Theorem distrprg 7714

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

Step	Hyp	Ref	Expression
1		distrlem1prl 7708	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
2		distrlem5prl 7712	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
3	1, 2	eqssd 3212	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘(𝐴 ·P (𝐵 +P 𝐶))) = (1st ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
4		distrlem1pru 7709	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
5		distrlem5pru 7713	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))) ⊆ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
6	4, 5	eqssd 3212	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) = (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
7		simp1 1000	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → 𝐴 ∈ P)
8		simp2 1001	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → 𝐵 ∈ P)
9		simp3 1002	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → 𝐶 ∈ P)
10		addclpr 7663	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P)
11	8, 9, 10	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P)
12		mulclpr 7698	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝐵 +P 𝐶) ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
13	7, 11, 12	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
14		mulclpr 7698	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
15	7, 8, 14	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
16		mulclpr 7698	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
17	7, 9, 16	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
18		addclpr 7663	. . . 4 ⊢ (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ∈ P)
19	15, 17, 18	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)) ∈ P)
20		preqlu 7598	. . 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))))))
21	13, 19, 20	syl2anc 411	. 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))))))
22	3, 6, 21	mpbir2and 947	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) = ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ‘cfv 5277  (class class class)co 5954  1^st c1st 6234  2^nd c2nd 6235  Pcnp 7417   +_P cpp 7419   ·_P cmp 7420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-eprel 4341  df-id 4345  df-po 4348  df-iso 4349  df-iord 4418  df-on 4420  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-recs 6401  df-irdg 6466  df-1o 6512  df-2o 6513  df-oadd 6516  df-omul 6517  df-er 6630  df-ec 6632  df-qs 6636  df-ni 7430  df-pli 7431  df-mi 7432  df-lti 7433  df-plpq 7470  df-mpq 7471  df-enq 7473  df-nqqs 7474  df-plqqs 7475  df-mqqs 7476  df-1nqqs 7477  df-rq 7478  df-ltnqqs 7479  df-enq0 7550  df-nq0 7551  df-0nq0 7552  df-plq0 7553  df-mq0 7554  df-inp 7592  df-iplp 7594  df-imp 7595

This theorem is referenced by:  ltmprr  7768  mulcmpblnrlemg  7866  mulasssrg  7884  distrsrg  7885  m1m1sr  7887  1idsr  7894  recexgt0sr  7899  mulgt0sr  7904  mulextsr1lem  7906  recidpirqlemcalc  7983