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Theorem distrlem1pr 11066
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 11059 . . . . 5 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
2 df-mp 11025 . . . . . 6 ·P = (𝑦P, 𝑧P ↦ {𝑓 ∣ ∃𝑔𝑦𝑧 𝑓 = (𝑔 ·Q )})
3 mulclnq 10988 . . . . . 6 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
42, 3genpelv 11041 . . . . 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 11024 . . . . . . . . . . 11 +P = (𝑤P, 𝑥P ↦ {𝑓 ∣ ∃𝑔𝑤𝑥 𝑓 = (𝑔 +Q )})
8 addclnq 10986 . . . . . . . . . . 11 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
97, 8genpelv 11041 . . . . . . . . . 10 ((𝐵P𝐶P) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦𝐵𝑧𝐶 𝑣 = (𝑦 +Q 𝑧)))
1093adant1 1130 . . . . . . . . 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 772 . . . . . . . . . . . 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 7440 . . . . . . . . . . . . . . 15 (𝑣 = (𝑦 +Q 𝑧) → (𝑥 ·Q 𝑣) = (𝑥 ·Q (𝑦 +Q 𝑧)))
1514eqeq2d 2747 . . . . . . . . . . . . . 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 11002 . . . . . . . . . . . . 13 (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))
1816, 17eqtrdi 2792 . . . . . . . . . . . 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 11061 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
21203adant3 1132 . . . . . . . . . . . . 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 11061 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
24233adant2 1131 . . . . . . . . . . . . 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 766 . . . . . . . . . . . . 13 (((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑦𝐵)
272, 3genpprecl 11042 . . . . . . . . . . . . . . . 16 ((𝐴P𝐵P) → ((𝑥𝐴𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵)))
28273adant3 1132 . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . 13 (((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑧𝐶)
332, 3genpprecl 11042 . . . . . . . . . . . . . . . 16 ((𝐴P𝐶P) → ((𝑥𝐴𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)))
34333adant2 1131 . . . . . . . . . . . . . . 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 11042 . . . . . . . . . . . . 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 838 . . . . . . . . . . 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 2840 . . . . . . . . . 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 3211 . . . . . . . 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 3211 . . 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 3988 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 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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