Theorem distrlem1pr 11082

Description: Lemma for distributive law for positive reals. (Contributed by NM, 1-May-1996.) (Revised by Mario Carneiro, 13-Jun-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
distrlem1pr	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ⊆ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))

Proof of Theorem distrlem1pr

Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addclpr 11075	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P)
2		df-mp 11041	. . . . . 6 ⊢ ·P = (𝑦 ∈ P, 𝑧 ∈ P ↦ {𝑓 ∣ ∃𝑔 ∈ 𝑦 ∃ℎ ∈ 𝑧 𝑓 = (𝑔 ·Q ℎ)})
3		mulclnq 11004	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·Q ℎ) ∈ Q)
4	2, 3	genpelv 11057	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵 +P 𝐶) ∈ P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣)))
5	1, 4	sylan2 605	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣)))
6	5	3impb 1132	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ∃𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣)))
7		df-plp 11040	. . . . . . . . . . 11 ⊢ +P = (𝑤 ∈ P, 𝑥 ∈ P ↦ {𝑓 ∣ ∃𝑔 ∈ 𝑤 ∃ℎ ∈ 𝑥 𝑓 = (𝑔 +Q ℎ)})
8		addclnq 11002	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
9	7, 8	genpelv 11057	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑣 = (𝑦 +Q 𝑧)))
10	9	3adant1 1148	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑣 = (𝑦 +Q 𝑧)))
11	10	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑣 = (𝑦 +Q 𝑧)))
12		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) → 𝑤 = (𝑥 ·Q 𝑣))
13		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑣 = (𝑦 +Q 𝑧))
14		oveq2 7416	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑦 +Q 𝑧) → (𝑥 ·Q 𝑣) = (𝑥 ·Q (𝑦 +Q 𝑧)))
15	14	eqeq2d 2771	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑦 +Q 𝑧) → (𝑤 = (𝑥 ·Q 𝑣) ↔ 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧))))
16	15	biimpac 484	. . . . . . . . . . . . 13 ⊢ ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧)))
17		distrnq 11018	. . . . . . . . . . . . 13 ⊢ (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))
18	16, 17	eqtrdi 2811	. . . . . . . . . . . 12 ⊢ ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
19	12, 13, 18	syl2an 608	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
20		mulclpr 11077	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
21	20	3adant3 1150	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
22	21	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐵) ∈ P)
23		mulclpr 11077	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
24	23	3adant2 1149	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
25	24	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐶) ∈ P)
26		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑦 ∈ 𝐵)
27	2, 3	genpprecl 11058	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵)))
28	27	3adant3 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵)))
29	28	impl 461	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
30	29	adantlrr 734	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑦 ∈ 𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
31	26, 30	sylan2 605	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
32		simplr 781	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑧 ∈ 𝐶)
33	2, 3	genpprecl 11058	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)))
34	33	3adant2 1149	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)))
35	34	impl 461	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
36	35	adantlrr 734	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑧 ∈ 𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
37	32, 36	sylan2 605	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
38	7, 8	genpprecl 11058	. . . . . . . . . . . . 13 ⊢ (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → (((𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵) ∧ (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
39	38	imp 412	. . . . . . . . . . . 12 ⊢ ((((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) ∧ ((𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵) ∧ (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
40	22, 25, 31, 37, 39	syl22anc 852	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
41	19, 40	eqeltrd 2860	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))
42	41	exp32 426	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) → ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
43	42	rexlimdvv 3218	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
44	11, 43	sylbid 243	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (𝐵 +P 𝐶) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
45	44	exp32 426	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ 𝐴 → (𝑤 = (𝑥 ·Q 𝑣) → (𝑣 ∈ (𝐵 +P 𝐶) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
46	45	com34 92	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ 𝐴 → (𝑣 ∈ (𝐵 +P 𝐶) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
47	46	impd 416	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ 𝐴 ∧ 𝑣 ∈ (𝐵 +P 𝐶)) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
48	47	rexlimdvv 3218	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑥 ∈ 𝐴 ∃𝑣 ∈ (𝐵 +P 𝐶)𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
49	6, 48	sylbid 243	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
50	49	ssrdv 3936	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ⊆ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ⊆ wss 3898  (class class class)co 7408   +_Q cplq 10912   ·_Q cmq 10913  Pcnp 10916   +_P cpp 10918   ·_P cmp 10919

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-oadd 8458  df-omul 8459  df-er 8695  df-ni 10929  df-pli 10930  df-mi 10931  df-lti 10932  df-plpq 10965  df-mpq 10966  df-ltpq 10967  df-enq 10968  df-nq 10969  df-erq 10970  df-plq 10971  df-mq 10972  df-1nq 10973  df-rq 10974  df-ltnq 10975  df-np 11038  df-plp 11040  df-mp 11041

This theorem is used by:  distrpr  11085