → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑞 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑞)))
62, 5mpdan 413 . . . 4 (𝐴P → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (1st𝐴)∃𝑞 ∈ (1st𝐵)𝑤 = (𝑧 ·Q 𝑞)))
71recexprlemell 7278 . . . . . . . 8 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
8 ltrelnq 7021 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
98brel 4519 . . . . . . . . . . . . 13 (𝑞 <Q 𝑦 → (𝑞Q𝑦Q))
109simprd 113 . . . . . . . . . . . 12 (𝑞 <Q 𝑦𝑦Q)
11 prop 7131 . . . . . . . . . . . . . . . . . 18 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 elprnql 7137 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
1311, 12sylan 278 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (1st𝐴)) → 𝑧Q)
14 ltmnqi 7059 . . . . . . . . . . . . . . . . . 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 271 . . . . . . . . . . . . . . 15 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → (𝑞 <Q 𝑦 → (𝑧 ·Q 𝑞) <Q (𝑧 ·Q 𝑦)))
18 prltlu 7143 . . . . . . . . . . . . . . . . . . 19 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑧 <Q (*Q𝑦))
1911, 18syl3an1 1214 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝑧 ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑧 <Q (*Q𝑦))
20193expia 1148 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (1st𝐴)) → ((*Q𝑦) ∈ (2nd𝐴) → 𝑧 <Q (*Q𝑦)))
2120adantr 271 . . . . . . . . . . . . . . . 16 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → ((*Q𝑦) ∈ (2nd𝐴) → 𝑧 <Q (*Q𝑦)))
22 ltmnqi 7059 . . . . . . . . . . . . . . . . . . . . 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 271 . . . . . . . . . . . . . . . . . . 19 ((𝑦Q𝑧Q) → (𝑧 <Q (*Q𝑦) → (𝑦 ·Q 𝑧) <Q (𝑦 ·Q (*Q𝑦))))
25 mulcomnqg 7039 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
26 recidnq 7049 . . . . . . . . . . . . . . . . . . . . 21 (𝑦Q → (𝑦 ·Q (*Q𝑦)) = 1Q)
2726adantr 271 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q (*Q𝑦)) = 1Q)
2825, 27breq12d 3880 . . . . . . . . . . . . . . . . . . 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 265 . . . . . . . . . . . . . . . . 17 ((𝑧Q𝑦Q) → (𝑧 <Q (*Q𝑦) → (𝑧 ·Q 𝑦) <Q 1Q))
3113, 30sylan 278 . . . . . . . . . . . . . . . 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 329 . . . . . . . . . . . . . 14 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → ((𝑧 ·Q 𝑞) <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q 1Q)))
34 ltsonq 7054 . . . . . . . . . . . . . . 15 <Q Or Q
3534, 8sotri 4860 . . . . . . . . . . . . . 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 360 . . . . . . . . . . . 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 1754 . . . . . . . 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 3870 . . . . . . . 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 356 . . . . 5 (𝐴P → ((𝑧 ∈ (1st𝐴) ∧ 𝑞 ∈ (1st𝐵)) → (𝑤 = (𝑧 ·Q 𝑞) → 𝑤 <Q 1Q)))
4746rexlimdvv 2509 . . . 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 7211 . . . 4 (1st ‘1P) = {𝑤𝑤 <Q 1Q}
5049abeq2i 2205 . . 3 (𝑤 ∈ (1st ‘1P) ↔ 𝑤 <Q 1Q)
5148, 50syl6ibr 161 . 2 (𝐴P → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) → 𝑤 ∈ (1st ‘1P)))
5251ssrdv 3045 1 (𝐴P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1296  wex 1433  wcel 1445  {cab 2081  wrex 2371  wss 3013  cop 3469   class class class wbr 3867  cfv 5049  (class class class)co 5690  1st c1st 5947  2nd c2nd 5948  Qcnq 6936  1Qc1q 6937   ·Q cmq 6939  *Qcrq 6940   <Q cltq 6941  Pcnp 6947  1Pc1p 6948   ·P cmp 6950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009  df-inp 7122  df-i1p 7123  df-imp 7125
This theorem is referenced by:  recexprlemex  7293
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