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Theorem mulnqprlemfu 7588
Description: Lemma for mulnqpr 7589. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
Assertion
Ref Expression
mulnqprlemfu ((𝐴Q𝐵Q) → (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
Distinct variable groups:   𝐴,𝑙,𝑢   𝐵,𝑙,𝑢

Proof of Theorem mulnqprlemfu
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 mulnqprlemrl 7585 . . . . . 6 ((𝐴Q𝐵Q) → (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
2 ltsonq 7410 . . . . . . . . 9 <Q Or Q
3 mulclnq 7388 . . . . . . . . 9 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) ∈ Q)
4 sonr 4329 . . . . . . . . 9 (( <Q Or Q ∧ (𝐴 ·Q 𝐵) ∈ Q) → ¬ (𝐴 ·Q 𝐵) <Q (𝐴 ·Q 𝐵))
52, 3, 4sylancr 414 . . . . . . . 8 ((𝐴Q𝐵Q) → ¬ (𝐴 ·Q 𝐵) <Q (𝐴 ·Q 𝐵))
6 ltrelnq 7377 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
76brel 4690 . . . . . . . . . . 11 ((𝐴 ·Q 𝐵) <Q (𝐴 ·Q 𝐵) → ((𝐴 ·Q 𝐵) ∈ Q ∧ (𝐴 ·Q 𝐵) ∈ Q))
87simpld 112 . . . . . . . . . 10 ((𝐴 ·Q 𝐵) <Q (𝐴 ·Q 𝐵) → (𝐴 ·Q 𝐵) ∈ Q)
9 elex 2760 . . . . . . . . . 10 ((𝐴 ·Q 𝐵) ∈ Q → (𝐴 ·Q 𝐵) ∈ V)
108, 9syl 14 . . . . . . . . 9 ((𝐴 ·Q 𝐵) <Q (𝐴 ·Q 𝐵) → (𝐴 ·Q 𝐵) ∈ V)
11 breq1 4018 . . . . . . . . 9 (𝑙 = (𝐴 ·Q 𝐵) → (𝑙 <Q (𝐴 ·Q 𝐵) ↔ (𝐴 ·Q 𝐵) <Q (𝐴 ·Q 𝐵)))
1210, 11elab3 2901 . . . . . . . 8 ((𝐴 ·Q 𝐵) ∈ {𝑙𝑙 <Q (𝐴 ·Q 𝐵)} ↔ (𝐴 ·Q 𝐵) <Q (𝐴 ·Q 𝐵))
135, 12sylnibr 678 . . . . . . 7 ((𝐴Q𝐵Q) → ¬ (𝐴 ·Q 𝐵) ∈ {𝑙𝑙 <Q (𝐴 ·Q 𝐵)})
14 ltnqex 7561 . . . . . . . . 9 {𝑙𝑙 <Q (𝐴 ·Q 𝐵)} ∈ V
15 gtnqex 7562 . . . . . . . . 9 {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢} ∈ V
1614, 15op1st 6160 . . . . . . . 8 (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) = {𝑙𝑙 <Q (𝐴 ·Q 𝐵)}
1716eleq2i 2254 . . . . . . 7 ((𝐴 ·Q 𝐵) ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ↔ (𝐴 ·Q 𝐵) ∈ {𝑙𝑙 <Q (𝐴 ·Q 𝐵)})
1813, 17sylnibr 678 . . . . . 6 ((𝐴Q𝐵Q) → ¬ (𝐴 ·Q 𝐵) ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
191, 18ssneldd 3170 . . . . 5 ((𝐴Q𝐵Q) → ¬ (𝐴 ·Q 𝐵) ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
2019adantr 276 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)) → ¬ (𝐴 ·Q 𝐵) ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
21 nqprlu 7559 . . . . . . . 8 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
22 nqprlu 7559 . . . . . . . 8 (𝐵Q → ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P)
23 mulclpr 7584 . . . . . . . 8 ((⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ∈ P)
2421, 22, 23syl2an 289 . . . . . . 7 ((𝐴Q𝐵Q) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ∈ P)
25 prop 7487 . . . . . . 7 ((⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ∈ P → ⟨(1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)), (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))⟩ ∈ P)
2624, 25syl 14 . . . . . 6 ((𝐴Q𝐵Q) → ⟨(1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)), (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))⟩ ∈ P)
27 vex 2752 . . . . . . . 8 𝑟 ∈ V
28 breq2 4019 . . . . . . . 8 (𝑢 = 𝑟 → ((𝐴 ·Q 𝐵) <Q 𝑢 ↔ (𝐴 ·Q 𝐵) <Q 𝑟))
2914, 15op2nd 6161 . . . . . . . 8 (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) = {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}
3027, 28, 29elab2 2897 . . . . . . 7 (𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ↔ (𝐴 ·Q 𝐵) <Q 𝑟)
3130biimpi 120 . . . . . 6 (𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) → (𝐴 ·Q 𝐵) <Q 𝑟)
32 prloc 7503 . . . . . 6 ((⟨(1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)), (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))⟩ ∈ P ∧ (𝐴 ·Q 𝐵) <Q 𝑟) → ((𝐴 ·Q 𝐵) ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ∨ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3326, 31, 32syl2an 289 . . . . 5 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)) → ((𝐴 ·Q 𝐵) ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ∨ 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3433orcomd 730 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)) → (𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ∨ (𝐴 ·Q 𝐵) ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3520, 34ecased 1359 . . 3 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)) → 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
3635ex 115 . 2 ((𝐴Q𝐵Q) → (𝑟 ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) → 𝑟 ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3736ssrdv 3173 1 ((𝐴Q𝐵Q) → (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 709  wcel 2158  {cab 2173  Vcvv 2749  wss 3141  cop 3607   class class class wbr 4015   Or wor 4307  cfv 5228  (class class class)co 5888  1st c1st 6152  2nd c2nd 6153  Qcnq 7292   ·Q cmq 7295   <Q cltq 7297  Pcnp 7303   ·P cmp 7306
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-2o 6431  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-enq0 7436  df-nq0 7437  df-0nq0 7438  df-plq0 7439  df-mq0 7440  df-inp 7478  df-imp 7481
This theorem is referenced by:  mulnqpr  7589
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