Theorem distrlem1pru 7524

Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)

Assertion

Ref	Expression
distrlem1pru	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))

Proof of Theorem distrlem1pru

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 7478	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +P 𝐶) ∈ P)
2		df-imp 7410	. . . . . 6 ⊢ ·P = (𝑦 ∈ P, 𝑧 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1st ‘𝑦) ∧ ℎ ∈ (1st ‘𝑧) ∧ 𝑓 = (𝑔 ·Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2nd ‘𝑦) ∧ ℎ ∈ (2nd ‘𝑧) ∧ 𝑓 = (𝑔 ·Q ℎ))}⟩)
3		mulclnq 7317	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·Q ℎ) ∈ Q)
4	2, 3	genpelvu 7454	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵 +P 𝐶) ∈ P) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (2nd ‘𝐴)∃𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
5	1, 4	sylan2 284	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (2nd ‘𝐴)∃𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
6	5	3impb 1189	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ↔ ∃𝑥 ∈ (2nd ‘𝐴)∃𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣)))
7		df-iplp 7409	. . . . . . . . . . 11 ⊢ +P = (𝑤 ∈ P, 𝑥 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1st ‘𝑤) ∧ ℎ ∈ (1st ‘𝑥) ∧ 𝑓 = (𝑔 +Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2nd ‘𝑤) ∧ ℎ ∈ (2nd ‘𝑥) ∧ 𝑓 = (𝑔 +Q ℎ))}⟩)
8		addclnq 7316	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
9	7, 8	genpelvu 7454	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (2nd ‘𝐵)∃𝑧 ∈ (2nd ‘𝐶)𝑣 = (𝑦 +Q 𝑧)))
10	9	3adant1 1005	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (2nd ‘𝐵)∃𝑧 ∈ (2nd ‘𝐶)𝑣 = (𝑦 +Q 𝑧)))
11	10	adantr 274	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) ↔ ∃𝑦 ∈ (2nd ‘𝐵)∃𝑧 ∈ (2nd ‘𝐶)𝑣 = (𝑦 +Q 𝑧)))
12		prop 7416	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
13		elprnqu 7423	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
14	12, 13	sylan 281	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
15	14	3ad2antl1 1149	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑥 ∈ (2nd ‘𝐴)) → 𝑥 ∈ Q)
16	15	adantrr 471	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → 𝑥 ∈ Q)
17	16	adantr 274	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑥 ∈ Q)
18		prop 7416	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
19		elprnqu 7423	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (2nd ‘𝐵)) → 𝑦 ∈ Q)
20	18, 19	sylan 281	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (2nd ‘𝐵)) → 𝑦 ∈ Q)
21		prop 7416	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ P → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P)
22		elprnqu 7423	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ∈ P ∧ 𝑧 ∈ (2nd ‘𝐶)) → 𝑧 ∈ Q)
23	21, 22	sylan 281	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐶)) → 𝑧 ∈ Q)
24	20, 23	anim12i 336	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ P ∧ 𝑦 ∈ (2nd ‘𝐵)) ∧ (𝐶 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐶))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
25	24	an4s 578	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
26	25	3adantl1 1143	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
27	26	ad2ant2r 501	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
28		3anass 972	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) ↔ (𝑥 ∈ Q ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q)))
29	17, 27, 28	sylanbrc 414	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q))
30		simprr 522	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → 𝑤 = (𝑥 ·Q 𝑣))
31		simpr 109	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑣 = (𝑦 +Q 𝑧))
32	30, 31	anim12i 336	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)))
33		oveq2 5850	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑦 +Q 𝑧) → (𝑥 ·Q 𝑣) = (𝑥 ·Q (𝑦 +Q 𝑧)))
34	33	eqeq2d 2177	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑦 +Q 𝑧) → (𝑤 = (𝑥 ·Q 𝑣) ↔ 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧))))
35	34	biimpac 296	. . . . . . . . . . . . 13 ⊢ ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧)))
36		distrnqg 7328	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
37	36	eqeq2d 2177	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑤 = (𝑥 ·Q (𝑦 +Q 𝑧)) ↔ 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))))
38	35, 37	syl5ib 153	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((𝑤 = (𝑥 ·Q 𝑣) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))))
39	29, 32, 38	sylc 62	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
40		mulclpr 7513	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
41	40	3adant3 1007	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐵) ∈ P)
42	41	ad2antrr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐵) ∈ P)
43		mulclpr 7513	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
44	43	3adant2 1006	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·P 𝐶) ∈ P)
45	44	ad2antrr 480	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝐴 ·P 𝐶) ∈ P)
46		simpll 519	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑦 ∈ (2nd ‘𝐵))
47	2, 3	genppreclu 7456	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵))))
48	47	3adant3 1007	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑦 ∈ (2nd ‘𝐵)) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵))))
49	48	impl 378	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑥 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ (2nd ‘𝐵)) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵)))
50	49	adantlrr 475	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑦 ∈ (2nd ‘𝐵)) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵)))
51	46, 50	sylan2 284	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵)))
52		simplr 520	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧)) → 𝑧 ∈ (2nd ‘𝐶))
53	2, 3	genppreclu 7456	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶))))
54	53	3adant2 1006	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐶)) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶))))
55	54	impl 378	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑥 ∈ (2nd ‘𝐴)) ∧ 𝑧 ∈ (2nd ‘𝐶)) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶)))
56	55	adantlrr 475	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ 𝑧 ∈ (2nd ‘𝐶)) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶)))
57	52, 56	sylan2 284	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶)))
58	7, 8	genppreclu 7456	. . . . . . . . . . . . 13 ⊢ (((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) → (((𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵)) ∧ (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
59	58	imp 123	. . . . . . . . . . . 12 ⊢ ((((𝐴 ·P 𝐵) ∈ P ∧ (𝐴 ·P 𝐶) ∈ P) ∧ ((𝑥 ·Q 𝑦) ∈ (2nd ‘(𝐴 ·P 𝐵)) ∧ (𝑥 ·Q 𝑧) ∈ (2nd ‘(𝐴 ·P 𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
60	42, 45, 51, 57, 59	syl22anc 1229	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
61	39, 60	eqeltrd 2243	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) ∧ ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) ∧ 𝑣 = (𝑦 +Q 𝑧))) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))
62	61	exp32 363	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → ((𝑦 ∈ (2nd ‘𝐵) ∧ 𝑧 ∈ (2nd ‘𝐶)) → (𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
63	62	rexlimdvv 2590	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (∃𝑦 ∈ (2nd ‘𝐵)∃𝑧 ∈ (2nd ‘𝐶)𝑣 = (𝑦 +Q 𝑧) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
64	11, 63	sylbid 149	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (2nd ‘𝐴) ∧ 𝑤 = (𝑥 ·Q 𝑣))) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
65	64	exp32 363	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (2nd ‘𝐴) → (𝑤 = (𝑥 ·Q 𝑣) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))))
66	65	com34 83	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (2nd ‘𝐴) → (𝑣 ∈ (2nd ‘(𝐵 +P 𝐶)) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))))
67	66	impd 252	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ (2nd ‘𝐴) ∧ 𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))) → (𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))))
68	67	rexlimdvv 2590	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑥 ∈ (2nd ‘𝐴)∃𝑣 ∈ (2nd ‘(𝐵 +P 𝐶))𝑤 = (𝑥 ·Q 𝑣) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
69	6, 68	sylbid 149	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) → 𝑤 ∈ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶)))))
70	69	ssrdv 3148	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))) ⊆ (2nd ‘((𝐴 ·P 𝐵) +P (𝐴 ·P 𝐶))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  ∃wrex 2445   ⊆ wss 3116  ⟨cop 3579  ‘cfv 5188  (class class class)co 5842  1^st c1st 6106  2^nd c2nd 6107  Qcnq 7221   +_Q cplq 7223   ·_Q cmq 7224  Pcnp 7232   +_P cpp 7234   ·_P cmp 7235

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409  df-imp 7410

This theorem is referenced by:  distrprg  7529