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Theorem distrlem1pr 11063
Description: Lemma for distributive law for positive reals. (Contributed by NM, 1-May-1996.) (Revised by Mario Carneiro, 13-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
distrlem1pr ((𝐴P𝐵P𝐶P) → (𝐴 ·P (𝐵 +P 𝐶)) ⊆ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))

Proof of Theorem distrlem1pr
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addclpr 11056 . . . . 5 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
2 df-mp 11022 . . . . . 6 ·P = (𝑦P, 𝑧P ↦ {𝑓 ∣ ∃𝑔𝑦𝑧 𝑓 = (𝑔 ·Q )})
3 mulclnq 10985 . . . . . 6 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
42, 3genpelv 11038 . . . . 5 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣)))
51, 4sylan2 593 . . . 4 ((𝐴P ∧ (𝐵P𝐶P)) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣)))
653impb 1114 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣)))
7 df-plp 11021 . . . . . . . . . . 11 +P = (𝑤P, 𝑥P ↦ {𝑓 ∣ ∃𝑔𝑤𝑥 𝑓 = (𝑔 +Q )})
8 addclnq 10983 . . . . . . . . . . 11 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
97, 8genpelv 11038 . . . . . . . . . 10 ((𝐵P𝐶P) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦𝐵𝑧𝐶 𝑣 = (𝑦 +Q 𝑧)))
1093adant1 1129 . . . . . . . . 9 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦𝐵𝑧𝐶 𝑣 = (𝑦 +Q 𝑧)))
1110adantr 480 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦𝐵𝑧𝐶 𝑣 = (𝑦 +Q 𝑧)))
12 simprr 773 . . . . . . . . . . . 12 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → 𝑤 = (𝑥 ·Q 𝑣))
13 simpr 484 . . . . . . . . . . . 12 (((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑣 = (𝑦 +Q 𝑧))
14 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑣 = (𝑦 +Q 𝑧) → (𝑥 ·Q 𝑣) = (𝑥 ·Q (𝑦 +Q 𝑧)))
1514eqeq2d 2746 . . . . . . . . . . . . . 14 (𝑣 = (𝑦 +Q 𝑧) → (𝑤 = (𝑥 ·Q 𝑣) ↔ 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧))))
1615biimpac 478 . . . . . . . . . . . . 13 ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧)))
17 distrnq 10999 . . . . . . . . . . . . 13 (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))
1816, 17eqtrdi 2791 . . . . . . . . . . . 12 ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
1912, 13, 18syl2an 596 . . . . . . . . . . 11 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
20 mulclpr 11058 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
21203adant3 1131 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
2221ad2antrr 726 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐵) ∈ P)
23 mulclpr 11058 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
24233adant2 1130 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
2524ad2antrr 726 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐶) ∈ P)
26 simpll 767 . . . . . . . . . . . . 13 (((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑦𝐵)
272, 3genpprecl 11039 . . . . . . . . . . . . . . . 16 ((𝐴P𝐵P) → ((𝑥𝐴𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵)))
28273adant3 1131 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥𝐴𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵)))
2928impl 455 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥𝐴) ∧ 𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
3029adantlrr 721 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
3126, 30sylan2 593 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
32 simplr 769 . . . . . . . . . . . . 13 (((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑧𝐶)
332, 3genpprecl 11039 . . . . . . . . . . . . . . . 16 ((𝐴P𝐶P) → ((𝑥𝐴𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)))
34333adant2 1130 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥𝐴𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)))
3534impl 455 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥𝐴) ∧ 𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
3635adantlrr 721 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
3732, 36sylan2 593 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
387, 8genpprecl 11039 . . . . . . . . . . . . 13 (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → (((𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵) ∧ (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
3938imp 406 . . . . . . . . . . . 12 ((((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) ∧ ((𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵) ∧ (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
4022, 25, 31, 37, 39syl22anc 839 . . . . . . . . . . 11 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
4119, 40eqeltrd 2839 . . . . . . . . . 10 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
4241exp32 420 . . . . . . . . 9 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → ((𝑦𝐵𝑧𝐶) → (𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
4342rexlimdvv 3210 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → (∃𝑦𝐵𝑧𝐶 𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
4411, 43sylbid 240 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (𝐵 +P 𝐶) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
4544exp32 420 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑥𝐴 → (𝑤 = (𝑥 ·Q 𝑣) → (𝑣 ∈ (𝐵 +P 𝐶) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
4645com34 91 . . . . 5 ((𝐴P𝐵P𝐶P) → (𝑥𝐴 → (𝑣 ∈ (𝐵 +P 𝐶) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
4746impd 410 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
4847rexlimdvv 3210 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
496, 48sylbid 240 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
5049ssrdv 4001 1 ((𝐴P𝐵P𝐶P) → (𝐴 ·P (𝐵 +P 𝐶)) ⊆ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  wss 3963  (class class class)co 7431   +Q cplq 10893   ·Q cmq 10894  Pcnp 10897   +P cpp 10899   ·P cmp 10900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-ni 10910  df-pli 10911  df-mi 10912  df-lti 10913  df-plpq 10946  df-mpq 10947  df-ltpq 10948  df-enq 10949  df-nq 10950  df-erq 10951  df-plq 10952  df-mq 10953  df-1nq 10954  df-rq 10955  df-ltnq 10956  df-np 11019  df-plp 11021  df-mp 11022
This theorem is referenced by:  distrpr  11066
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