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Theorem distrlem4prl 7240

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

**Assertion**
Ref	Expression
distrlem4prl	⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))

Distinct variable groups: 𝑥,𝑦,𝑧,𝑓,𝐴 𝑥,𝐵,𝑦,𝑧,𝑓 𝑥,𝐶,𝑦,𝑧,𝑓

Proof of Theorem distrlem4prl Dummy variables 𝑤 𝑣 𝑢 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltmnqg 7057	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q) → (𝑤 <_Q 𝑣 ↔ (𝑢 ·_Q 𝑤) <_Q (𝑢 ·_Q 𝑣)))
2	1	adantl 272	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) ∧ (𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q)) → (𝑤 <_Q 𝑣 ↔ (𝑢 ·_Q 𝑤) <_Q (𝑢 ·_Q 𝑣)))
3		simp1 946	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → 𝐴 ∈ P)
4		simpll 497	. . . . . . 7 ⊢ (((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶))) → 𝑥 ∈ (1^st ‘𝐴))
5		prop 7131	. . . . . . . 8 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
6		elprnql 7137	. . . . . . . 8 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
7	5, 6	sylan 278	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
8	3, 4, 7	syl2an 284	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → 𝑥 ∈ Q)
9		simprl 499	. . . . . . 7 ⊢ (((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶))) → 𝑓 ∈ (1^st ‘𝐴))
10		elprnql 7137	. . . . . . . 8 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
11	5, 10	sylan 278	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
12	3, 9, 11	syl2an 284	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → 𝑓 ∈ Q)
13		simpl2 950	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → 𝐵 ∈ P)
14		simprlr 506	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → 𝑦 ∈ (1^st ‘𝐵))
15		prop 7131	. . . . . . . 8 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
16		elprnql 7137	. . . . . . . 8 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
17	15, 16	sylan 278	. . . . . . 7 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
18	13, 14, 17	syl2anc 404	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → 𝑦 ∈ Q)
19		mulcomnqg 7039	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q) → (𝑤 ·_Q 𝑣) = (𝑣 ·_Q 𝑤))
20	19	adantl 272	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) ∧ (𝑤 ∈ Q ∧ 𝑣 ∈ Q)) → (𝑤 ·_Q 𝑣) = (𝑣 ·_Q 𝑤))
21	2, 8, 12, 18, 20	caovord2d 5852	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑥 <_Q 𝑓 ↔ (𝑥 ·_Q 𝑦) <_Q (𝑓 ·_Q 𝑦)))
22		ltanqg 7056	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q) → (𝑤 <_Q 𝑣 ↔ (𝑢 +_Q 𝑤) <_Q (𝑢 +_Q 𝑣)))
23	22	adantl 272	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) ∧ (𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q)) → (𝑤 <_Q 𝑣 ↔ (𝑢 +_Q 𝑤) <_Q (𝑢 +_Q 𝑣)))
24		mulclnq 7032	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ·_Q 𝑦) ∈ Q)
25	8, 18, 24	syl2anc 404	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑥 ·_Q 𝑦) ∈ Q)
26		mulclnq 7032	. . . . . . 7 ⊢ ((𝑓 ∈ Q ∧ 𝑦 ∈ Q) → (𝑓 ·_Q 𝑦) ∈ Q)
27	12, 18, 26	syl2anc 404	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑓 ·_Q 𝑦) ∈ Q)
28		simpl3 951	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → 𝐶 ∈ P)
29		simprrr 508	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → 𝑧 ∈ (1^st ‘𝐶))
30		prop 7131	. . . . . . . . 9 ⊢ (𝐶 ∈ P → ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P)
31		elprnql 7137	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐶)) → 𝑧 ∈ Q)
32	30, 31	sylan 278	. . . . . . . 8 ⊢ ((𝐶 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐶)) → 𝑧 ∈ Q)
33	28, 29, 32	syl2anc 404	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → 𝑧 ∈ Q)
34		mulclnq 7032	. . . . . . 7 ⊢ ((𝑓 ∈ Q ∧ 𝑧 ∈ Q) → (𝑓 ·_Q 𝑧) ∈ Q)
35	12, 33, 34	syl2anc 404	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑓 ·_Q 𝑧) ∈ Q)
36		addcomnqg 7037	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q) → (𝑤 +_Q 𝑣) = (𝑣 +_Q 𝑤))
37	36	adantl 272	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) ∧ (𝑤 ∈ Q ∧ 𝑣 ∈ Q)) → (𝑤 +_Q 𝑣) = (𝑣 +_Q 𝑤))
38	23, 25, 27, 35, 37	caovord2d 5852	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → ((𝑥 ·_Q 𝑦) <_Q (𝑓 ·_Q 𝑦) ↔ ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) <_Q ((𝑓 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧))))
39	21, 38	bitrd 187	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑥 <_Q 𝑓 ↔ ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) <_Q ((𝑓 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧))))
40		simpl1 949	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → 𝐴 ∈ P)
41		addclpr 7193	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +_P 𝐶) ∈ P)
42	41	3adant1 964	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +_P 𝐶) ∈ P)
43	42	adantr 271	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝐵 +_P 𝐶) ∈ P)
44		mulclpr 7228	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ (𝐵 +_P 𝐶) ∈ P) → (𝐴 ·_P (𝐵 +_P 𝐶)) ∈ P)
45	40, 43, 44	syl2anc 404	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝐴 ·_P (𝐵 +_P 𝐶)) ∈ P)
46		distrnqg 7043	. . . . . . 7 ⊢ ((𝑓 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑓 ·_Q (𝑦 +_Q 𝑧)) = ((𝑓 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)))
47	12, 18, 33, 46	syl3anc 1181	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑓 ·_Q (𝑦 +_Q 𝑧)) = ((𝑓 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)))
48		simprrl 507	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → 𝑓 ∈ (1^st ‘𝐴))
49		df-iplp 7124	. . . . . . . . . 10 ⊢ +_P = (𝑢 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑤 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1^st ‘𝑢) ∧ ℎ ∈ (1^st ‘𝑣) ∧ 𝑤 = (𝑔 +_Q ℎ))}, {𝑤 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2^nd ‘𝑢) ∧ ℎ ∈ (2^nd ‘𝑣) ∧ 𝑤 = (𝑔 +_Q ℎ))}⟩)
50		addclnq 7031	. . . . . . . . . 10 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
51	49, 50	genpprecll 7170	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑦 +_Q 𝑧) ∈ (1^st ‘(𝐵 +_P 𝐶))))
52	51	imp 123	. . . . . . . 8 ⊢ (((𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ (𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶))) → (𝑦 +_Q 𝑧) ∈ (1^st ‘(𝐵 +_P 𝐶)))
53	13, 28, 14, 29, 52	syl22anc 1182	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑦 +_Q 𝑧) ∈ (1^st ‘(𝐵 +_P 𝐶)))
54		df-imp 7125	. . . . . . . . 9 ⊢ ·_P = (𝑢 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑤 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1^st ‘𝑢) ∧ ℎ ∈ (1^st ‘𝑣) ∧ 𝑤 = (𝑔 ·_Q ℎ))}, {𝑤 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2^nd ‘𝑢) ∧ ℎ ∈ (2^nd ‘𝑣) ∧ 𝑤 = (𝑔 ·_Q ℎ))}⟩)
55		mulclnq 7032	. . . . . . . . 9 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·_Q ℎ) ∈ Q)
56	54, 55	genpprecll 7170	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (𝐵 +_P 𝐶) ∈ P) → ((𝑓 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑧) ∈ (1^st ‘(𝐵 +_P 𝐶))) → (𝑓 ·_Q (𝑦 +_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
57	56	imp 123	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ (𝐵 +_P 𝐶) ∈ P) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑧) ∈ (1^st ‘(𝐵 +_P 𝐶)))) → (𝑓 ·_Q (𝑦 +_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))
58	40, 43, 48, 53, 57	syl22anc 1182	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑓 ·_Q (𝑦 +_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))
59	47, 58	eqeltrrd 2172	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → ((𝑓 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))
60		prop 7131	. . . . . 6 ⊢ ((𝐴 ·_P (𝐵 +_P 𝐶)) ∈ P → ⟨(1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))), (2^nd ‘(𝐴 ·_P (𝐵 +_P 𝐶)))⟩ ∈ P)
61		prcdnql 7140	. . . . . 6 ⊢ ((⟨(1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))), (2^nd ‘(𝐴 ·_P (𝐵 +_P 𝐶)))⟩ ∈ P ∧ ((𝑓 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))) → (((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) <_Q ((𝑓 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
62	60, 61	sylan 278	. . . . 5 ⊢ (((𝐴 ·_P (𝐵 +_P 𝐶)) ∈ P ∧ ((𝑓 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))) → (((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) <_Q ((𝑓 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
63	45, 59, 62	syl2anc 404	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) <_Q ((𝑓 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
64	39, 63	sylbid 149	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑥 <_Q 𝑓 → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
65	2, 12, 8, 33, 20	caovord2d 5852	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑓 <_Q 𝑥 ↔ (𝑓 ·_Q 𝑧) <_Q (𝑥 ·_Q 𝑧)))
66		mulclnq 7032	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 ·_Q 𝑧) ∈ Q)
67	8, 33, 66	syl2anc 404	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑥 ·_Q 𝑧) ∈ Q)
68		ltanqg 7056	. . . . . 6 ⊢ (((𝑓 ·_Q 𝑧) ∈ Q ∧ (𝑥 ·_Q 𝑧) ∈ Q ∧ (𝑥 ·_Q 𝑦) ∈ Q) → ((𝑓 ·_Q 𝑧) <_Q (𝑥 ·_Q 𝑧) ↔ ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) <_Q ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧))))
69	35, 67, 25, 68	syl3anc 1181	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → ((𝑓 ·_Q 𝑧) <_Q (𝑥 ·_Q 𝑧) ↔ ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) <_Q ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧))))
70	65, 69	bitrd 187	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑓 <_Q 𝑥 ↔ ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) <_Q ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧))))
71		distrnqg 7043	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 ·_Q (𝑦 +_Q 𝑧)) = ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)))
72	8, 18, 33, 71	syl3anc 1181	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑥 ·_Q (𝑦 +_Q 𝑧)) = ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)))
73		simprll 505	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → 𝑥 ∈ (1^st ‘𝐴))
74	54, 55	genpprecll 7170	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (𝐵 +_P 𝐶) ∈ P) → ((𝑥 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑧) ∈ (1^st ‘(𝐵 +_P 𝐶))) → (𝑥 ·_Q (𝑦 +_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
75	74	imp 123	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ (𝐵 +_P 𝐶) ∈ P) ∧ (𝑥 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑧) ∈ (1^st ‘(𝐵 +_P 𝐶)))) → (𝑥 ·_Q (𝑦 +_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))
76	40, 43, 73, 53, 75	syl22anc 1182	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑥 ·_Q (𝑦 +_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))
77	72, 76	eqeltrrd 2172	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))
78		prcdnql 7140	. . . . . 6 ⊢ ((⟨(1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))), (2^nd ‘(𝐴 ·_P (𝐵 +_P 𝐶)))⟩ ∈ P ∧ ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))) → (((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) <_Q ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)) → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
79	60, 78	sylan 278	. . . . 5 ⊢ (((𝐴 ·_P (𝐵 +_P 𝐶)) ∈ P ∧ ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))) → (((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) <_Q ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)) → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
80	45, 77, 79	syl2anc 404	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) <_Q ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)) → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
81	70, 80	sylbid 149	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑓 <_Q 𝑥 → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
82	64, 81	jaod 675	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → ((𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥) → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
83		ltsonq 7054	. . . . 5 ⊢ <_Q Or Q
84		nqtri3or 7052	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 <_Q 𝑓 ∨ 𝑥 = 𝑓 ∨ 𝑓 <_Q 𝑥))
85	83, 84	sotritrieq 4176	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥)))
86	8, 12, 85	syl2anc 404	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥)))
87		oveq1 5697	. . . . . . 7 ⊢ (𝑥 = 𝑓 → (𝑥 ·_Q 𝑧) = (𝑓 ·_Q 𝑧))
88	87	oveq2d 5706	. . . . . 6 ⊢ (𝑥 = 𝑓 → ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)) = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)))
89	72, 88	sylan9eq 2147	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) ∧ 𝑥 = 𝑓) → (𝑥 ·_Q (𝑦 +_Q 𝑧)) = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)))
90	76	adantr 271	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) ∧ 𝑥 = 𝑓) → (𝑥 ·_Q (𝑦 +_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))
91	89, 90	eqeltrrd 2172	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) ∧ 𝑥 = 𝑓) → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))
92	91	ex 114	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (𝑥 = 𝑓 → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
93	86, 92	sylbird 169	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → (¬ (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥) → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶)))))
94		ltdcnq 7053	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID 𝑥 <_Q 𝑓)
95		ltdcnq 7053	. . . . . 6 ⊢ ((𝑓 ∈ Q ∧ 𝑥 ∈ Q) → DECID 𝑓 <_Q 𝑥)
96	95	ancoms 265	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID 𝑓 <_Q 𝑥)
97		dcor 884	. . . . 5 ⊢ (DECID 𝑥 <_Q 𝑓 → (DECID 𝑓 <_Q 𝑥 → DECID (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥)))
98	94, 96, 97	sylc 62	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥))
99	8, 12, 98	syl2anc 404	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → DECID (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥))
100		df-dc 784	. . 3 ⊢ (DECID (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥) ↔ ((𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥) ∨ ¬ (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥)))
101	99, 100	sylib 121	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → ((𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥) ∨ ¬ (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥)))
102	82, 93, 101	mpjaod 676	1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) ∧ ((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶)))) → ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)) ∈ (1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 667 DECID wdc 783 ∧ w3a 927 = wceq 1296 ∈ wcel 1445 ⟨cop 3469 class class class wbr 3867 ‘cfv 5049 (class class class)co 5690 1^st c1st 5947 2^nd c2nd 5948 Qcnq 6936 +_Q cplq 6938 ·_Q cmq 6939 <_Q cltq 6941 Pcnp 6947 +_P cpp 6949 ·_P cmp 6950

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431

This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-eprel 4140 df-id 4144 df-po 4147 df-iso 4148 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-irdg 6173 df-1o 6219 df-2o 6220 df-oadd 6223 df-omul 6224 df-er 6332 df-ec 6334 df-qs 6338 df-ni 6960 df-pli 6961 df-mi 6962 df-lti 6963 df-plpq 7000 df-mpq 7001 df-enq 7003 df-nqqs 7004 df-plqqs 7005 df-mqqs 7006 df-1nqqs 7007 df-rq 7008 df-ltnqqs 7009 df-enq0 7080 df-nq0 7081 df-0nq0 7082 df-plq0 7083 df-mq0 7084 df-inp 7122 df-iplp 7124 df-imp 7125

This theorem is referenced by: distrlem5prl 7242

W3C validator