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Theorem mulnqprlemfl 7134
Description: Lemma for mulnqpr 7136. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 18-Jul-2021.)
Assertion
Ref Expression
mulnqprlemfl ((𝐴Q𝐵Q) → (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{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 mulnqprlemfl
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 mulnqprlemru 7133 . . . . . 6 ((𝐴Q𝐵Q) → (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
2 ltsonq 6957 . . . . . . . . 9 <Q Or Q
3 mulclnq 6935 . . . . . . . . 9 ((𝐴Q𝐵Q) → (𝐴 ·Q 𝐵) ∈ Q)
4 sonr 4144 . . . . . . . . 9 (( <Q Or Q ∧ (𝐴 ·Q 𝐵) ∈ Q) → ¬ (𝐴 ·Q 𝐵) <Q (𝐴 ·Q 𝐵))
52, 3, 4sylancr 405 . . . . . . . 8 ((𝐴Q𝐵Q) → ¬ (𝐴 ·Q 𝐵) <Q (𝐴 ·Q 𝐵))
6 ltrelnq 6924 . . . . . . . . . . . 12 <Q ⊆ (Q × Q)
76brel 4490 . . . . . . . . . . 11 ((𝐴 ·Q 𝐵) <Q (𝐴 ·Q 𝐵) → ((𝐴 ·Q 𝐵) ∈ Q ∧ (𝐴 ·Q 𝐵) ∈ Q))
87simpld 110 . . . . . . . . . 10 ((𝐴 ·Q 𝐵) <Q (𝐴 ·Q 𝐵) → (𝐴 ·Q 𝐵) ∈ Q)
9 elex 2630 . . . . . . . . . 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 breq2 3849 . . . . . . . . 9 (𝑢 = (𝐴 ·Q 𝐵) → ((𝐴 ·Q 𝐵) <Q 𝑢 ↔ (𝐴 ·Q 𝐵) <Q (𝐴 ·Q 𝐵)))
1210, 11elab3 2767 . . . . . . . 8 ((𝐴 ·Q 𝐵) ∈ {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢} ↔ (𝐴 ·Q 𝐵) <Q (𝐴 ·Q 𝐵))
135, 12sylnibr 637 . . . . . . 7 ((𝐴Q𝐵Q) → ¬ (𝐴 ·Q 𝐵) ∈ {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢})
14 ltnqex 7108 . . . . . . . . 9 {𝑙𝑙 <Q (𝐴 ·Q 𝐵)} ∈ V
15 gtnqex 7109 . . . . . . . . 9 {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢} ∈ V
1614, 15op2nd 5918 . . . . . . . 8 (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) = {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}
1716eleq2i 2154 . . . . . . 7 ((𝐴 ·Q 𝐵) ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ↔ (𝐴 ·Q 𝐵) ∈ {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢})
1813, 17sylnibr 637 . . . . . 6 ((𝐴Q𝐵Q) → ¬ (𝐴 ·Q 𝐵) ∈ (2nd ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩))
191, 18ssneldd 3028 . . . . 5 ((𝐴Q𝐵Q) → ¬ (𝐴 ·Q 𝐵) ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
2019adantr 270 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)) → ¬ (𝐴 ·Q 𝐵) ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
21 nqprlu 7106 . . . . . . 7 (𝐴Q → ⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ∈ P)
22 nqprlu 7106 . . . . . . 7 (𝐵Q → ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩ ∈ P)
23 mulclpr 7131 . . . . . . 7 ((⟨{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 283 . . . . . 6 ((𝐴Q𝐵Q) → (⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩) ∈ P)
25 prop 7034 . . . . . 6 ((⟨{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 . . . . 5 ((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 2622 . . . . . . 7 𝑟 ∈ V
28 breq1 3848 . . . . . . 7 (𝑙 = 𝑟 → (𝑙 <Q (𝐴 ·Q 𝐵) ↔ 𝑟 <Q (𝐴 ·Q 𝐵)))
2914, 15op1st 5917 . . . . . . 7 (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) = {𝑙𝑙 <Q (𝐴 ·Q 𝐵)}
3027, 28, 29elab2 2763 . . . . . 6 (𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ↔ 𝑟 <Q (𝐴 ·Q 𝐵))
3130biimpi 118 . . . . 5 (𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) → 𝑟 <Q (𝐴 ·Q 𝐵))
32 prloc 7050 . . . . 5 ((⟨(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}⟩))⟩ ∈ P𝑟 <Q (𝐴 ·Q 𝐵)) → (𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ∨ (𝐴 ·Q 𝐵) ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3326, 31, 32syl2an 283 . . . 4 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)) → (𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)) ∨ (𝐴 ·Q 𝐵) ∈ (2nd ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3420, 33ecased 1285 . . 3 (((𝐴Q𝐵Q) ∧ 𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩)) → 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩)))
3534ex 113 . 2 ((𝐴Q𝐵Q) → (𝑟 ∈ (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) → 𝑟 ∈ (1st ‘(⟨{𝑙𝑙 <Q 𝐴}, {𝑢𝐴 <Q 𝑢}⟩ ·P ⟨{𝑙𝑙 <Q 𝐵}, {𝑢𝐵 <Q 𝑢}⟩))))
3635ssrdv 3031 1 ((𝐴Q𝐵Q) → (1st ‘⟨{𝑙𝑙 <Q (𝐴 ·Q 𝐵)}, {𝑢 ∣ (𝐴 ·Q 𝐵) <Q 𝑢}⟩) ⊆ (1st ‘(⟨{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 102  wo 664  wcel 1438  {cab 2074  Vcvv 2619  wss 2999  cop 3449   class class class wbr 3845   Or wor 4122  cfv 5015  (class class class)co 5652  1st c1st 5909  2nd c2nd 5910  Qcnq 6839   ·Q cmq 6842   <Q cltq 6844  Pcnp 6850   ·P cmp 6853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-2o 6182  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-mpq 6904  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-mqqs 6909  df-1nqqs 6910  df-rq 6911  df-ltnqqs 6912  df-enq0 6983  df-nq0 6984  df-0nq0 6985  df-plq0 6986  df-mq0 6987  df-inp 7025  df-imp 7028
This theorem is referenced by:  mulnqpr  7136
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