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Theorem distrlem1pr 9703
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 9696 . . . . 5 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
2 df-mp 9662 . . . . . 6 ·P = (𝑦P, 𝑧P ↦ {𝑓 ∣ ∃𝑔𝑦𝑧 𝑓 = (𝑔 ·Q )})
3 mulclnq 9625 . . . . . 6 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
42, 3genpelv 9678 . . . . 5 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣)))
51, 4sylan2 489 . . . 4 ((𝐴P ∧ (𝐵P𝐶P)) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣)))
653impb 1251 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣)))
7 df-plp 9661 . . . . . . . . . . 11 +P = (𝑤P, 𝑥P ↦ {𝑓 ∣ ∃𝑔𝑤𝑥 𝑓 = (𝑔 +Q )})
8 addclnq 9623 . . . . . . . . . . 11 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
97, 8genpelv 9678 . . . . . . . . . 10 ((𝐵P𝐶P) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦𝐵𝑧𝐶 𝑣 = (𝑦 +Q 𝑧)))
1093adant1 1071 . . . . . . . . 9 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦𝐵𝑧𝐶 𝑣 = (𝑦 +Q 𝑧)))
1110adantr 479 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦𝐵𝑧𝐶 𝑣 = (𝑦 +Q 𝑧)))
12 simprr 791 . . . . . . . . . . . 12 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → 𝑤 = (𝑥 ·Q 𝑣))
13 simpr 475 . . . . . . . . . . . 12 (((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑣 = (𝑦 +Q 𝑧))
14 oveq2 6535 . . . . . . . . . . . . . . 15 (𝑣 = (𝑦 +Q 𝑧) → (𝑥 ·Q 𝑣) = (𝑥 ·Q (𝑦 +Q 𝑧)))
1514eqeq2d 2619 . . . . . . . . . . . . . 14 (𝑣 = (𝑦 +Q 𝑧) → (𝑤 = (𝑥 ·Q 𝑣) ↔ 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧))))
1615biimpac 501 . . . . . . . . . . . . 13 ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧)))
17 distrnq 9639 . . . . . . . . . . . . 13 (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))
1816, 17syl6eq 2659 . . . . . . . . . . . 12 ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
1912, 13, 18syl2an 492 . . . . . . . . . . 11 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
20 mulclpr 9698 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
21203adant3 1073 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
2221ad2antrr 757 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐵) ∈ P)
23 mulclpr 9698 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
24233adant2 1072 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
2524ad2antrr 757 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐶) ∈ P)
26 simpll 785 . . . . . . . . . . . . 13 (((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑦𝐵)
272, 3genpprecl 9679 . . . . . . . . . . . . . . . 16 ((𝐴P𝐵P) → ((𝑥𝐴𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵)))
28273adant3 1073 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥𝐴𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵)))
2928impl 647 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥𝐴) ∧ 𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
3029adantlrr 752 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑦𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
3126, 30sylan2 489 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
32 simplr 787 . . . . . . . . . . . . 13 (((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑧𝐶)
332, 3genpprecl 9679 . . . . . . . . . . . . . . . 16 ((𝐴P𝐶P) → ((𝑥𝐴𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)))
34333adant2 1072 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P𝐶P) → ((𝑥𝐴𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)))
3534impl 647 . . . . . . . . . . . . . 14 ((((𝐴P𝐵P𝐶P) ∧ 𝑥𝐴) ∧ 𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
3635adantlrr 752 . . . . . . . . . . . . 13 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑧𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
3732, 36sylan2 489 . . . . . . . . . . . 12 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
387, 8genpprecl 9679 . . . . . . . . . . . . 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 443 . . . . . . . . . . . 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 1318 . . . . . . . . . . 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 2687 . . . . . . . . . 10 ((((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦𝐵𝑧𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
4241exp32 628 . . . . . . . . 9 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → ((𝑦𝐵𝑧𝐶) → (𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
4342rexlimdvv 3018 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → (∃𝑦𝐵𝑧𝐶 𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
4411, 43sylbid 228 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ (𝑥𝐴𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (𝐵 +P 𝐶) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
4544exp32 628 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑥𝐴 → (𝑤 = (𝑥 ·Q 𝑣) → (𝑣 ∈ (𝐵 +P 𝐶) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
4645com34 88 . . . . 5 ((𝐴P𝐵P𝐶P) → (𝑥𝐴 → (𝑣 ∈ (𝐵 +P 𝐶) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
4746impd 445 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
4847rexlimdvv 3018 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑥𝐴𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
496, 48sylbid 228 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
5049ssrdv 3573 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wrex 2896  wss 3539  (class class class)co 6527   +Q cplq 9533   ·Q cmq 9534  Pcnp 9537   +P cpp 9539   ·P cmp 9540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-omul 7429  df-er 7606  df-ni 9550  df-pli 9551  df-mi 9552  df-lti 9553  df-plpq 9586  df-mpq 9587  df-ltpq 9588  df-enq 9589  df-nq 9590  df-erq 9591  df-plq 9592  df-mq 9593  df-1nq 9594  df-rq 9595  df-ltnq 9596  df-np 9659  df-plp 9661  df-mp 9662
This theorem is referenced by:  distrpr  9706
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