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Theorem distrlem5prl 7066
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem5prl ((𝐴P𝐵P𝐶P) → (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))

Proof of Theorem distrlem5prl
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulclpr 7052 . . . . 5 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
213adant3 961 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
3 mulclpr 7052 . . . . 5 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
433adant2 960 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
5 df-iplp 6948 . . . . 5 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
6 addclnq 6855 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
75, 6genpelvl 6992 . . . 4 (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → (𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ↔ ∃𝑣 ∈ (1st ‘(𝐴 ·P 𝐵))∃𝑢 ∈ (1st ‘(𝐴 ·P 𝐶))𝑤 = (𝑣 +Q 𝑢)))
82, 4, 7syl2anc 403 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ↔ ∃𝑣 ∈ (1st ‘(𝐴 ·P 𝐵))∃𝑢 ∈ (1st ‘(𝐴 ·P 𝐶))𝑤 = (𝑣 +Q 𝑢)))
9 df-imp 6949 . . . . . . . 8 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑣) ∧ 𝑥 = (𝑔 ·Q ))}, {𝑥Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑣) ∧ 𝑥 = (𝑔 ·Q ))}⟩)
10 mulclnq 6856 . . . . . . . 8 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
119, 10genpelvl 6992 . . . . . . 7 ((𝐴P𝐶P) → (𝑢 ∈ (1st ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧)))
12113adant2 960 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑢 ∈ (1st ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧)))
1312anbi2d 452 . . . . 5 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ 𝑢 ∈ (1st ‘(𝐴 ·P 𝐶))) ↔ (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧))))
14 df-imp 6949 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑣) ∧ 𝑓 = (𝑔 ·Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑣) ∧ 𝑓 = (𝑔 ·Q ))}⟩)
1514, 10genpelvl 6992 . . . . . . . 8 ((𝐴P𝐵P) → (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦)))
16153adant3 961 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦)))
17 distrlem4prl 7064 . . . . . . . . . . . . . . 15 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
18 oveq12 5603 . . . . . . . . . . . . . . . . . 18 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
1918eqeq2d 2096 . . . . . . . . . . . . . . . . 17 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
20 eleq1 2147 . . . . . . . . . . . . . . . . 17 (𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → (𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
2119, 20syl6bi 161 . . . . . . . . . . . . . . . 16 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → (𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))))
2221imp 122 . . . . . . . . . . . . . . 15 (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → (𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
2317, 22syl5ibrcom 155 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
2423exp4b 359 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶))) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
2524com3l 80 . . . . . . . . . . . 12 (((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶))) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
2625exp4b 359 . . . . . . . . . . 11 ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) → ((𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)) → (𝑣 = (𝑥 ·Q 𝑦) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))))
2726com23 77 . . . . . . . . . 10 ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) → (𝑣 = (𝑥 ·Q 𝑦) → ((𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))))
2827rexlimivv 2490 . . . . . . . . 9 (∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦) → ((𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))))))
2928rexlimdvv 2491 . . . . . . . 8 (∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
3029com3r 78 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
3116, 30sylbid 148 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) → (∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
3231impd 251 . . . . 5 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))))
3313, 32sylbid 148 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ 𝑢 ∈ (1st ‘(𝐴 ·P 𝐶))) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))))
3433rexlimdvv 2491 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑣 ∈ (1st ‘(𝐴 ·P 𝐵))∃𝑢 ∈ (1st ‘(𝐴 ·P 𝐶))𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
358, 34sylbid 148 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
3635ssrdv 3018 1 ((𝐴P𝐵P𝐶P) → (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 922   = wceq 1287  wcel 1436  wrex 2356  wss 2986  cfv 4972  (class class class)co 5594  1st c1st 5847   +Q cplq 6762   ·Q cmq 6763  Pcnp 6771   +P cpp 6773   ·P cmp 6774
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-eprel 4083  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-irdg 6070  df-1o 6116  df-2o 6117  df-oadd 6120  df-omul 6121  df-er 6225  df-ec 6227  df-qs 6231  df-ni 6784  df-pli 6785  df-mi 6786  df-lti 6787  df-plpq 6824  df-mpq 6825  df-enq 6827  df-nqqs 6828  df-plqqs 6829  df-mqqs 6830  df-1nqqs 6831  df-rq 6832  df-ltnqqs 6833  df-enq0 6904  df-nq0 6905  df-0nq0 6906  df-plq0 6907  df-mq0 6908  df-inp 6946  df-iplp 6948  df-imp 6949
This theorem is referenced by:  distrprg  7068
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