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Theorem distrlem5pru 6743
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem5pru ((𝐴P𝐵P𝐶P) → (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))

Proof of Theorem distrlem5pru
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulclpr 6728 . . . . 5 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
213adant3 935 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
3 mulclpr 6728 . . . . 5 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
433adant2 934 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
5 df-iplp 6624 . . . . 5 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
6 addclnq 6531 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
75, 6genpelvu 6669 . . . 4 (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → (𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ↔ ∃𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵))∃𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶))𝑤 = (𝑣 +Q 𝑢)))
82, 4, 7syl2anc 397 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ↔ ∃𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵))∃𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶))𝑤 = (𝑣 +Q 𝑢)))
9 df-imp 6625 . . . . . . . 8 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑣) ∧ 𝑥 = (𝑔 ·Q ))}, {𝑥Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑣) ∧ 𝑥 = (𝑔 ·Q ))}⟩)
10 mulclnq 6532 . . . . . . . 8 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
119, 10genpelvu 6669 . . . . . . 7 ((𝐴P𝐶P) → (𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐶)𝑢 = (𝑓 ·Q 𝑧)))
12113adant2 934 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐶)𝑢 = (𝑓 ·Q 𝑧)))
1312anbi2d 445 . . . . 5 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ∧ 𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶))) ↔ (𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ∧ ∃𝑓 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐶)𝑢 = (𝑓 ·Q 𝑧))))
14 df-imp 6625 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑣) ∧ 𝑓 = (𝑔 ·Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑣) ∧ 𝑓 = (𝑔 ·Q ))}⟩)
1514, 10genpelvu 6669 . . . . . . . 8 ((𝐴P𝐵P) → (𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑣 = (𝑥 ·Q 𝑦)))
16153adant3 935 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑣 = (𝑥 ·Q 𝑦)))
17 distrlem4pru 6741 . . . . . . . . . . . . . . 15 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
18 oveq12 5549 . . . . . . . . . . . . . . . . . 18 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
1918eqeq2d 2067 . . . . . . . . . . . . . . . . 17 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
20 eleq1 2116 . . . . . . . . . . . . . . . . 17 (𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
2119, 20syl6bi 156 . . . . . . . . . . . . . . . 16 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))))
2221imp 119 . . . . . . . . . . . . . . 15 (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
2317, 22syl5ibrcom 150 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
2423exp4b 353 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶))) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
2524com3l 79 . . . . . . . . . . . 12 (((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶))) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
2625exp4b 353 . . . . . . . . . . 11 ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) → ((𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑣 = (𝑥 ·Q 𝑦) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))))
2726com23 76 . . . . . . . . . 10 ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) → (𝑣 = (𝑥 ·Q 𝑦) → ((𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))))
2827rexlimivv 2455 . . . . . . . . 9 (∃𝑥 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑣 = (𝑥 ·Q 𝑦) → ((𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))))))
2928rexlimdvv 2456 . . . . . . . 8 (∃𝑥 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐶)𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
3029com3r 77 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (∃𝑥 ∈ (2nd𝐴)∃𝑦 ∈ (2nd𝐵)𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐶)𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
3116, 30sylbid 143 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) → (∃𝑓 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐶)𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
3231impd 246 . . . . 5 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ∧ ∃𝑓 ∈ (2nd𝐴)∃𝑧 ∈ (2nd𝐶)𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))))
3313, 32sylbid 143 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵)) ∧ 𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶))) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))))
3433rexlimdvv 2456 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑣 ∈ (2nd ‘(𝐴 ·P 𝐵))∃𝑢 ∈ (2nd ‘(𝐴 ·P 𝐶))𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
358, 34sylbid 143 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) → 𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
3635ssrdv 2979 1 ((𝐴P𝐵P𝐶P) → (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  wrex 2324  wss 2945  cfv 4930  (class class class)co 5540  2nd c2nd 5794   +Q cplq 6438   ·Q cmq 6439  Pcnp 6447   +P cpp 6449   ·P cmp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624  df-imp 6625
This theorem is referenced by:  distrprg  6744
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