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Theorem distrlem1prl 7523

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 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 ∈ (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 7317	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·_Q ℎ) ∈ Q)
4	2, 3	genpelvl 7453	. . . . 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 284	. . . 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 1189	. . 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 7409	. . . . . . . . . . 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 7316	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
9	7, 8	genpelvl 7453	. . . . . . . . . 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 1005	. . . . . . . . 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 274	. . . . . . . 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 7416	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
13		elprnql 7422	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
14	12, 13	sylan 281	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
15	14	3ad2antl1 1149	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
16	15	adantrr 471	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) → 𝑥 ∈ Q)
17	16	adantr 274	. . . . . . . . . . . . 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 7416	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
19		elprnql 7422	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
20	18, 19	sylan 281	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
21		prop 7416	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ P → ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P)
22		elprnql 7422	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐶)) → 𝑧 ∈ Q)
23	21, 22	sylan 281	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐶)) → 𝑧 ∈ Q)
24	20, 23	anim12i 336	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝐶 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐶))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
25	24	an4s 578	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
26	25	3adantl1 1143	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
27	26	ad2ant2r 501	. . . . . . . . . . . . 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 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 ∈ (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 522	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ 𝑤 = (𝑥 ·_Q 𝑣))) → 𝑤 = (𝑥 ·_Q 𝑣))
31		simpr 109	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧)) → 𝑣 = (𝑦 +_Q 𝑧))
32	30, 31	anim12i 336	. . . . . . . . . . . 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 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 ∈ (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 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 ∈ (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 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 ∈ (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 519	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧)) → 𝑦 ∈ (1^st ‘𝐵))
47	2, 3	genpprecll 7455	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) → (𝑥 ·_Q 𝑦) ∈ (1^st ‘(𝐴 ·_P 𝐵))))
48	47	3adant3 1007	. . . . . . . . . . . . . . 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 378	. . . . . . . . . . . . . 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 475	. . . . . . . . . . . . 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 284	. . . . . . . . . . . 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 520	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧)) → 𝑧 ∈ (1^st ‘𝐶))
53	2, 3	genpprecll 7455	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑥 ·_Q 𝑧) ∈ (1^st ‘(𝐴 ·_P 𝐶))))
54	53	3adant2 1006	. . . . . . . . . . . . . . 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 378	. . . . . . . . . . . . . 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 475	. . . . . . . . . . . . 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 284	. . . . . . . . . . . 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 7455	. . . . . . . . . . . . 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 123	. . . . . . . . . . . 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 1229	. . . . . . . . . . 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 2243	. . . . . . . . . 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 363	. . . . . . . . 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 2590	. . . . . . . 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 149	. . . . . . 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 363	. . . . . 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 252	. . . 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 2590	. . 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 149	. 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 3148	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 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

W3C validator