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Theorem distrlem5prl 7610
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 7596 . . . . 5 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
213adant3 1019 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
3 mulclpr 7596 . . . . 5 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
433adant2 1018 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
5 df-iplp 7492 . . . . 5 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
6 addclnq 7399 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
75, 6genpelvl 7536 . . . 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 411 . . 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 7493 . . . . . . . 8 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑣) ∧ 𝑥 = (𝑔 ·Q ))}, {𝑥Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑣) ∧ 𝑥 = (𝑔 ·Q ))}⟩)
10 mulclnq 7400 . . . . . . . 8 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
119, 10genpelvl 7536 . . . . . . 7 ((𝐴P𝐶P) → (𝑢 ∈ (1st ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧)))
12113adant2 1018 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑢 ∈ (1st ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧)))
1312anbi2d 464 . . . . 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 7493 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑣) ∧ 𝑓 = (𝑔 ·Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑣) ∧ 𝑓 = (𝑔 ·Q ))}⟩)
1514, 10genpelvl 7536 . . . . . . . 8 ((𝐴P𝐵P) → (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦)))
16153adant3 1019 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦)))
17 distrlem4prl 7608 . . . . . . . . . . . . . . 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 5901 . . . . . . . . . . . . . . . . . 18 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
1918eqeq2d 2201 . . . . . . . . . . . . . . . . 17 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
20 eleq1 2252 . . . . . . . . . . . . . . . . 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, 20biimtrdi 163 . . . . . . . . . . . . . . . 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 124 . . . . . . . . . . . . . . 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 157 . . . . . . . . . . . . . 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 367 . . . . . . . . . . . . 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 81 . . . . . . . . . . . 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 367 . . . . . . . . . . 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 78 . . . . . . . . . 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 2613 . . . . . . . . 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 2614 . . . . . . . 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 79 . . . . . . 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 150 . . . . . 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 254 . . . . 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 150 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ 𝑢 ∈ (1st ‘(𝐴 ·P 𝐶))) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))))
3433rexlimdvv 2614 . . 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 150 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
3635ssrdv 3176 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 104  wb 105  w3a 980   = wceq 1364  wcel 2160  wrex 2469  wss 3144  cfv 5232  (class class class)co 5892  1st c1st 6158   +Q cplq 7306   ·Q cmq 7307  Pcnp 7315   +P cpp 7317   ·P cmp 7318
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-1o 6436  df-2o 6437  df-oadd 6440  df-omul 6441  df-er 6554  df-ec 6556  df-qs 6560  df-ni 7328  df-pli 7329  df-mi 7330  df-lti 7331  df-plpq 7368  df-mpq 7369  df-enq 7371  df-nqqs 7372  df-plqqs 7373  df-mqqs 7374  df-1nqqs 7375  df-rq 7376  df-ltnqqs 7377  df-enq0 7448  df-nq0 7449  df-0nq0 7450  df-plq0 7451  df-mq0 7452  df-inp 7490  df-iplp 7492  df-imp 7493
This theorem is referenced by:  distrprg  7612
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