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Theorem distrlem1prl 7390

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 7345	. . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +_P 𝐶) ∈ P)
2		df-imp 7277	. . . . . 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 7184	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·_Q ℎ) ∈ Q)
4	2, 3	genpelvl 7320	. . . . 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 1177	. . 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 7276	. . . . . . . . . . 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 7183	. . . . . . . . . . 11 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
9	7, 8	genpelvl 7320	. . . . . . . . . 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 999	. . . . . . . . 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 7283	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
13		elprnql 7289	. . . . . . . . . . . . . . . . 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 1143	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
16	15	adantrr 470	. . . . . . . . . . . . . 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 7283	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
19		elprnql 7289	. . . . . . . . . . . . . . . . . 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 7283	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐶 ∈ P → ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P)
22		elprnql 7289	. . . . . . . . . . . . . . . . . 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 577	. . . . . . . . . . . . . . 15 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
26	25	3adantl1 1137	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶))) → (𝑦 ∈ Q ∧ 𝑧 ∈ Q))
27	26	ad2ant2r 500	. . . . . . . . . . . . 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 966	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) ↔ (𝑥 ∈ Q ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q)))
29	17, 27, 28	sylanbrc 413	. . . . . . . . . . . 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 521	. . . . . . . . . . . . 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 5782	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = (𝑦 +_Q 𝑧) → (𝑥 ·_Q 𝑣) = (𝑥 ·_Q (𝑦 +_Q 𝑧)))
34	33	eqeq2d 2151	. . . . . . . . . . . . . 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 7195	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 ·_Q (𝑦 +_Q 𝑧)) = ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)))
37	36	eqeq2d 2151	. . . . . . . . . . . . 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 7380	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
41	40	3adant3 1001	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
42	41	ad2antrr 479	. . . . . . . . . . . 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 7380	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
44	43	3adant2 1000	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴 ·_P 𝐶) ∈ P)
45	44	ad2antrr 479	. . . . . . . . . . . 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 518	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧)) → 𝑦 ∈ (1^st ‘𝐵))
47	2, 3	genpprecll 7322	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) → (𝑥 ·_Q 𝑦) ∈ (1^st ‘(𝐴 ·_P 𝐵))))
48	47	3adant3 1001	. . . . . . . . . . . . . . 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 377	. . . . . . . . . . . . . 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 474	. . . . . . . . . . . . 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 519	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) ∧ 𝑣 = (𝑦 +_Q 𝑧)) → 𝑧 ∈ (1^st ‘𝐶))
53	2, 3	genpprecll 7322	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P) → ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑥 ·_Q 𝑧) ∈ (1^st ‘(𝐴 ·_P 𝐶))))
54	53	3adant2 1000	. . . . . . . . . . . . . . 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 377	. . . . . . . . . . . . . 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 474	. . . . . . . . . . . . 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 7322	. . . . . . . . . . . . 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 1217	. . . . . . . . . . 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 2216	. . . . . . . . . 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 362	. . . . . . . . 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 2556	. . . . . . . 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 362	. . . . . 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 2556	. . 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 3103	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 962 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 ⊆ wss 3071 ⟨cop 3530 ‘cfv 5123 (class class class)co 5774 1^st c1st 6036 2^nd c2nd 6037 Qcnq 7088 +_Q cplq 7090 ·_Q cmq 7091 Pcnp 7099 +_P cpp 7101 ·_P cmp 7102

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-iplp 7276 df-imp 7277

This theorem is referenced by: distrprg 7396

W3C validator