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Theorem distrprg 7440
 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 ((𝐴P𝐵P𝐶P) → (𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))

Proof of Theorem distrprg
StepHypRef Expression
1 distrlem1prl 7434 . . 3 ((𝐴P𝐵P𝐶P) → (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
2 distrlem5prl 7438 . . 3 ((𝐴P𝐵P𝐶P) → (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
31, 2eqssd 3120 . 2 ((𝐴P𝐵P𝐶P) → (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) = (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
4 distrlem1pru 7435 . . 3 ((𝐴P𝐵P𝐶P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
5 distrlem5pru 7439 . . 3 ((𝐴P𝐵P𝐶P) → (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
64, 5eqssd 3120 . 2 ((𝐴P𝐵P𝐶P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) = (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
7 simp1 982 . . . 4 ((𝐴P𝐵P𝐶P) → 𝐴P)
8 simp2 983 . . . . 5 ((𝐴P𝐵P𝐶P) → 𝐵P)
9 simp3 984 . . . . 5 ((𝐴P𝐵P𝐶P) → 𝐶P)
10 addclpr 7389 . . . . 5 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
118, 9, 10syl2anc 409 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
12 mulclpr 7424 . . . 4 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
137, 11, 12syl2anc 409 . . 3 ((𝐴P𝐵P𝐶P) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
14 mulclpr 7424 . . . . 5 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
157, 8, 14syl2anc 409 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐵) ∈ P)
16 mulclpr 7424 . . . . 5 ((𝐴P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
177, 9, 16syl2anc 409 . . . 4 ((𝐴P𝐵P𝐶P) → (𝐴 ·P 𝐶) ∈ P)
18 addclpr 7389 . . . 4 (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ∈ P)
1915, 17, 18syl2anc 409 . . 3 ((𝐴P𝐵P𝐶P) → ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ∈ P)
20 preqlu 7324 . . 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 409 . 2 ((𝐴P𝐵P𝐶P) → ((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 929 1 ((𝐴P𝐵P𝐶P) → (𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ‘cfv 5132  (class class class)co 5783  1st c1st 6045  2nd c2nd 6046  Pcnp 7143   +P cpp 7145   ·P cmp 7146 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461  ax-iinf 4511 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-tr 4036  df-eprel 4220  df-id 4224  df-po 4227  df-iso 4228  df-iord 4297  df-on 4299  df-suc 4302  df-iom 4514  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-ov 5786  df-oprab 5787  df-mpo 5788  df-1st 6047  df-2nd 6048  df-recs 6211  df-irdg 6276  df-1o 6322  df-2o 6323  df-oadd 6326  df-omul 6327  df-er 6438  df-ec 6440  df-qs 6444  df-ni 7156  df-pli 7157  df-mi 7158  df-lti 7159  df-plpq 7196  df-mpq 7197  df-enq 7199  df-nqqs 7200  df-plqqs 7201  df-mqqs 7202  df-1nqqs 7203  df-rq 7204  df-ltnqqs 7205  df-enq0 7276  df-nq0 7277  df-0nq0 7278  df-plq0 7279  df-mq0 7280  df-inp 7318  df-iplp 7320  df-imp 7321 This theorem is referenced by:  ltmprr  7494  mulcmpblnrlemg  7592  mulasssrg  7610  distrsrg  7611  m1m1sr  7613  1idsr  7620  recexgt0sr  7625  mulgt0sr  7630  mulextsr1lem  7632  recidpirqlemcalc  7709
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