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Theorem distrlem5prl 6641
 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 6627 . . . . 5 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
213adant3 924 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
3 mulclpr 6627 . . . . 5 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
433adant2 923 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
5 df-iplp 6523 . . . . 5 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
6 addclnq 6430 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
75, 6genpelvl 6567 . . . 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 391 . . 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 6524 . . . . . . . 8 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑣) ∧ 𝑥 = (𝑔 ·Q ))}, {𝑥Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑣) ∧ 𝑥 = (𝑔 ·Q ))}⟩)
10 mulclnq 6431 . . . . . . . 8 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
119, 10genpelvl 6567 . . . . . . 7 ((𝐴P𝐶P) → (𝑢 ∈ (1st ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧)))
12113adant2 923 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑢 ∈ (1st ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧)))
1312anbi2d 437 . . . . 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 6524 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑣) ∧ 𝑓 = (𝑔 ·Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑣) ∧ 𝑓 = (𝑔 ·Q ))}⟩)
1514, 10genpelvl 6567 . . . . . . . 8 ((𝐴P𝐵P) → (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦)))
16153adant3 924 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦)))
17 distrlem4prl 6639 . . . . . . . . . . . . . . 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 5484 . . . . . . . . . . . . . . . . . 18 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
1918eqeq2d 2051 . . . . . . . . . . . . . . . . 17 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
20 eleq1 2100 . . . . . . . . . . . . . . . . 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 152 . . . . . . . . . . . . . . . 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 115 . . . . . . . . . . . . . . 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 146 . . . . . . . . . . . . . 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 349 . . . . . . . . . . . . 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 75 . . . . . . . . . . . 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 349 . . . . . . . . . . 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 72 . . . . . . . . . 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 2435 . . . . . . . . 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 2436 . . . . . . . 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 73 . . . . . . 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 139 . . . . . 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 242 . . . . 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 139 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ 𝑢 ∈ (1st ‘(𝐴 ·P 𝐶))) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))))
3433rexlimdvv 2436 . . 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 139 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
3635ssrdv 2948 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 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∃wrex 2304   ⊆ wss 2914  ‘cfv 4865  (class class class)co 5475  1st c1st 5728   +Q cplq 6337   ·Q cmq 6338  Pcnp 6346   +P cpp 6348   ·P cmp 6349 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4142  ax-setind 4232  ax-iinf 4274 This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4075  df-on 4077  df-suc 4080  df-iom 4277  df-xp 4314  df-rel 4315  df-cnv 4316  df-co 4317  df-dm 4318  df-rn 4319  df-res 4320  df-ima 4321  df-iota 4830  df-fun 4867  df-fn 4868  df-f 4869  df-f1 4870  df-fo 4871  df-f1o 4872  df-fv 4873  df-ov 5478  df-oprab 5479  df-mpt2 5480  df-1st 5730  df-2nd 5731  df-recs 5883  df-irdg 5920  df-1o 5964  df-2o 5965  df-oadd 5968  df-omul 5969  df-er 6069  df-ec 6071  df-qs 6075  df-ni 6359  df-pli 6360  df-mi 6361  df-lti 6362  df-plpq 6399  df-mpq 6400  df-enq 6402  df-nqqs 6403  df-plqqs 6404  df-mqqs 6405  df-1nqqs 6406  df-rq 6407  df-ltnqqs 6408  df-enq0 6479  df-nq0 6480  df-0nq0 6481  df-plq0 6482  df-mq0 6483  df-inp 6521  df-iplp 6523  df-imp 6524 This theorem is referenced by:  distrprg  6643
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