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Theorem distrlem1pr 9726
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 9719 . . . . 5 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
2 df-mp 9685 . . . . . 6 ·P = (𝑦P, 𝑧P ↦ {𝑓 ∣ ∃𝑔𝑦𝑧 𝑓 = (𝑔 ·Q )})
3 mulclnq 9648 . . . . . 6 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
42, 3genpelv 9701 . . . . 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 1252 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣)))
7 df-plp 9684 . . . . . . . . . . 11 +P = (𝑤P, 𝑥P ↦ {𝑓 ∣ ∃𝑔𝑤𝑥 𝑓 = (𝑔 +Q )})
8 addclnq 9646 . . . . . . . . . . 11 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
97, 8genpelv 9701 . . . . . . . . . 10 ((𝐵P𝐶P) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦𝐵𝑧𝐶 𝑣 = (𝑦 +Q 𝑧)))
1093adant1 1072 . . . . . . . . 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 792 . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . . 15 (𝑣 = (𝑦 +Q 𝑧) → (𝑥 ·Q 𝑣) = (𝑥 ·Q (𝑦 +Q 𝑧)))
1514eqeq2d 2620 . . . . . . . . . . . . . 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 9662 . . . . . . . . . . . . 13 (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))
1816, 17syl6eq 2660 . . . . . . . . . . . 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 9721 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
21203adant3 1074 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
2221ad2antrr 758 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐵) ∈ P)
23 mulclpr 9721 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
24233adant2 1073 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
2524ad2antrr 758 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐶) ∈ P)
26 simpll 786 . . . . . . . . . . . . 13 (((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑦𝐵)
272, 3genpprecl 9702 . . . . . . . . . . . . . . . 16 ((𝐴P𝐵P) → ((𝑥𝐴𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵)))
28273adant3 1074 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥𝐴𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵)))
2928impl 648 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥𝐴) ∧ 𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
3029adantlrr 753 . . . . . . . . . . . . 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 788 . . . . . . . . . . . . 13 (((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑧𝐶)
332, 3genpprecl 9702 . . . . . . . . . . . . . . . 16 ((𝐴P𝐶P) → ((𝑥𝐴𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)))
34333adant2 1073 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥𝐴𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)))
3534impl 648 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥𝐴) ∧ 𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
3635adantlrr 753 . . . . . . . . . . . . 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 9702 . . . . . . . . . . . . 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 1319 . . . . . . . . . . 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 2688 . . . . . . . . . 10 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
4241exp32 629 . . . . . . . . 9 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → ((𝑦𝐵𝑧𝐶) → (𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
4342rexlimdvv 3019 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → (∃𝑦𝐵𝑧𝐶 𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
4411, 43sylbid 229 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (𝐵 +P 𝐶) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
4544exp32 629 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑥𝐴 → (𝑤 = (𝑥 ·Q 𝑣) → (𝑣 ∈ (𝐵 +P 𝐶) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
4645com34 89 . . . . 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 3019 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
496, 48sylbid 229 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
5049ssrdv 3574 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wrex 2897  wss 3540  (class class class)co 6549   +Q cplq 9556   ·Q cmq 9557  Pcnp 9560   +P cpp 9562   ·P cmp 9563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-mq 9616  df-1nq 9617  df-rq 9618  df-ltnq 9619  df-np 9682  df-plp 9684  df-mp 9685
This theorem is referenced by:  distrpr  9729
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