Theorem distrlem1pr 10985

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 10978	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P)
2		df-mp 10944	. . . . . 6 ⊢ ·P = (𝑦 ∈ P, 𝑧 ∈ P ↦ {𝑓 ∣ ∃𝑔 ∈ 𝑦 ∃ℎ ∈ 𝑧 𝑓 = (𝑔 ·Q ℎ)})
3		mulclnq 10907	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·Q ℎ) ∈ Q)
4	2, 3	genpelv 10960	. . . . 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 593	. . . 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 1114	. . 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 10943	. . . . . . . . . . 11 ⊢ +P = (𝑤 ∈ P, 𝑥 ∈ P ↦ {𝑓 ∣ ∃𝑔 ∈ 𝑤 ∃ℎ ∈ 𝑥 𝑓 = (𝑔 +Q ℎ)})
8		addclnq 10905	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
9	7, 8	genpelv 10960	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑣 = (𝑦 +Q 𝑧)))
10	9	3adant1 1130	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (𝐵 +P 𝐶) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 𝑣 = (𝑦 +Q 𝑧)))
11	10	adantr 480	. . . . . . . 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 772	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑤 = (𝑥 ·Q 𝑣))) → 𝑤 = (𝑥 ·Q 𝑣))
13		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑣 = (𝑦 +Q 𝑧))
14		oveq2 7398	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑦 +Q 𝑧) → (𝑥 ·Q 𝑣) = (𝑥 ·Q (𝑦 +Q 𝑧)))
15	14	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑦 +Q 𝑧) → (𝑤 = (𝑥 ·Q 𝑣) ↔ 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧))))
16	15	biimpac 478	. . . . . . . . . . . . 13 ⊢ ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧)))
17		distrnq 10921	. . . . . . . . . . . . 13 ⊢ (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))
18	16, 17	eqtrdi 2781	. . . . . . . . . . . 12 ⊢ ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
19	12, 13, 18	syl2an 596	. . . . . . . . . . 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 10980	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
21	20	3adant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
22	21	ad2antrr 726	. . . . . . . . . . . 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 10980	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
24	23	3adant2 1131	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
25	24	ad2antrr 726	. . . . . . . . . . . 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 766	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑦 ∈ 𝐵)
27	2, 3	genpprecl 10961	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵)))
28	27	3adant3 1132	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵)))
29	28	impl 455	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) → (𝑥 ·Q 𝑦) ∈ (𝐴 ·P 𝐵))
30	29	adantlrr 721	. . . . . . . . . . . . 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 593	. . . . . . . . . . . 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 768	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑧 ∈ 𝐶)
33	2, 3	genpprecl 10961	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)))
34	33	3adant2 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶)))
35	34	impl 455	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 ∈ 𝐶) → (𝑥 ·Q 𝑧) ∈ (𝐴 ·P 𝐶))
36	35	adantlrr 721	. . . . . . . . . . . . 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 593	. . . . . . . . . . . 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 10961	. . . . . . . . . . . . 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 406	. . . . . . . . . . . 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 838	. . . . . . . . . . 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 2829	. . . . . . . . . 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 420	. . . . . . . . 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 3194	. . . . . . . 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 240	. . . . . . 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 420	. . . . . 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 91	. . . . 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 410	. . . 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 3194	. . 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 240	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ (𝐴 ·P (𝐵 +P 𝐶)) → 𝑤 ∈ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
50	49	ssrdv 3955	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ⊆ ((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3054   ⊆ wss 3917  (class class class)co 7390   +_Q cplq 10815   ·_Q cmq 10816  Pcnp 10819   +_P cpp 10821   ·_P cmp 10822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-omul 8442  df-er 8674  df-ni 10832  df-pli 10833  df-mi 10834  df-lti 10835  df-plpq 10868  df-mpq 10869  df-ltpq 10870  df-enq 10871  df-nq 10872  df-erq 10873  df-plq 10874  df-mq 10875  df-1nq 10876  df-rq 10877  df-ltnq 10878  df-np 10941  df-plp 10943  df-mp 10944

This theorem is referenced by:  distrpr  10988