ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recexprlemss1l GIF version

Theorem recexprlemss1l 7411
Description: The lower cut of 𝐴 ·P 𝐵 is a subset of the lower cut of one. Lemma for recexpr 7414. (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 7408 . . . . 5 (𝐴P𝐵P)
3 df-imp 7245 . . . . . 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 7152 . . . . . 6 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
53, 4genpelvl 7288 . . . . 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 7398 . . . . . . . 8 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
8 ltrelnq 7141 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
98brel 4561 . . . . . . . . . . . . 13 (𝑞 <Q 𝑦 → (𝑞Q𝑦Q))
109simprd 113 . . . . . . . . . . . 12 (𝑞 <Q 𝑦𝑦Q)
11 prop 7251 . . . . . . . . . . . . . . . . . 18 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
12 elprnql 7257 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
1311, 12sylan 281 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (1st𝐴)) → 𝑧Q)
14 ltmnqi 7179 . . . . . . . . . . . . . . . . . 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 7263 . . . . . . . . . . . . . . . . . . 19 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑧 <Q (*Q𝑦))
1911, 18syl3an1 1234 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝑧 ∈ (1st𝐴) ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑧 <Q (*Q𝑦))
20193expia 1168 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧 ∈ (1st𝐴)) → ((*Q𝑦) ∈ (2nd𝐴) → 𝑧 <Q (*Q𝑦)))
2120adantr 274 . . . . . . . . . . . . . . . 16 (((𝐴P𝑧 ∈ (1st𝐴)) ∧ 𝑦Q) → ((*Q𝑦) ∈ (2nd𝐴) → 𝑧 <Q (*Q𝑦)))
22 ltmnqi 7179 . . . . . . . . . . . . . . . . . . . . 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 7159 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
26 recidnq 7169 . . . . . . . . . . . . . . . . . . . . 21 (𝑦Q → (𝑦 ·Q (*Q𝑦)) = 1Q)
2726adantr 274 . . . . . . . . . . . . . . . . . . . 20 ((𝑦Q𝑧Q) → (𝑦 ·Q (*Q𝑦)) = 1Q)
2825, 27breq12d 3912 . . . . . . . . . . . . . . . . . . 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 7174 . . . . . . . . . . . . . . 15 <Q Or Q
3534, 8sotri 4904 . . . . . . . . . . . . . 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 1775 . . . . . . . 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 3902 . . . . . . . 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 2533 . . . 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 7331 . . . 4 (1st ‘1P) = {𝑤𝑤 <Q 1Q}
5049abeq2i 2228 . . 3 (𝑤 ∈ (1st ‘1P) ↔ 𝑤 <Q 1Q)
5148, 50syl6ibr 161 . 2 (𝐴P → (𝑤 ∈ (1st ‘(𝐴 ·P 𝐵)) → 𝑤 ∈ (1st ‘1P)))
5251ssrdv 3073 1 (𝐴P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wex 1453  wcel 1465  {cab 2103  wrex 2394  wss 3041  cop 3500   class class class wbr 3899  cfv 5093  (class class class)co 5742  1st c1st 6004  2nd c2nd 6005  Qcnq 7056  1Qc1q 7057   ·Q cmq 7059  *Qcrq 7060   <Q cltq 7061  Pcnp 7067  1Pc1p 7068   ·P cmp 7070
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-inp 7242  df-i1p 7243  df-imp 7245
This theorem is referenced by:  recexprlemex  7413
  Copyright terms: Public domain W3C validator