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Theorem distrlem5pr 10164
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 10157 . . . . 5 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
213adant3 1168 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
3 mulclpr 10157 . . . . 5 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
433adant2 1167 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
5 df-plp 10120 . . . . 5 +P = (𝑥P, 𝑦P ↦ {𝑓 ∣ ∃𝑔𝑥𝑦 𝑓 = (𝑔 +Q )})
6 addclnq 10082 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
75, 6genpelv 10137 . . . 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 581 . . 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 10121 . . . . . . . 8 ·P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑔𝑤𝑣 𝑥 = (𝑔 ·Q )})
10 mulclnq 10084 . . . . . . . 8 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
119, 10genpelv 10137 . . . . . . 7 ((𝐴P𝐶P) → (𝑢 ∈ (𝐴 ·P 𝐶) ↔ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧)))
12113adant2 1167 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑢 ∈ (𝐴 ·P 𝐶) ↔ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧)))
1312anbi2d 624 . . . . 5 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ 𝑢 ∈ (𝐴 ·P 𝐶)) ↔ (𝑣 ∈ (𝐴 ·P 𝐵) ∧ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧))))
14 df-mp 10121 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ {𝑓 ∣ ∃𝑔𝑤𝑣 𝑓 = (𝑔 ·Q )})
1514, 10genpelv 10137 . . . . . . . 8 ((𝐴P𝐵P) → (𝑣 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦)))
16153adant3 1168 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦)))
17 distrlem4pr 10163 . . . . . . . . . . . . . . 15 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
18 oveq12 6914 . . . . . . . . . . . . . . . . . 18 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
1918eqeq2d 2835 . . . . . . . . . . . . . . . . 17 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
20 eleq1 2894 . . . . . . . . . . . . . . . . 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 245 . . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . . . 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 239 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
2423exp4b 423 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶)) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
2524com3l 89 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶)) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
2625exp4b 423 . . . . . . . . . . 11 ((𝑥𝐴𝑦𝐵) → ((𝑓𝐴𝑧𝐶) → (𝑣 = (𝑥 ·Q 𝑦) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))))
2726com23 86 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐵) → (𝑣 = (𝑥 ·Q 𝑦) → ((𝑓𝐴𝑧𝐶) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))))
2827rexlimivv 3246 . . . . . . . . 9 (∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦) → ((𝑓𝐴𝑧𝐶) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))))
2928rexlimdvv 3247 . . . . . . . 8 (∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
3029com3r 87 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (∃𝑥𝐴𝑦𝐵 𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
3116, 30sylbid 232 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (𝐴 ·P 𝐵) → (∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))))
3231impd 400 . . . . 5 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ ∃𝑓𝐴𝑧𝐶 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))
3313, 32sylbid 232 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (𝐴 ·P 𝐵) ∧ 𝑢 ∈ (𝐴 ·P 𝐶)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)))))
3433rexlimdvv 3247 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑣 ∈ (𝐴 ·P 𝐵)∃𝑢 ∈ (𝐴 ·P 𝐶)𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
358, 34sylbid 232 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) → 𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶))))
3635ssrdv 3833 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 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wrex 3118  wss 3798  (class class class)co 6905   +Q cplq 9992   ·Q cmq 9993  Pcnp 9996   +P cpp 9998   ·P cmp 9999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-omul 7831  df-er 8009  df-ni 10009  df-pli 10010  df-mi 10011  df-lti 10012  df-plpq 10045  df-mpq 10046  df-ltpq 10047  df-enq 10048  df-nq 10049  df-erq 10050  df-plq 10051  df-mq 10052  df-1nq 10053  df-rq 10054  df-ltnq 10055  df-np 10118  df-plp 10120  df-mp 10121
This theorem is referenced by:  distrpr  10165
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