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Theorem distrlem5prl 7406
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem5prl ((𝐴P𝐵P𝐶P) → (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))

Proof of Theorem distrlem5prl
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulclpr 7392 . . . . 5 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
213adant3 1001 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
3 mulclpr 7392 . . . . 5 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
433adant2 1000 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
5 df-iplp 7288 . . . . 5 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
6 addclnq 7195 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
75, 6genpelvl 7332 . . . 4 (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → (𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ↔ ∃𝑣 ∈ (1st ‘(𝐴 ·P 𝐵))∃𝑢 ∈ (1st ‘(𝐴 ·P 𝐶))𝑤 = (𝑣 +Q 𝑢)))
82, 4, 7syl2anc 408 . . 3 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ↔ ∃𝑣 ∈ (1st ‘(𝐴 ·P 𝐵))∃𝑢 ∈ (1st ‘(𝐴 ·P 𝐶))𝑤 = (𝑣 +Q 𝑢)))
9 df-imp 7289 . . . . . . . 8 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑣) ∧ 𝑥 = (𝑔 ·Q ))}, {𝑥Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑣) ∧ 𝑥 = (𝑔 ·Q ))}⟩)
10 mulclnq 7196 . . . . . . . 8 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
119, 10genpelvl 7332 . . . . . . 7 ((𝐴P𝐶P) → (𝑢 ∈ (1st ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧)))
12113adant2 1000 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑢 ∈ (1st ‘(𝐴 ·P 𝐶)) ↔ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧)))
1312anbi2d 459 . . . . 5 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ 𝑢 ∈ (1st ‘(𝐴 ·P 𝐶))) ↔ (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧))))
14 df-imp 7289 . . . . . . . . 9 ·P = (𝑤P, 𝑣P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑤) ∧ ∈ (1st𝑣) ∧ 𝑓 = (𝑔 ·Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑤) ∧ ∈ (2nd𝑣) ∧ 𝑓 = (𝑔 ·Q ))}⟩)
1514, 10genpelvl 7332 . . . . . . . 8 ((𝐴P𝐵P) → (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦)))
16153adant3 1001 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ↔ ∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦)))
17 distrlem4prl 7404 . . . . . . . . . . . . . . 15 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
18 oveq12 5783 . . . . . . . . . . . . . . . . . 18 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑣 +Q 𝑢) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
1918eqeq2d 2151 . . . . . . . . . . . . . . . . 17 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
20 eleq1 2202 . . . . . . . . . . . . . . . . 17 (𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → (𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
2119, 20syl6bi 162 . . . . . . . . . . . . . . . 16 ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → (𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))))
2221imp 123 . . . . . . . . . . . . . . 15 (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → (𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
2317, 22syl5ibrcom 156 . . . . . . . . . . . . . 14 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) ∧ 𝑤 = (𝑣 +Q 𝑢)) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
2423exp4b 364 . . . . . . . . . . . . 13 ((𝐴P𝐵P𝐶P) → (((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶))) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
2524com3l 81 . . . . . . . . . . . 12 (((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶))) → ((𝑣 = (𝑥 ·Q 𝑦) ∧ 𝑢 = (𝑓 ·Q 𝑧)) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
2625exp4b 364 . . . . . . . . . . 11 ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) → ((𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)) → (𝑣 = (𝑥 ·Q 𝑦) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))))
2726com23 78 . . . . . . . . . 10 ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) → (𝑣 = (𝑥 ·Q 𝑦) → ((𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))))
2827rexlimivv 2555 . . . . . . . . 9 (∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦) → ((𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)) → (𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))))))
2928rexlimdvv 2556 . . . . . . . 8 (∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧) → ((𝐴P𝐵P𝐶P) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
3029com3r 79 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (∃𝑥 ∈ (1st𝐴)∃𝑦 ∈ (1st𝐵)𝑣 = (𝑥 ·Q 𝑦) → (∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
3116, 30sylbid 149 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) → (∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))))
3231impd 252 . . . . 5 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ ∃𝑓 ∈ (1st𝐴)∃𝑧 ∈ (1st𝐶)𝑢 = (𝑓 ·Q 𝑧)) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))))
3313, 32sylbid 149 . . . 4 ((𝐴P𝐵P𝐶P) → ((𝑣 ∈ (1st ‘(𝐴 ·P 𝐵)) ∧ 𝑢 ∈ (1st ‘(𝐴 ·P 𝐶))) → (𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))))
3433rexlimdvv 2556 . . 3 ((𝐴P𝐵P𝐶P) → (∃𝑣 ∈ (1st ‘(𝐴 ·P 𝐵))∃𝑢 ∈ (1st ‘(𝐴 ·P 𝐶))𝑤 = (𝑣 +Q 𝑢) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
358, 34sylbid 149 . 2 ((𝐴P𝐵P𝐶P) → (𝑤 ∈ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) → 𝑤 ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
3635ssrdv 3103 1 ((𝐴P𝐵P𝐶P) → (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wcel 1480  wrex 2417  wss 3071  cfv 5123  (class class class)co 5774  1st c1st 6036   +Q cplq 7102   ·Q cmq 7103  Pcnp 7111   +P cpp 7113   ·P cmp 7114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-enq0 7244  df-nq0 7245  df-0nq0 7246  df-plq0 7247  df-mq0 7248  df-inp 7286  df-iplp 7288  df-imp 7289
This theorem is referenced by:  distrprg  7408
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