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Theorem distrlem1prl 7621

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 7576	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +_P 𝐶) ∈ P)
2		df-imp 7508	. . . . . 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 7415	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·_Q ℎ) ∈ Q)
4	2, 3	genpelvl 7551	. . . . 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 1201	. . 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 7507	. . . . . . . . . . 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 7414	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
9	7, 8	genpelvl 7551	. . . . . . . . . 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 1017	. . . . . . . . 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 7514	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
13		elprnql 7520	. . . . . . . . . . . . . . . . 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 1161	. . . . . . . . . . . . . . 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 7514	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
19		elprnql 7520	. . . . . . . . . . . . . . . . . 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 7514	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ P → ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P)
22		elprnql 7520	. . . . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
26	25	3adantl1 1155	. . . . . . . . . . . . . 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 984	. . . . . . . . . . . . 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 531	. . . . . . . . . . . . 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 5912	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑦 +_Q 𝑧) → (𝑥 ·_Q 𝑣) = (𝑥 ·_Q (𝑦 +_Q 𝑧)))
34	33	eqeq2d 2201	. . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 ·_Q (𝑦 +_Q 𝑧)) = ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)))
37	36	eqeq2d 2201	. . . . . . . . . . . . 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 7611	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
41	40	3adant3 1019	. . . . . . . . . . . . 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 7611	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
44	43	3adant2 1018	. . . . . . . . . . . . 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 7553	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) → (𝑥 ·_Q 𝑦) ∈ (1^st ‘(𝐴 ·_P 𝐵))))
48	47	3adant3 1019	. . . . . . . . . . . . . . 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 528	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧)) → 𝑧 ∈ (1^st ‘𝐶))
53	2, 3	genpprecll 7553	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑥 ·_Q 𝑧) ∈ (1^st ‘(𝐴 ·_P 𝐶))))
54	53	3adant2 1018	. . . . . . . . . . . . . . 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 7553	. . . . . . . . . . . . 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 1250	. . . . . . . . . . 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 2266	. . . . . . . . . 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 2614	. . . . . . . 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 2614	. . 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 3180	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 980 = wceq 1364 ∈ wcel 2160 ∃wrex 2469 ⊆ wss 3148 ⟨cop 3617 ‘cfv 5242 (class class class)co 5904 1^st c1st 6171 2^nd c2nd 6172 Qcnq 7319 +_Q cplq 7321 ·_Q cmq 7322 Pcnp 7330 +_P cpp 7332 ·_P cmp 7333

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4140 ax-sep 4143 ax-nul 4151 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-iinf 4612

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2758 df-sbc 2982 df-csb 3077 df-dif 3150 df-un 3152 df-in 3154 df-ss 3161 df-nul 3442 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-iun 3910 df-br 4026 df-opab 4087 df-mpt 4088 df-tr 4124 df-eprel 4314 df-id 4318 df-po 4321 df-iso 4322 df-iord 4391 df-on 4393 df-suc 4396 df-iom 4615 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-ov 5907 df-oprab 5908 df-mpo 5909 df-1st 6173 df-2nd 6174 df-recs 6338 df-irdg 6403 df-1o 6449 df-2o 6450 df-oadd 6453 df-omul 6454 df-er 6567 df-ec 6569 df-qs 6573 df-ni 7343 df-pli 7344 df-mi 7345 df-lti 7346 df-plpq 7383 df-mpq 7384 df-enq 7386 df-nqqs 7387 df-plqqs 7388 df-mqqs 7389 df-1nqqs 7390 df-rq 7391 df-ltnqqs 7392 df-enq0 7463 df-nq0 7464 df-0nq0 7465 df-plq0 7466 df-mq0 7467 df-inp 7505 df-iplp 7507 df-imp 7508

This theorem is referenced by: distrprg 7627

W3C validator