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Theorem distrlem1prl 6557
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem1prl ((A P B P 𝐶 P) → (1st ‘(A ·P (B +P 𝐶))) ⊆ (1st ‘((A ·P B) +P (A ·P 𝐶))))

Proof of Theorem distrlem1prl
Dummy variables x y z w v f g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addclpr 6519 . . . . 5 ((B P 𝐶 P) → (B +P 𝐶) P)
2 df-imp 6451 . . . . . 6 ·P = (y P, z P ↦ ⟨{f Qg Q Q (g (1sty) (1stz) f = (g ·Q ))}, {f Qg Q Q (g (2ndy) (2ndz) f = (g ·Q ))}⟩)
3 mulclnq 6360 . . . . . 6 ((g Q Q) → (g ·Q ) Q)
42, 3genpelvl 6494 . . . . 5 ((A P (B +P 𝐶) P) → (w (1st ‘(A ·P (B +P 𝐶))) ↔ x (1stA)v (1st ‘(B +P 𝐶))w = (x ·Q v)))
51, 4sylan2 270 . . . 4 ((A P (B P 𝐶 P)) → (w (1st ‘(A ·P (B +P 𝐶))) ↔ x (1stA)v (1st ‘(B +P 𝐶))w = (x ·Q v)))
653impb 1099 . . 3 ((A P B P 𝐶 P) → (w (1st ‘(A ·P (B +P 𝐶))) ↔ x (1stA)v (1st ‘(B +P 𝐶))w = (x ·Q v)))
7 df-iplp 6450 . . . . . . . . . . 11 +P = (w P, x P ↦ ⟨{f Qg Q Q (g (1stw) (1stx) f = (g +Q ))}, {f Qg Q Q (g (2ndw) (2ndx) f = (g +Q ))}⟩)
8 addclnq 6359 . . . . . . . . . . 11 ((g Q Q) → (g +Q ) Q)
97, 8genpelvl 6494 . . . . . . . . . 10 ((B P 𝐶 P) → (v (1st ‘(B +P 𝐶)) ↔ y (1stB)z (1st𝐶)v = (y +Q z)))
1093adant1 921 . . . . . . . . 9 ((A P B P 𝐶 P) → (v (1st ‘(B +P 𝐶)) ↔ y (1stB)z (1st𝐶)v = (y +Q z)))
1110adantr 261 . . . . . . . 8 (((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) → (v (1st ‘(B +P 𝐶)) ↔ y (1stB)z (1st𝐶)v = (y +Q z)))
12 prop 6457 . . . . . . . . . . . . . . . . 17 (A P → ⟨(1stA), (2ndA)⟩ P)
13 elprnql 6463 . . . . . . . . . . . . . . . . 17 ((⟨(1stA), (2ndA)⟩ P x (1stA)) → x Q)
1412, 13sylan 267 . . . . . . . . . . . . . . . 16 ((A P x (1stA)) → x Q)
15143ad2antl1 1065 . . . . . . . . . . . . . . 15 (((A P B P 𝐶 P) x (1stA)) → x Q)
1615adantrr 448 . . . . . . . . . . . . . 14 (((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) → x Q)
1716adantr 261 . . . . . . . . . . . . 13 ((((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) ((y (1stB) z (1st𝐶)) v = (y +Q z))) → x Q)
18 prop 6457 . . . . . . . . . . . . . . . . . 18 (B P → ⟨(1stB), (2ndB)⟩ P)
19 elprnql 6463 . . . . . . . . . . . . . . . . . 18 ((⟨(1stB), (2ndB)⟩ P y (1stB)) → y Q)
2018, 19sylan 267 . . . . . . . . . . . . . . . . 17 ((B P y (1stB)) → y Q)
21 prop 6457 . . . . . . . . . . . . . . . . . 18 (𝐶 P → ⟨(1st𝐶), (2nd𝐶)⟩ P)
22 elprnql 6463 . . . . . . . . . . . . . . . . . 18 ((⟨(1st𝐶), (2nd𝐶)⟩ P z (1st𝐶)) → z Q)
2321, 22sylan 267 . . . . . . . . . . . . . . . . 17 ((𝐶 P z (1st𝐶)) → z Q)
2420, 23anim12i 321 . . . . . . . . . . . . . . . 16 (((B P y (1stB)) (𝐶 P z (1st𝐶))) → (y Q z Q))
2524an4s 522 . . . . . . . . . . . . . . 15 (((B P 𝐶 P) (y (1stB) z (1st𝐶))) → (y Q z Q))
26253adantl1 1059 . . . . . . . . . . . . . 14 (((A P B P 𝐶 P) (y (1stB) z (1st𝐶))) → (y Q z Q))
2726ad2ant2r 478 . . . . . . . . . . . . 13 ((((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) ((y (1stB) z (1st𝐶)) v = (y +Q z))) → (y Q z Q))
28 3anass 888 . . . . . . . . . . . . 13 ((x Q y Q z Q) ↔ (x Q (y Q z Q)))
2917, 27, 28sylanbrc 394 . . . . . . . . . . . 12 ((((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) ((y (1stB) z (1st𝐶)) v = (y +Q z))) → (x Q y Q z Q))
30 simprr 484 . . . . . . . . . . . . 13 (((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) → w = (x ·Q v))
31 simpr 103 . . . . . . . . . . . . 13 (((y (1stB) z (1st𝐶)) v = (y +Q z)) → v = (y +Q z))
3230, 31anim12i 321 . . . . . . . . . . . 12 ((((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) ((y (1stB) z (1st𝐶)) v = (y +Q z))) → (w = (x ·Q v) v = (y +Q z)))
33 oveq2 5463 . . . . . . . . . . . . . . 15 (v = (y +Q z) → (x ·Q v) = (x ·Q (y +Q z)))
3433eqeq2d 2048 . . . . . . . . . . . . . 14 (v = (y +Q z) → (w = (x ·Q v) ↔ w = (x ·Q (y +Q z))))
3534biimpac 282 . . . . . . . . . . . . 13 ((w = (x ·Q v) v = (y +Q z)) → w = (x ·Q (y +Q z)))
36 distrnqg 6371 . . . . . . . . . . . . . 14 ((x Q y Q z Q) → (x ·Q (y +Q z)) = ((x ·Q y) +Q (x ·Q z)))
3736eqeq2d 2048 . . . . . . . . . . . . 13 ((x Q y Q z Q) → (w = (x ·Q (y +Q z)) ↔ w = ((x ·Q y) +Q (x ·Q z))))
3835, 37syl5ib 143 . . . . . . . . . . . 12 ((x Q y Q z Q) → ((w = (x ·Q v) v = (y +Q z)) → w = ((x ·Q y) +Q (x ·Q z))))
3929, 32, 38sylc 56 . . . . . . . . . . 11 ((((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) ((y (1stB) z (1st𝐶)) v = (y +Q z))) → w = ((x ·Q y) +Q (x ·Q z)))
40 mulclpr 6552 . . . . . . . . . . . . . 14 ((A P B P) → (A ·P B) P)
41403adant3 923 . . . . . . . . . . . . 13 ((A P B P 𝐶 P) → (A ·P B) P)
4241ad2antrr 457 . . . . . . . . . . . 12 ((((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) ((y (1stB) z (1st𝐶)) v = (y +Q z))) → (A ·P B) P)
43 mulclpr 6552 . . . . . . . . . . . . . 14 ((A P 𝐶 P) → (A ·P 𝐶) P)
44433adant2 922 . . . . . . . . . . . . 13 ((A P B P 𝐶 P) → (A ·P 𝐶) P)
4544ad2antrr 457 . . . . . . . . . . . 12 ((((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) ((y (1stB) z (1st𝐶)) v = (y +Q z))) → (A ·P 𝐶) P)
46 simpll 481 . . . . . . . . . . . . 13 (((y (1stB) z (1st𝐶)) v = (y +Q z)) → y (1stB))
472, 3genpprecll 6496 . . . . . . . . . . . . . . . 16 ((A P B P) → ((x (1stA) y (1stB)) → (x ·Q y) (1st ‘(A ·P B))))
48473adant3 923 . . . . . . . . . . . . . . 15 ((A P B P 𝐶 P) → ((x (1stA) y (1stB)) → (x ·Q y) (1st ‘(A ·P B))))
4948impl 362 . . . . . . . . . . . . . 14 ((((A P B P 𝐶 P) x (1stA)) y (1stB)) → (x ·Q y) (1st ‘(A ·P B)))
5049adantlrr 452 . . . . . . . . . . . . 13 ((((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) y (1stB)) → (x ·Q y) (1st ‘(A ·P B)))
5146, 50sylan2 270 . . . . . . . . . . . 12 ((((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) ((y (1stB) z (1st𝐶)) v = (y +Q z))) → (x ·Q y) (1st ‘(A ·P B)))
52 simplr 482 . . . . . . . . . . . . 13 (((y (1stB) z (1st𝐶)) v = (y +Q z)) → z (1st𝐶))
532, 3genpprecll 6496 . . . . . . . . . . . . . . . 16 ((A P 𝐶 P) → ((x (1stA) z (1st𝐶)) → (x ·Q z) (1st ‘(A ·P 𝐶))))
54533adant2 922 . . . . . . . . . . . . . . 15 ((A P B P 𝐶 P) → ((x (1stA) z (1st𝐶)) → (x ·Q z) (1st ‘(A ·P 𝐶))))
5554impl 362 . . . . . . . . . . . . . 14 ((((A P B P 𝐶 P) x (1stA)) z (1st𝐶)) → (x ·Q z) (1st ‘(A ·P 𝐶)))
5655adantlrr 452 . . . . . . . . . . . . 13 ((((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) z (1st𝐶)) → (x ·Q z) (1st ‘(A ·P 𝐶)))
5752, 56sylan2 270 . . . . . . . . . . . 12 ((((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) ((y (1stB) z (1st𝐶)) v = (y +Q z))) → (x ·Q z) (1st ‘(A ·P 𝐶)))
587, 8genpprecll 6496 . . . . . . . . . . . . 13 (((A ·P B) P (A ·P 𝐶) P) → (((x ·Q y) (1st ‘(A ·P B)) (x ·Q z) (1st ‘(A ·P 𝐶))) → ((x ·Q y) +Q (x ·Q z)) (1st ‘((A ·P B) +P (A ·P 𝐶)))))
5958imp 115 . . . . . . . . . . . 12 ((((A ·P B) P (A ·P 𝐶) P) ((x ·Q y) (1st ‘(A ·P B)) (x ·Q z) (1st ‘(A ·P 𝐶)))) → ((x ·Q y) +Q (x ·Q z)) (1st ‘((A ·P B) +P (A ·P 𝐶))))
6042, 45, 51, 57, 59syl22anc 1135 . . . . . . . . . . 11 ((((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) ((y (1stB) z (1st𝐶)) v = (y +Q z))) → ((x ·Q y) +Q (x ·Q z)) (1st ‘((A ·P B) +P (A ·P 𝐶))))
6139, 60eqeltrd 2111 . . . . . . . . . 10 ((((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) ((y (1stB) z (1st𝐶)) v = (y +Q z))) → w (1st ‘((A ·P B) +P (A ·P 𝐶))))
6261exp32 347 . . . . . . . . 9 (((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) → ((y (1stB) z (1st𝐶)) → (v = (y +Q z) → w (1st ‘((A ·P B) +P (A ·P 𝐶))))))
6362rexlimdvv 2433 . . . . . . . 8 (((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) → (y (1stB)z (1st𝐶)v = (y +Q z) → w (1st ‘((A ·P B) +P (A ·P 𝐶)))))
6411, 63sylbid 139 . . . . . . 7 (((A P B P 𝐶 P) (x (1stA) w = (x ·Q v))) → (v (1st ‘(B +P 𝐶)) → w (1st ‘((A ·P B) +P (A ·P 𝐶)))))
6564exp32 347 . . . . . 6 ((A P B P 𝐶 P) → (x (1stA) → (w = (x ·Q v) → (v (1st ‘(B +P 𝐶)) → w (1st ‘((A ·P B) +P (A ·P 𝐶)))))))
6665com34 77 . . . . 5 ((A P B P 𝐶 P) → (x (1stA) → (v (1st ‘(B +P 𝐶)) → (w = (x ·Q v) → w (1st ‘((A ·P B) +P (A ·P 𝐶)))))))
6766impd 242 . . . 4 ((A P B P 𝐶 P) → ((x (1stA) v (1st ‘(B +P 𝐶))) → (w = (x ·Q v) → w (1st ‘((A ·P B) +P (A ·P 𝐶))))))
6867rexlimdvv 2433 . . 3 ((A P B P 𝐶 P) → (x (1stA)v (1st ‘(B +P 𝐶))w = (x ·Q v) → w (1st ‘((A ·P B) +P (A ·P 𝐶)))))
696, 68sylbid 139 . 2 ((A P B P 𝐶 P) → (w (1st ‘(A ·P (B +P 𝐶))) → w (1st ‘((A ·P B) +P (A ·P 𝐶)))))
7069ssrdv 2945 1 ((A P B P 𝐶 P) → (1st ‘(A ·P (B +P 𝐶))) ⊆ (1st ‘((A ·P B) +P (A ·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  cop 3370  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   +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 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-iplp 6450  df-imp 6451
This theorem is referenced by:  distrprg  6563
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