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Theorem distrlem5pr 10714
Description: Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
distrlem5pr ((𝐴P𝐵P𝐶P) → ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ⊆ (𝐴 ·P (𝐵 +P 𝐶)))

Proof of Theorem distrlem5pr
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulclpr 10707 . . . . 5 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
213adant3 1130 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
3 mulclpr 10707 . . . 4 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
4 df-plp 10670 . . . . 5 +P = (𝑥P, 𝑦P ↦ {𝑓 ∣ ∃𝑔𝑥𝑦 𝑓 = (𝑔 +Q )})
5 addclnq 10632 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
64, 5genpelv 10687 . . . 4 (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → (𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ↔ ∃𝑣 ∈ (𝐴 ·P 𝐵)∃𝑢 ∈ (𝐴 ·P 𝐶)𝑤 = (𝑣 +Q 𝑢)))
72, 3, 63imp3i2an 1343 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ↔ ∃𝑣 ∈ (𝐴 ·P 𝐵)∃𝑢 ∈ (𝐴 ·P 𝐶)𝑤 = (𝑣 +Q 𝑢)))
8 df-mp 10671 . . . . . . . 8 ·P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑔𝑤𝑣 𝑥 = (𝑔 ·Q )})
9 mulclnq 10634 . . . . . . . 8 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
108, 9genpelv 10687 . . . . . . 7 ((𝐴P𝐶P) → (𝑢 ∈ (𝐴 ·P 𝐶) ↔ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧)))
11103adant2 1129 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑢 ∈ (𝐴 ·P 𝐶) ↔ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧)))
1211anbi2d 628 . . . . 5 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ 𝑢 ∈ (𝐴 ·P 𝐶)) ↔ (𝑣 ∈ (𝐴 ·P 𝐵) ∧ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧))))
13 df-mp 10671 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ {𝑓 ∣ ∃𝑔𝑤𝑣 𝑓 = (𝑔 ·Q )})
1413, 9genpelv 10687 . . . . . . . 8 ((𝐴P𝐵P) → (𝑣 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦)))
15143adant3 1130 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦)))
16 distrlem4pr 10713 . . . . . . . . . . . . . . 15 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
17 oveq12 7264 . . . . . . . . . . . . . . . . . 18 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
1817eqeq2d 2749 . . . . . . . . . . . . . . . . 17 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
19 eleq1 2826 . . . . . . . . . . . . . . . . 17 (𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
2018, 19syl6bi 252 . . . . . . . . . . . . . . . 16 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))))
2120imp 406 . . . . . . . . . . . . . . 15 (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
2216, 21syl5ibrcom 246 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
2322exp4b 430 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶)) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
2423com3l 89 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶)) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
2524exp4b 430 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ((𝑓𝐴𝑧𝐶) → (𝑣 = (𝑥 ·Q 𝑦) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))))
2625com23 86 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (𝑣 = (𝑥 ·Q 𝑦) → ((𝑓𝐴𝑧𝐶) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))))
2726rexlimivv 3220 . . . . . . . . 9 (∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦) → ((𝑓𝐴𝑧𝐶) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))))
2827rexlimdvv 3221 . . . . . . . 8 (∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
2928com3r 87 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
3015, 29sylbid 239 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (𝐴 ·P 𝐵) → (∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
3130impd 410 . . . . 5 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))
3212, 31sylbid 239 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ 𝑢 ∈ (𝐴 ·P 𝐶)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))
3332rexlimdvv 3221 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑣 ∈ (𝐴 ·P 𝐵)∃𝑢 ∈ (𝐴 ·P 𝐶)𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
347, 33sylbid 239 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
3534ssrdv 3923 1 ((𝐴P𝐵P𝐶P) → ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ⊆ (𝐴 ·P (𝐵 +P 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1539  wcel 2108  wrex 3064  wss 3883  (class class class)co 7255   +Q cplq 10542   ·Q cmq 10543  Pcnp 10546   +P cpp 10548   ·P cmp 10549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-omul 8272  df-er 8456  df-ni 10559  df-pli 10560  df-mi 10561  df-lti 10562  df-plpq 10595  df-mpq 10596  df-ltpq 10597  df-enq 10598  df-nq 10599  df-erq 10600  df-plq 10601  df-mq 10602  df-1nq 10603  df-rq 10604  df-ltnq 10605  df-np 10668  df-plp 10670  df-mp 10671
This theorem is referenced by:  distrpr  10715
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