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Theorem distrlem5pr 11068
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 11061 . . . . 5 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
213adant3 1132 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
3 mulclpr 11061 . . . 4 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
4 df-plp 11024 . . . . 5 +P = (𝑥P, 𝑦P ↦ {𝑓 ∣ ∃𝑔𝑥𝑦 𝑓 = (𝑔 +Q )})
5 addclnq 10986 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
64, 5genpelv 11041 . . . 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 1345 . . 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 11025 . . . . . . . 8 ·P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑔𝑤𝑣 𝑥 = (𝑔 ·Q )})
9 mulclnq 10988 . . . . . . . 8 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
108, 9genpelv 11041 . . . . . . 7 ((𝐴P𝐶P) → (𝑢 ∈ (𝐴 ·P 𝐶) ↔ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧)))
11103adant2 1131 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑢 ∈ (𝐴 ·P 𝐶) ↔ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧)))
1211anbi2d 630 . . . . 5 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ 𝑢 ∈ (𝐴 ·P 𝐶)) ↔ (𝑣 ∈ (𝐴 ·P 𝐵) ∧ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧))))
13 df-mp 11025 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ {𝑓 ∣ ∃𝑔𝑤𝑣 𝑓 = (𝑔 ·Q )})
1413, 9genpelv 11041 . . . . . . . 8 ((𝐴P𝐵P) → (𝑣 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦)))
15143adant3 1132 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦)))
16 distrlem4pr 11067 . . . . . . . . . . . . . . 15 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
17 oveq12 7441 . . . . . . . . . . . . . . . . . 18 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
1817eqeq2d 2747 . . . . . . . . . . . . . . . . 17 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
19 eleq1 2828 . . . . . . . . . . . . . . . . 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, 19biimtrdi 253 . . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . 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 3200 . . . . . . . . 9 (∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦) → ((𝑓𝐴𝑧𝐶) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))))
2827rexlimdvv 3211 . . . . . . . 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 240 . . . . . 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 240 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ 𝑢 ∈ (𝐴 ·P 𝐶)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))
3332rexlimdvv 3211 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑣 ∈ (𝐴 ·P 𝐵)∃𝑢 ∈ (𝐴 ·P 𝐶)𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
347, 33sylbid 240 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
3534ssrdv 3988 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 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wrex 3069  wss 3950  (class class class)co 7432   +Q cplq 10896   ·Q cmq 10897  Pcnp 10900   +P cpp 10902   ·P cmp 10903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-omul 8512  df-er 8746  df-ni 10913  df-pli 10914  df-mi 10915  df-lti 10916  df-plpq 10949  df-mpq 10950  df-ltpq 10951  df-enq 10952  df-nq 10953  df-erq 10954  df-plq 10955  df-mq 10956  df-1nq 10957  df-rq 10958  df-ltnq 10959  df-np 11022  df-plp 11024  df-mp 11025
This theorem is referenced by:  distrpr  11069
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