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Theorem distrlem1pr 9885
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 9878 . . . . 5 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
2 df-mp 9844 . . . . . 6 ·P = (𝑦P, 𝑧P ↦ {𝑓 ∣ ∃𝑔𝑦𝑧 𝑓 = (𝑔 ·Q )})
3 mulclnq 9807 . . . . . 6 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
42, 3genpelv 9860 . . . . 5 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣)))
51, 4sylan2 490 . . . 4 ((𝐴P ∧ (𝐵P𝐶P)) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣)))
653impb 1279 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣)))
7 df-plp 9843 . . . . . . . . . . 11 +P = (𝑤P, 𝑥P ↦ {𝑓 ∣ ∃𝑔𝑤𝑥 𝑓 = (𝑔 +Q )})
8 addclnq 9805 . . . . . . . . . . 11 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
97, 8genpelv 9860 . . . . . . . . . 10 ((𝐵P𝐶P) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦𝐵𝑧𝐶 𝑣 = (𝑦 +Q 𝑧)))
1093adant1 1099 . . . . . . . . 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 811 . . . . . . . . . . . 12 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → 𝑤 = (𝑥 ·Q 𝑣))
13 simpr 476 . . . . . . . . . . . 12 (((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑣 = (𝑦 +Q 𝑧))
14 oveq2 6698 . . . . . . . . . . . . . . 15 (𝑣 = (𝑦 +Q 𝑧) → (𝑥 ·Q 𝑣) = (𝑥 ·Q (𝑦 +Q 𝑧)))
1514eqeq2d 2661 . . . . . . . . . . . . . 14 (𝑣 = (𝑦 +Q 𝑧) → (𝑤 = (𝑥 ·Q 𝑣) ↔ 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧))))
1615biimpac 502 . . . . . . . . . . . . 13 ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧)))
17 distrnq 9821 . . . . . . . . . . . . 13 (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))
1816, 17syl6eq 2701 . . . . . . . . . . . 12 ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
1912, 13, 18syl2an 493 . . . . . . . . . . 11 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
20 mulclpr 9880 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
21203adant3 1101 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
2221ad2antrr 762 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐵) ∈ P)
23 mulclpr 9880 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
24233adant2 1100 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
2524ad2antrr 762 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐶) ∈ P)
26 simpll 805 . . . . . . . . . . . . 13 (((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑦𝐵)
272, 3genpprecl 9861 . . . . . . . . . . . . . . . 16 ((𝐴P𝐵P) → ((𝑥𝐴𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵)))
28273adant3 1101 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥𝐴𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵)))
2928impl 649 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥𝐴) ∧ 𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
3029adantlrr 757 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
3126, 30sylan2 490 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
32 simplr 807 . . . . . . . . . . . . 13 (((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑧𝐶)
332, 3genpprecl 9861 . . . . . . . . . . . . . . . 16 ((𝐴P𝐶P) → ((𝑥𝐴𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)))
34333adant2 1100 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥𝐴𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)))
3534impl 649 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥𝐴) ∧ 𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
3635adantlrr 757 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
3732, 36sylan2 490 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
387, 8genpprecl 9861 . . . . . . . . . . . . 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 444 . . . . . . . . . . . 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 1367 . . . . . . . . . . 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 2730 . . . . . . . . . 10 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
4241exp32 630 . . . . . . . . 9 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → ((𝑦𝐵𝑧𝐶) → (𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
4342rexlimdvv 3066 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → (∃𝑦𝐵𝑧𝐶 𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
4411, 43sylbid 230 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (𝐵 +P 𝐶) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
4544exp32 630 . . . . . 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 446 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
4847rexlimdvv 3066 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
496, 48sylbid 230 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
5049ssrdv 3642 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  wss 3607  (class class class)co 6690   +Q cplq 9715   ·Q cmq 9716  Pcnp 9719   +P cpp 9721   ·P cmp 9722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-plp 9843  df-mp 9844
This theorem is referenced by:  distrpr  9888
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