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Theorem distrlem4prl 7767

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 7584	. . . . . . 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 1021	. . . . . . 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 7658	. . . . . . . 8 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
6		elprnql 7664	. . . . . . . 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 529	. . . . . . 7 ⊢ (((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶))) → 𝑓 ∈ (1^st ‘𝐴))
10		elprnql 7664	. . . . . . . 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 1025	. . . . . . 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 538	. . . . . . 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 7658	. . . . . . . 8 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
16		elprnql 7664	. . . . . . . 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 7566	. . . . . . 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 6174	. . . . 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 7583	. . . . . . 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 7559	. . . . . . 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 7559	. . . . . . 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 1026	. . . . . . . 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 540	. . . . . . . 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 7658	. . . . . . . . 9 ⊢ (𝐶 ∈ P → ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P)
31		elprnql 7664	. . . . . . . . 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 7559	. . . . . . 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 7564	. . . . . . 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 6174	. . . . 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 1024	. . . . . 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 7720	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +_P 𝐶) ∈ P)
42	41	3adant1 1039	. . . . . . 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 7755	. . . . . 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 7570	. . . . . . 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 1271	. . . . . 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 539	. . . . . . 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 7651	. . . . . . . . . 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 7558	. . . . . . . . . 10 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
51	49, 50	genpprecll 7697	. . . . . . . . 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 1272	. . . . . . 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 7652	. . . . . . . . 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 7559	. . . . . . . . 9 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·_Q ℎ) ∈ Q)
56	54, 55	genpprecll 7697	. . . . . . . 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 1272	. . . . . 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 2307	. . . . 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 7658	. . . . . 6 ⊢ ((𝐴 ·_P (𝐵 +_P 𝐶)) ∈ P → ⟨(1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))), (2^nd ‘(𝐴 ·_P (𝐵 +_P 𝐶)))⟩ ∈ P)
61		prcdnql 7667	. . . . . 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 6174	. . . . 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 7559	. . . . . . 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 7583	. . . . . 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 1271	. . . . 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 7570	. . . . . . 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 1271	. . . . . 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 537	. . . . . . 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 7697	. . . . . . . 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 1272	. . . . . 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 2307	. . . . 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 7667	. . . . . 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 722	. 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 7581	. . . . 5 ⊢ <_Q Or Q
84		nqtri3or 7579	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 <_Q 𝑓 ∨ 𝑥 = 𝑓 ∨ 𝑓 <_Q 𝑥))
85	83, 84	sotritrieq 4415	. . . 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 6007	. . . . . . 7 ⊢ (𝑥 = 𝑓 → (𝑥 ·_Q 𝑧) = (𝑓 ·_Q 𝑧))
88	87	oveq2d 6016	. . . . . 6 ⊢ (𝑥 = 𝑓 → ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)) = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)))
89	72, 88	sylan9eq 2282	. . . . 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 2307	. . . 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 7580	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID 𝑥 <_Q 𝑓)
95		ltdcnq 7580	. . . . . 6 ⊢ ((𝑓 ∈ Q ∧ 𝑥 ∈ Q) → DECID 𝑓 <_Q 𝑥)
96	95	ancoms 268	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID 𝑓 <_Q 𝑥)
97		dcor 941	. . . . 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 840	. . 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 723	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 713 DECID wdc 839 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ⟨cop 3669 class class class wbr 4082 ‘cfv 5317 (class class class)co 6000 1^st c1st 6282 2^nd c2nd 6283 Qcnq 7463 +_Q cplq 7465 ·_Q cmq 7466 <_Q cltq 7468 Pcnp 7474 +_P cpp 7476 ·_P cmp 7477

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-eprel 4379 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-irdg 6514 df-1o 6560 df-2o 6561 df-oadd 6564 df-omul 6565 df-er 6678 df-ec 6680 df-qs 6684 df-ni 7487 df-pli 7488 df-mi 7489 df-lti 7490 df-plpq 7527 df-mpq 7528 df-enq 7530 df-nqqs 7531 df-plqqs 7532 df-mqqs 7533 df-1nqqs 7534 df-rq 7535 df-ltnqqs 7536 df-enq0 7607 df-nq0 7608 df-0nq0 7609 df-plq0 7610 df-mq0 7611 df-inp 7649 df-iplp 7651 df-imp 7652

This theorem is referenced by: distrlem5prl 7769

W3C validator