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Theorem distrlem5pr 9728
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 9721 . . . . 5 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
213adant3 1074 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
3 mulclpr 9721 . . . . 5 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
433adant2 1073 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
5 df-plp 9684 . . . . 5 +P = (𝑥P, 𝑦P ↦ {𝑓 ∣ ∃𝑔𝑥𝑦 𝑓 = (𝑔 +Q )})
6 addclnq 9646 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
75, 6genpelv 9701 . . . 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)))
82, 4, 7syl2anc 691 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ↔ ∃𝑣 ∈ (𝐴 ·P 𝐵)∃𝑢 ∈ (𝐴 ·P 𝐶)𝑤 = (𝑣 +Q 𝑢)))
9 df-mp 9685 . . . . . . . 8 ·P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑔𝑤𝑣 𝑥 = (𝑔 ·Q )})
10 mulclnq 9648 . . . . . . . 8 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
119, 10genpelv 9701 . . . . . . 7 ((𝐴P𝐶P) → (𝑢 ∈ (𝐴 ·P 𝐶) ↔ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧)))
12113adant2 1073 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑢 ∈ (𝐴 ·P 𝐶) ↔ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧)))
1312anbi2d 736 . . . . 5 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ 𝑢 ∈ (𝐴 ·P 𝐶)) ↔ (𝑣 ∈ (𝐴 ·P 𝐵) ∧ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧))))
14 df-mp 9685 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ {𝑓 ∣ ∃𝑔𝑤𝑣 𝑓 = (𝑔 ·Q )})
1514, 10genpelv 9701 . . . . . . . 8 ((𝐴P𝐵P) → (𝑣 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦)))
16153adant3 1074 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦)))
17 distrlem4pr 9727 . . . . . . . . . . . . . . 15 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
18 oveq12 6558 . . . . . . . . . . . . . . . . . 18 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
1918eqeq2d 2620 . . . . . . . . . . . . . . . . 17 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
20 eleq1 2676 . . . . . . . . . . . . . . . . 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))))
2119, 20syl6bi 242 . . . . . . . . . . . . . . . 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)))))
2221imp 444 . . . . . . . . . . . . . . 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))))
2317, 22syl5ibrcom 236 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
2423exp4b 630 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶)) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
2524com3l 87 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶)) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
2625exp4b 630 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ((𝑓𝐴𝑧𝐶) → (𝑣 = (𝑥 ·Q 𝑦) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))))
2726com23 84 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (𝑣 = (𝑥 ·Q 𝑦) → ((𝑓𝐴𝑧𝐶) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))))
2827rexlimivv 3018 . . . . . . . . 9 (∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦) → ((𝑓𝐴𝑧𝐶) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))))
2928rexlimdvv 3019 . . . . . . . 8 (∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
3029com3r 85 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
3116, 30sylbid 229 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (𝐴 ·P 𝐵) → (∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
3231impd 446 . . . . 5 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))
3313, 32sylbid 229 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ 𝑢 ∈ (𝐴 ·P 𝐶)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))
3433rexlimdvv 3019 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑣 ∈ (𝐴 ·P 𝐵)∃𝑢 ∈ (𝐴 ·P 𝐶)𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
358, 34sylbid 229 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
3635ssrdv 3574 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 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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