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Theorem distrprg 6562
Description: Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrprg ((A P B P 𝐶 P) → (A ·P (B +P 𝐶)) = ((A ·P B) +P (A ·P 𝐶)))

Proof of Theorem distrprg
StepHypRef Expression
1 distrlem1prl 6556 . . 3 ((A P B P 𝐶 P) → (1st ‘(A ·P (B +P 𝐶))) ⊆ (1st ‘((A ·P B) +P (A ·P 𝐶))))
2 distrlem5prl 6560 . . 3 ((A P B P 𝐶 P) → (1st ‘((A ·P B) +P (A ·P 𝐶))) ⊆ (1st ‘(A ·P (B +P 𝐶))))
31, 2eqssd 2956 . 2 ((A P B P 𝐶 P) → (1st ‘(A ·P (B +P 𝐶))) = (1st ‘((A ·P B) +P (A ·P 𝐶))))
4 distrlem1pru 6557 . . 3 ((A P B P 𝐶 P) → (2nd ‘(A ·P (B +P 𝐶))) ⊆ (2nd ‘((A ·P B) +P (A ·P 𝐶))))
5 distrlem5pru 6561 . . 3 ((A P B P 𝐶 P) → (2nd ‘((A ·P B) +P (A ·P 𝐶))) ⊆ (2nd ‘(A ·P (B +P 𝐶))))
64, 5eqssd 2956 . 2 ((A P B P 𝐶 P) → (2nd ‘(A ·P (B +P 𝐶))) = (2nd ‘((A ·P B) +P (A ·P 𝐶))))
7 simp1 903 . . . 4 ((A P B P 𝐶 P) → A P)
8 simp2 904 . . . . 5 ((A P B P 𝐶 P) → B P)
9 simp3 905 . . . . 5 ((A P B P 𝐶 P) → 𝐶 P)
10 addclpr 6520 . . . . 5 ((B P 𝐶 P) → (B +P 𝐶) P)
118, 9, 10syl2anc 391 . . . 4 ((A P B P 𝐶 P) → (B +P 𝐶) P)
12 mulclpr 6551 . . . 4 ((A P (B +P 𝐶) P) → (A ·P (B +P 𝐶)) P)
137, 11, 12syl2anc 391 . . 3 ((A P B P 𝐶 P) → (A ·P (B +P 𝐶)) P)
14 mulclpr 6551 . . . . 5 ((A P B P) → (A ·P B) P)
157, 8, 14syl2anc 391 . . . 4 ((A P B P 𝐶 P) → (A ·P B) P)
16 mulclpr 6551 . . . . 5 ((A P 𝐶 P) → (A ·P 𝐶) P)
177, 9, 16syl2anc 391 . . . 4 ((A P B P 𝐶 P) → (A ·P 𝐶) P)
18 addclpr 6520 . . . 4 (((A ·P B) P (A ·P 𝐶) P) → ((A ·P B) +P (A ·P 𝐶)) P)
1915, 17, 18syl2anc 391 . . 3 ((A P B P 𝐶 P) → ((A ·P B) +P (A ·P 𝐶)) P)
20 preqlu 6454 . . 3 (((A ·P (B +P 𝐶)) P ((A ·P B) +P (A ·P 𝐶)) P) → ((A ·P (B +P 𝐶)) = ((A ·P B) +P (A ·P 𝐶)) ↔ ((1st ‘(A ·P (B +P 𝐶))) = (1st ‘((A ·P B) +P (A ·P 𝐶))) (2nd ‘(A ·P (B +P 𝐶))) = (2nd ‘((A ·P B) +P (A ·P 𝐶))))))
2113, 19, 20syl2anc 391 . 2 ((A P B P 𝐶 P) → ((A ·P (B +P 𝐶)) = ((A ·P B) +P (A ·P 𝐶)) ↔ ((1st ‘(A ·P (B +P 𝐶))) = (1st ‘((A ·P B) +P (A ·P 𝐶))) (2nd ‘(A ·P (B +P 𝐶))) = (2nd ‘((A ·P B) +P (A ·P 𝐶))))))
223, 6, 21mpbir2and 850 1 ((A P B P 𝐶 P) → (A ·P (B +P 𝐶)) = ((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  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  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:  ltmprr  6612  mulcmpblnrlemg  6628  mulasssrg  6646  distrsrg  6647  m1m1sr  6649  1idsr  6656  recexgt0sr  6661  mulgt0sr  6664  mulextsr1lem  6666
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