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Theorem distrlem4prl 7847

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 7664	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q) → (𝑤 <_Q 𝑣 ↔ (𝑢 ·_Q 𝑤) <_Q (𝑢 ·_Q 𝑣)))
2	1	adantl 277	. . . . . 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 1024	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → 𝐴 ∈ P)
4		simpll 527	. . . . . . 7 ⊢ (((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶))) → 𝑥 ∈ (1^st ‘𝐴))
5		prop 7738	. . . . . . . 8 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
6		elprnql 7744	. . . . . . . 8 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
7	5, 6	sylan 283	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
8	3, 4, 7	syl2an 289	. . . . . 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 531	. . . . . . 7 ⊢ (((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶))) → 𝑓 ∈ (1^st ‘𝐴))
10		elprnql 7744	. . . . . . . 8 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
11	5, 10	sylan 283	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
12	3, 9, 11	syl2an 289	. . . . . 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 1028	. . . . . . 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 540	. . . . . . 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 7738	. . . . . . . 8 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
16		elprnql 7744	. . . . . . . 8 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
17	15, 16	sylan 283	. . . . . . 7 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
18	13, 14, 17	syl2anc 411	. . . . . 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 7646	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q) → (𝑤 ·_Q 𝑣) = (𝑣 ·_Q 𝑤))
20	19	adantl 277	. . . . . 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 6202	. . . . 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 7663	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q) → (𝑤 <_Q 𝑣 ↔ (𝑢 +_Q 𝑤) <_Q (𝑢 +_Q 𝑣)))
23	22	adantl 277	. . . . . 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 7639	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ·_Q 𝑦) ∈ Q)
25	8, 18, 24	syl2anc 411	. . . . . 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 7639	. . . . . . 7 ⊢ ((𝑓 ∈ Q ∧ 𝑦 ∈ Q) → (𝑓 ·_Q 𝑦) ∈ Q)
27	12, 18, 26	syl2anc 411	. . . . . 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 1029	. . . . . . . 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 542	. . . . . . . 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 7738	. . . . . . . . 9 ⊢ (𝐶 ∈ P → ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P)
31		elprnql 7744	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐶)) → 𝑧 ∈ Q)
32	30, 31	sylan 283	. . . . . . . 8 ⊢ ((𝐶 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐶)) → 𝑧 ∈ Q)
33	28, 29, 32	syl2anc 411	. . . . . . 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 7639	. . . . . . 7 ⊢ ((𝑓 ∈ Q ∧ 𝑧 ∈ Q) → (𝑓 ·_Q 𝑧) ∈ Q)
35	12, 33, 34	syl2anc 411	. . . . . 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 7644	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q) → (𝑤 +_Q 𝑣) = (𝑣 +_Q 𝑤))
37	36	adantl 277	. . . . . 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 6202	. . . . 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 188	. . . 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 1027	. . . . . 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 7800	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +_P 𝐶) ∈ P)
42	41	3adant1 1042	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +_P 𝐶) ∈ P)
43	42	adantr 276	. . . . . 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 7835	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ (𝐵 +_P 𝐶) ∈ P) → (𝐴 ·_P (𝐵 +_P 𝐶)) ∈ P)
45	40, 43, 44	syl2anc 411	. . . . 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 7650	. . . . . . 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 1274	. . . . . 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 541	. . . . . . 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 7731	. . . . . . . . . 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 7638	. . . . . . . . . 10 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
51	49, 50	genpprecll 7777	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑦 +_Q 𝑧) ∈ (1^st ‘(𝐵 +_P 𝐶))))
52	51	imp 124	. . . . . . . 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 1275	. . . . . . 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 7732	. . . . . . . . 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 7639	. . . . . . . . 9 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·_Q ℎ) ∈ Q)
56	54, 55	genpprecll 7777	. . . . . . . 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 124	. . . . . . 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 1275	. . . . . 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 2309	. . . . 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 7738	. . . . . 6 ⊢ ((𝐴 ·_P (𝐵 +_P 𝐶)) ∈ P → ⟨(1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))), (2^nd ‘(𝐴 ·_P (𝐵 +_P 𝐶)))⟩ ∈ P)
61		prcdnql 7747	. . . . . 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 283	. . . . 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 411	. . . 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 150	. . 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 6202	. . . . 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 7639	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 ·_Q 𝑧) ∈ Q)
67	8, 33, 66	syl2anc 411	. . . . . 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 7663	. . . . . 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 1274	. . . . 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 188	. . . 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 7650	. . . . . . 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 1274	. . . . . 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 539	. . . . . . 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 7777	. . . . . . . 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 124	. . . . . . 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 1275	. . . . . 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 2309	. . . . 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 7747	. . . . . 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 283	. . . . 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 411	. . . 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 150	. . 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 725	. 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 7661	. . . . 5 ⊢ <_Q Or Q
84		nqtri3or 7659	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 <_Q 𝑓 ∨ 𝑥 = 𝑓 ∨ 𝑓 <_Q 𝑥))
85	83, 84	sotritrieq 4428	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥)))
86	8, 12, 85	syl2anc 411	. . 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 6035	. . . . . . 7 ⊢ (𝑥 = 𝑓 → (𝑥 ·_Q 𝑧) = (𝑓 ·_Q 𝑧))
88	87	oveq2d 6044	. . . . . 6 ⊢ (𝑥 = 𝑓 → ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)) = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)))
89	72, 88	sylan9eq 2284	. . . . 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 276	. . . . 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 2309	. . . 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 115	. . 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 170	. 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 7660	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID 𝑥 <_Q 𝑓)
95		ltdcnq 7660	. . . . . 6 ⊢ ((𝑓 ∈ Q ∧ 𝑥 ∈ Q) → DECID 𝑓 <_Q 𝑥)
96	95	ancoms 268	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID 𝑓 <_Q 𝑥)
97		dcor 944	. . . . 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 411	. . 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 843	. . 3 ⊢ (DECID (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥) ↔ ((𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥) ∨ ¬ (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥)))
101	99, 100	sylib 122	. 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 726	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 104 ↔ wb 105 ∨ wo 716 DECID wdc 842 ∧ w3a 1005 = wceq 1398 ∈ wcel 2202 ⟨cop 3676 class class class wbr 4093 ‘cfv 5333 (class class class)co 6028 1^st c1st 6310 2^nd c2nd 6311 Qcnq 7543 +_Q cplq 7545 ·_Q cmq 7546 <_Q cltq 7548 Pcnp 7554 +_P cpp 7556 ·_P cmp 7557

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 7567 df-pli 7568 df-mi 7569 df-lti 7570 df-plpq 7607 df-mpq 7608 df-enq 7610 df-nqqs 7611 df-plqqs 7612 df-mqqs 7613 df-1nqqs 7614 df-rq 7615 df-ltnqqs 7616 df-enq0 7687 df-nq0 7688 df-0nq0 7689 df-plq0 7690 df-mq0 7691 df-inp 7729 df-iplp 7731 df-imp 7732

This theorem is referenced by: distrlem5prl 7849

W3C validator