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Theorem distrlem1prl 7862

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

**Assertion**
Ref	Expression
distrlem1prl	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))) ⊆ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶))))

Proof of Theorem distrlem1prl 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 7817	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +_P 𝐶) ∈ P)
2		df-imp 7749	. . . . . 6 ⊢ ·_P = (𝑦 ∈ P, 𝑧 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1^st ‘𝑦) ∧ ℎ ∈ (1^st ‘𝑧) ∧ 𝑓 = (𝑔 ·_Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2^nd ‘𝑦) ∧ ℎ ∈ (2^nd ‘𝑧) ∧ 𝑓 = (𝑔 ·_Q ℎ))}⟩)
3		mulclnq 7656	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·_Q ℎ) ∈ Q)
4	2, 3	genpelvl 7792	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵 +_P 𝐶) ∈ P) → (𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))) ↔ ∃𝑥 ∈ (1^st ‘𝐴)∃𝑣 ∈ (1^st ‘(𝐵 +_P 𝐶))𝑤 = (𝑥 ·_Q 𝑣)))
5	1, 4	sylan2 286	. . . 4 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))) ↔ ∃𝑥 ∈ (1^st ‘𝐴)∃𝑣 ∈ (1^st ‘(𝐵 +_P 𝐶))𝑤 = (𝑥 ·_Q 𝑣)))
6	5	3impb 1226	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))) ↔ ∃𝑥 ∈ (1^st ‘𝐴)∃𝑣 ∈ (1^st ‘(𝐵 +_P 𝐶))𝑤 = (𝑥 ·_Q 𝑣)))
7		df-iplp 7748	. . . . . . . . . . 11 ⊢ +_P = (𝑤 ∈ P, 𝑥 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1^st ‘𝑤) ∧ ℎ ∈ (1^st ‘𝑥) ∧ 𝑓 = (𝑔 +_Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2^nd ‘𝑤) ∧ ℎ ∈ (2^nd ‘𝑥) ∧ 𝑓 = (𝑔 +_Q ℎ))}⟩)
8		addclnq 7655	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
9	7, 8	genpelvl 7792	. . . . . . . . . 10 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (1^st ‘(𝐵 +_P 𝐶)) ↔ ∃𝑦 ∈ (1^st ‘𝐵)∃𝑧 ∈ (1^st ‘𝐶)𝑣 = (𝑦 +_Q 𝑧)))
10	9	3adant1 1042	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑣 ∈ (1^st ‘(𝐵 +_P 𝐶)) ↔ ∃𝑦 ∈ (1^st ‘𝐵)∃𝑧 ∈ (1^st ‘𝐶)𝑣 = (𝑦 +_Q 𝑧)))
11	10	adantr 276	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) → (𝑣 ∈ (1^st ‘(𝐵 +_P 𝐶)) ↔ ∃𝑦 ∈ (1^st ‘𝐵)∃𝑧 ∈ (1^st ‘𝐶)𝑣 = (𝑦 +_Q 𝑧)))
12		prop 7755	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
13		elprnql 7761	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
14	12, 13	sylan 283	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
15	14	3ad2antl1 1186	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
16	15	adantrr 479	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) → 𝑥 ∈ Q)
17	16	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) ∧ ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧))) → 𝑥 ∈ Q)
18		prop 7755	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
19		elprnql 7761	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
20	18, 19	sylan 283	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
21		prop 7755	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ P → ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P)
22		elprnql 7761	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐶)) → 𝑧 ∈ Q)
23	21, 22	sylan 283	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐶)) → 𝑧 ∈ Q)
24	20, 23	anim12i 338	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝐶 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐶))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
25	24	an4s 592	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
26	25	3adantl1 1180	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
27	26	ad2ant2r 509	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) ∧ ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
28		3anass 1009	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) ↔ (𝑥 ∈ Q ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q)))
29	17, 27, 28	sylanbrc 417	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) ∧ ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧))) → (𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q))
30		simprr 533	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) → 𝑤 = (𝑥 ·_Q 𝑣))
31		simpr 110	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧)) → 𝑣 = (𝑦 +_Q 𝑧))
32	30, 31	anim12i 338	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) ∧ ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧))) → (𝑤 = (𝑥 ·_Q 𝑣) ∧ 𝑣 = (𝑦 +_Q 𝑧)))
33		oveq2 6036	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑦 +_Q 𝑧) → (𝑥 ·_Q 𝑣) = (𝑥 ·_Q (𝑦 +_Q 𝑧)))
34	33	eqeq2d 2243	. . . . . . . . . . . . . 14 ⊢ (𝑣 = (𝑦 +_Q 𝑧) → (𝑤 = (𝑥 ·_Q 𝑣) ↔ 𝑤 = (𝑥 ·_Q (𝑦 +_Q 𝑧))))
35	34	biimpac 298	. . . . . . . . . . . . 13 ⊢ ((𝑤 = (𝑥 ·_Q 𝑣) ∧ 𝑣 = (𝑦 +_Q 𝑧)) → 𝑤 = (𝑥 ·_Q (𝑦 +_Q 𝑧)))
36		distrnqg 7667	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 ·_Q (𝑦 +_Q 𝑧)) = ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)))
37	36	eqeq2d 2243	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑤 = (𝑥 ·_Q (𝑦 +_Q 𝑧)) ↔ 𝑤 = ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧))))
38	35, 37	imbitrid 154	. . . . . . . . . . . 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 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) ∧ ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧))) → 𝑤 = ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)))
40		mulclpr 7852	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
41	40	3adant3 1044	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
42	41	ad2antrr 488	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) ∧ ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧))) → (𝐴 ·_P 𝐵) ∈ P)
43		mulclpr 7852	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
44	43	3adant2 1043	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
45	44	ad2antrr 488	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) ∧ ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧))) → (𝐴 ·_P 𝐶) ∈ P)
46		simpll 527	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧)) → 𝑦 ∈ (1^st ‘𝐵))
47	2, 3	genpprecll 7794	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) → (𝑥 ·_Q 𝑦) ∈ (1^st ‘(𝐴 ·_P 𝐵))))
48	47	3adant3 1044	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) → (𝑥 ·_Q 𝑦) ∈ (1^st ‘(𝐴 ·_P 𝐵))))
49	48	impl 380	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑥 ∈ (1^st ‘𝐴)) ∧ 𝑦 ∈ (1^st ‘𝐵)) → (𝑥 ·_Q 𝑦) ∈ (1^st ‘(𝐴 ·_P 𝐵)))
50	49	adantlrr 483	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) ∧ 𝑦 ∈ (1^st ‘𝐵)) → (𝑥 ·_Q 𝑦) ∈ (1^st ‘(𝐴 ·_P 𝐵)))
51	46, 50	sylan2 286	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) ∧ ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧))) → (𝑥 ·_Q 𝑦) ∈ (1^st ‘(𝐴 ·_P 𝐵)))
52		simplr 529	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧)) → 𝑧 ∈ (1^st ‘𝐶))
53	2, 3	genpprecll 7794	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑥 ·_Q 𝑧) ∈ (1^st ‘(𝐴 ·_P 𝐶))))
54	53	3adant2 1043	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑥 ·_Q 𝑧) ∈ (1^st ‘(𝐴 ·_P 𝐶))))
55	54	impl 380	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑥 ∈ (1^st ‘𝐴)) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑥 ·_Q 𝑧) ∈ (1^st ‘(𝐴 ·_P 𝐶)))
56	55	adantlrr 483	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑥 ·_Q 𝑧) ∈ (1^st ‘(𝐴 ·_P 𝐶)))
57	52, 56	sylan2 286	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) ∧ ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧))) → (𝑥 ·_Q 𝑧) ∈ (1^st ‘(𝐴 ·_P 𝐶)))
58	7, 8	genpprecll 7794	. . . . . . . . . . . . 13 ⊢ (((𝐴 ·_P 𝐵) ∈ P ∧ (𝐴 ·_P 𝐶) ∈ P) → (((𝑥 ·_Q 𝑦) ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ (𝑥 ·_Q 𝑧) ∈ (1^st ‘(𝐴 ·_P 𝐶))) → ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)) ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶)))))
59	58	imp 124	. . . . . . . . . . . 12 ⊢ ((((𝐴 ·_P 𝐵) ∈ P ∧ (𝐴 ·_P 𝐶) ∈ P) ∧ ((𝑥 ·_Q 𝑦) ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ (𝑥 ·_Q 𝑧) ∈ (1^st ‘(𝐴 ·_P 𝐶)))) → ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)) ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶))))
60	42, 45, 51, 57, 59	syl22anc 1275	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) ∧ ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧))) → ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)) ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶))))
61	39, 60	eqeltrd 2308	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) ∧ ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧))) → 𝑤 ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶))))
62	61	exp32 365	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) → ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑣 = (𝑦 +_Q 𝑧) → 𝑤 ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶))))))
63	62	rexlimdvv 2658	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) → (∃𝑦 ∈ (1^st ‘𝐵)∃𝑧 ∈ (1^st ‘𝐶)𝑣 = (𝑦 +_Q 𝑧) → 𝑤 ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶)))))
64	11, 63	sylbid 150	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) → (𝑣 ∈ (1^st ‘(𝐵 +_P 𝐶)) → 𝑤 ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶)))))
65	64	exp32 365	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (1^st ‘𝐴) → (𝑤 = (𝑥 ·_Q 𝑣) → (𝑣 ∈ (1^st ‘(𝐵 +_P 𝐶)) → 𝑤 ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶)))))))
66	65	com34 83	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑥 ∈ (1^st ‘𝐴) → (𝑣 ∈ (1^st ‘(𝐵 +_P 𝐶)) → (𝑤 = (𝑥 ·_Q 𝑣) → 𝑤 ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶)))))))
67	66	impd 254	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑣 ∈ (1^st ‘(𝐵 +_P 𝐶))) → (𝑤 = (𝑥 ·_Q 𝑣) → 𝑤 ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶))))))
68	67	rexlimdvv 2658	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (∃𝑥 ∈ (1^st ‘𝐴)∃𝑣 ∈ (1^st ‘(𝐵 +_P 𝐶))𝑤 = (𝑥 ·_Q 𝑣) → 𝑤 ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶)))))
69	6, 68	sylbid 150	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝑤 ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))) → 𝑤 ∈ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶)))))
70	69	ssrdv 3234	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))) ⊆ (1^st ‘((𝐴 ·_P 𝐵) +_P (𝐴 ·_P 𝐶))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1005 = wceq 1398 ∈ wcel 2202 ∃wrex 2512 ⊆ wss 3201 ⟨cop 3676 ‘cfv 5333 (class class class)co 6028 1^st c1st 6310 2^nd c2nd 6311 Qcnq 7560 +_Q cplq 7562 ·_Q cmq 7563 Pcnp 7571 +_P cpp 7573 ·_P cmp 7574

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-eprel 4392 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-1o 6625 df-2o 6626 df-oadd 6629 df-omul 6630 df-er 6745 df-ec 6747 df-qs 6751 df-ni 7584 df-pli 7585 df-mi 7586 df-lti 7587 df-plpq 7624 df-mpq 7625 df-enq 7627 df-nqqs 7628 df-plqqs 7629 df-mqqs 7630 df-1nqqs 7631 df-rq 7632 df-ltnqqs 7633 df-enq0 7704 df-nq0 7705 df-0nq0 7706 df-plq0 7707 df-mq0 7708 df-inp 7746 df-iplp 7748 df-imp 7749

This theorem is referenced by: distrprg 7868

W3C validator