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Theorem distrlem5pru 6563
 Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem5pru ((A P B P 𝐶 P) → (2nd ‘((A ·P B) +P (A ·P 𝐶))) ⊆ (2nd ‘(A ·P (B +P 𝐶))))

Proof of Theorem distrlem5pru
Dummy variables x y z w v u f g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulclpr 6553 . . . . 5 ((A P B P) → (A ·P B) P)
213adant3 923 . . . 4 ((A P B P 𝐶 P) → (A ·P B) P)
3 mulclpr 6553 . . . . 5 ((A P 𝐶 P) → (A ·P 𝐶) P)
433adant2 922 . . . 4 ((A P B P 𝐶 P) → (A ·P 𝐶) P)
5 df-iplp 6451 . . . . 5 +P = (x P, y P ↦ ⟨{f Qg Q Q (g (1stx) (1sty) f = (g +Q ))}, {f Qg Q Q (g (2ndx) (2ndy) f = (g +Q ))}⟩)
6 addclnq 6359 . . . . 5 ((g Q Q) → (g +Q ) Q)
75, 6genpelvu 6496 . . . 4 (((A ·P B) P (A ·P 𝐶) P) → (w (2nd ‘((A ·P B) +P (A ·P 𝐶))) ↔ v (2nd ‘(A ·P B))u (2nd ‘(A ·P 𝐶))w = (v +Q u)))
82, 4, 7syl2anc 391 . . 3 ((A P B P 𝐶 P) → (w (2nd ‘((A ·P B) +P (A ·P 𝐶))) ↔ v (2nd ‘(A ·P B))u (2nd ‘(A ·P 𝐶))w = (v +Q u)))
9 df-imp 6452 . . . . . . . 8 ·P = (w P, v P ↦ ⟨{x Qg Q Q (g (1stw) (1stv) x = (g ·Q ))}, {x Qg Q Q (g (2ndw) (2ndv) x = (g ·Q ))}⟩)
10 mulclnq 6360 . . . . . . . 8 ((g Q Q) → (g ·Q ) Q)
119, 10genpelvu 6496 . . . . . . 7 ((A P 𝐶 P) → (u (2nd ‘(A ·P 𝐶)) ↔ f (2ndA)z (2nd𝐶)u = (f ·Q z)))
12113adant2 922 . . . . . 6 ((A P B P 𝐶 P) → (u (2nd ‘(A ·P 𝐶)) ↔ f (2ndA)z (2nd𝐶)u = (f ·Q z)))
1312anbi2d 437 . . . . 5 ((A P B P 𝐶 P) → ((v (2nd ‘(A ·P B)) u (2nd ‘(A ·P 𝐶))) ↔ (v (2nd ‘(A ·P B)) f (2ndA)z (2nd𝐶)u = (f ·Q z))))
14 df-imp 6452 . . . . . . . . 9 ·P = (w P, v P ↦ ⟨{f Qg Q Q (g (1stw) (1stv) f = (g ·Q ))}, {f Qg Q Q (g (2ndw) (2ndv) f = (g ·Q ))}⟩)
1514, 10genpelvu 6496 . . . . . . . 8 ((A P B P) → (v (2nd ‘(A ·P B)) ↔ x (2ndA)y (2ndB)v = (x ·Q y)))
16153adant3 923 . . . . . . 7 ((A P B P 𝐶 P) → (v (2nd ‘(A ·P B)) ↔ x (2ndA)y (2ndB)v = (x ·Q y)))
17 distrlem4pru 6561 . . . . . . . . . . . . . . 15 (((A P B P 𝐶 P) ((x (2ndA) y (2ndB)) (f (2ndA) z (2nd𝐶)))) → ((x ·Q y) +Q (f ·Q z)) (2nd ‘(A ·P (B +P 𝐶))))
18 oveq12 5464 . . . . . . . . . . . . . . . . . 18 ((v = (x ·Q y) u = (f ·Q z)) → (v +Q u) = ((x ·Q y) +Q (f ·Q z)))
1918eqeq2d 2048 . . . . . . . . . . . . . . . . 17 ((v = (x ·Q y) u = (f ·Q z)) → (w = (v +Q u) ↔ w = ((x ·Q y) +Q (f ·Q z))))
20 eleq1 2097 . . . . . . . . . . . . . . . . 17 (w = ((x ·Q y) +Q (f ·Q z)) → (w (2nd ‘(A ·P (B +P 𝐶))) ↔ ((x ·Q y) +Q (f ·Q z)) (2nd ‘(A ·P (B +P 𝐶)))))
2119, 20syl6bi 152 . . . . . . . . . . . . . . . 16 ((v = (x ·Q y) u = (f ·Q z)) → (w = (v +Q u) → (w (2nd ‘(A ·P (B +P 𝐶))) ↔ ((x ·Q y) +Q (f ·Q z)) (2nd ‘(A ·P (B +P 𝐶))))))
2221imp 115 . . . . . . . . . . . . . . 15 (((v = (x ·Q y) u = (f ·Q z)) w = (v +Q u)) → (w (2nd ‘(A ·P (B +P 𝐶))) ↔ ((x ·Q y) +Q (f ·Q z)) (2nd ‘(A ·P (B +P 𝐶)))))
2317, 22syl5ibrcom 146 . . . . . . . . . . . . . 14 (((A P B P 𝐶 P) ((x (2ndA) y (2ndB)) (f (2ndA) z (2nd𝐶)))) → (((v = (x ·Q y) u = (f ·Q z)) w = (v +Q u)) → w (2nd ‘(A ·P (B +P 𝐶)))))
2423exp4b 349 . . . . . . . . . . . . 13 ((A P B P 𝐶 P) → (((x (2ndA) y (2ndB)) (f (2ndA) z (2nd𝐶))) → ((v = (x ·Q y) u = (f ·Q z)) → (w = (v +Q u) → w (2nd ‘(A ·P (B +P 𝐶)))))))
2524com3l 75 . . . . . . . . . . . 12 (((x (2ndA) y (2ndB)) (f (2ndA) z (2nd𝐶))) → ((v = (x ·Q y) u = (f ·Q z)) → ((A P B P 𝐶 P) → (w = (v +Q u) → w (2nd ‘(A ·P (B +P 𝐶)))))))
2625exp4b 349 . . . . . . . . . . 11 ((x (2ndA) y (2ndB)) → ((f (2ndA) z (2nd𝐶)) → (v = (x ·Q y) → (u = (f ·Q z) → ((A P B P 𝐶 P) → (w = (v +Q u) → w (2nd ‘(A ·P (B +P 𝐶)))))))))
2726com23 72 . . . . . . . . . 10 ((x (2ndA) y (2ndB)) → (v = (x ·Q y) → ((f (2ndA) z (2nd𝐶)) → (u = (f ·Q z) → ((A P B P 𝐶 P) → (w = (v +Q u) → w (2nd ‘(A ·P (B +P 𝐶)))))))))
2827rexlimivv 2432 . . . . . . . . 9 (x (2ndA)y (2ndB)v = (x ·Q y) → ((f (2ndA) z (2nd𝐶)) → (u = (f ·Q z) → ((A P B P 𝐶 P) → (w = (v +Q u) → w (2nd ‘(A ·P (B +P 𝐶))))))))
2928rexlimdvv 2433 . . . . . . . 8 (x (2ndA)y (2ndB)v = (x ·Q y) → (f (2ndA)z (2nd𝐶)u = (f ·Q z) → ((A P B P 𝐶 P) → (w = (v +Q u) → w (2nd ‘(A ·P (B +P 𝐶)))))))
3029com3r 73 . . . . . . 7 ((A P B P 𝐶 P) → (x (2ndA)y (2ndB)v = (x ·Q y) → (f (2ndA)z (2nd𝐶)u = (f ·Q z) → (w = (v +Q u) → w (2nd ‘(A ·P (B +P 𝐶)))))))
3116, 30sylbid 139 . . . . . 6 ((A P B P 𝐶 P) → (v (2nd ‘(A ·P B)) → (f (2ndA)z (2nd𝐶)u = (f ·Q z) → (w = (v +Q u) → w (2nd ‘(A ·P (B +P 𝐶)))))))
3231impd 242 . . . . 5 ((A P B P 𝐶 P) → ((v (2nd ‘(A ·P B)) f (2ndA)z (2nd𝐶)u = (f ·Q z)) → (w = (v +Q u) → w (2nd ‘(A ·P (B +P 𝐶))))))
3313, 32sylbid 139 . . . 4 ((A P B P 𝐶 P) → ((v (2nd ‘(A ·P B)) u (2nd ‘(A ·P 𝐶))) → (w = (v +Q u) → w (2nd ‘(A ·P (B +P 𝐶))))))
3433rexlimdvv 2433 . . 3 ((A P B P 𝐶 P) → (v (2nd ‘(A ·P B))u (2nd ‘(A ·P 𝐶))w = (v +Q u) → w (2nd ‘(A ·P (B +P 𝐶)))))
358, 34sylbid 139 . 2 ((A P B P 𝐶 P) → (w (2nd ‘((A ·P B) +P (A ·P 𝐶))) → w (2nd ‘(A ·P (B +P 𝐶)))))
3635ssrdv 2945 1 ((A P B P 𝐶 P) → (2nd ‘((A ·P B) +P (A ·P 𝐶))) ⊆ (2nd ‘(A ·P (B +P 𝐶))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∃wrex 2301   ⊆ wss 2911  ‘cfv 4845  (class class class)co 5455  2nd c2nd 5708   +Q cplq 6266   ·Q cmq 6267  Pcnp 6275   +P cpp 6277   ·P cmp 6278 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254 This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-iplp 6451  df-imp 6452 This theorem is referenced by:  distrprg  6564
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