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Theorem distrlem5prl 7242
 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 7228 . . . . 5 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
213adant3 966 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
3 mulclpr 7228 . . . . 5 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
433adant2 965 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
5 df-iplp 7124 . . . . 5 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
6 addclnq 7031 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
75, 6genpelvl 7168 . . . 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 404 . . 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 7125 . . . . . . . 8 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑣) ∧ 𝑥 = (𝑔 ·Q ))}, {𝑥Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑣) ∧ 𝑥 = (𝑔 ·Q ))}⟩)
10 mulclnq 7032 . . . . . . . 8 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
119, 10genpelvl 7168 . . . . . . 7 ((𝐴P𝐶P) → (𝑢 ∈ (1st ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧)))
12113adant2 965 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑢 ∈ (1st ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧)))
1312anbi2d 453 . . . . 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 7125 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑣) ∧ 𝑓 = (𝑔 ·Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑣) ∧ 𝑓 = (𝑔 ·Q ))}⟩)
1514, 10genpelvl 7168 . . . . . . . 8 ((𝐴P𝐵P) → (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦)))
16153adant3 966 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦)))
17 distrlem4prl 7240 . . . . . . . . . . . . . . 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 5699 . . . . . . . . . . . . . . . . . 18 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
1918eqeq2d 2106 . . . . . . . . . . . . . . . . 17 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
20 eleq1 2157 . . . . . . . . . . . . . . . . 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 162 . . . . . . . . . . . . . . . 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 123 . . . . . . . . . . . . . . 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 156 . . . . . . . . . . . . . 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 360 . . . . . . . . . . . . 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 360 . . . . . . . . . . 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 2508 . . . . . . . . 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 2509 . . . . . . . 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 149 . . . . . 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 252 . . . . 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 149 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ 𝑢 ∈ (1st ‘(𝐴 ·P 𝐶))) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))))
3433rexlimdvv 2509 . . 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 149 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
3635ssrdv 3045 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 103   ↔ wb 104   ∧ w3a 927   = wceq 1296   ∈ wcel 1445  ∃wrex 2371   ⊆ wss 3013  ‘cfv 5049  (class class class)co 5690  1st c1st 5947   +Q cplq 6938   ·Q cmq 6939  Pcnp 6947   +P cpp 6949   ·P cmp 6950 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431 This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-2o 6220  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009  df-enq0 7080  df-nq0 7081  df-0nq0 7082  df-plq0 7083  df-mq0 7084  df-inp 7122  df-iplp 7124  df-imp 7125 This theorem is referenced by:  distrprg  7244
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