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Theorem distrlem4prl 7045

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 6862	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q) → (𝑤 <_Q 𝑣 ↔ (𝑢 ·_Q 𝑤) <_Q (𝑢 ·_Q 𝑣)))
2	1	adantl 271	. . . . . 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 939	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → 𝐴 ∈ P)
4		simpll 496	. . . . . . 7 ⊢ (((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶))) → 𝑥 ∈ (1^st ‘𝐴))
5		prop 6936	. . . . . . . 8 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
6		elprnql 6942	. . . . . . . 8 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
7	5, 6	sylan 277	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑥 ∈ Q)
8	3, 4, 7	syl2an 283	. . . . . 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 498	. . . . . . 7 ⊢ (((𝑥 ∈ (1^st ‘𝐴) ∧ 𝑦 ∈ (1^st ‘𝐵)) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑧 ∈ (1^st ‘𝐶))) → 𝑓 ∈ (1^st ‘𝐴))
10		elprnql 6942	. . . . . . . 8 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
11	5, 10	sylan 277	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
12	3, 9, 11	syl2an 283	. . . . . 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 943	. . . . . . 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 505	. . . . . . 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 6936	. . . . . . . 8 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
16		elprnql 6942	. . . . . . . 8 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
17	15, 16	sylan 277	. . . . . . 7 ⊢ ((𝐵 ∈ P ∧ 𝑦 ∈ (1^st ‘𝐵)) → 𝑦 ∈ Q)
18	13, 14, 17	syl2anc 403	. . . . . 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 6844	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q) → (𝑤 ·_Q 𝑣) = (𝑣 ·_Q 𝑤))
20	19	adantl 271	. . . . . 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 5748	. . . . 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 6861	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q ∧ 𝑢 ∈ Q) → (𝑤 <_Q 𝑣 ↔ (𝑢 +_Q 𝑤) <_Q (𝑢 +_Q 𝑣)))
23	22	adantl 271	. . . . . 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 6837	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ·_Q 𝑦) ∈ Q)
25	8, 18, 24	syl2anc 403	. . . . . 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 6837	. . . . . . 7 ⊢ ((𝑓 ∈ Q ∧ 𝑦 ∈ Q) → (𝑓 ·_Q 𝑦) ∈ Q)
27	12, 18, 26	syl2anc 403	. . . . . 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 944	. . . . . . . 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 507	. . . . . . . 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 6936	. . . . . . . . 9 ⊢ (𝐶 ∈ P → ⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P)
31		elprnql 6942	. . . . . . . . 9 ⊢ ((⟨(1^st ‘𝐶), (2^nd ‘𝐶)⟩ ∈ P ∧ 𝑧 ∈ (1^st ‘𝐶)) → 𝑧 ∈ Q)
32	30, 31	sylan 277	. . . . . . . 8 ⊢ ((𝐶 ∈ P ∧ 𝑧 ∈ (1^st ‘𝐶)) → 𝑧 ∈ Q)
33	28, 29, 32	syl2anc 403	. . . . . . 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 6837	. . . . . . 7 ⊢ ((𝑓 ∈ Q ∧ 𝑧 ∈ Q) → (𝑓 ·_Q 𝑧) ∈ Q)
35	12, 33, 34	syl2anc 403	. . . . . 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 6842	. . . . . . 7 ⊢ ((𝑤 ∈ Q ∧ 𝑣 ∈ Q) → (𝑤 +_Q 𝑣) = (𝑣 +_Q 𝑤))
37	36	adantl 271	. . . . . 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 5748	. . . . 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 186	. . . 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 942	. . . . . 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 6998	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +_P 𝐶) ∈ P)
42	41	3adant1 957	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵 +_P 𝐶) ∈ P)
43	42	adantr 270	. . . . . 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 7033	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ (𝐵 +_P 𝐶) ∈ P) → (𝐴 ·_P (𝐵 +_P 𝐶)) ∈ P)
45	40, 43, 44	syl2anc 403	. . . . 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 6848	. . . . . . 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 1170	. . . . . 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 506	. . . . . . 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 6929	. . . . . . . . . 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 6836	. . . . . . . . . 10 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
51	49, 50	genpprecll 6975	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝑦 ∈ (1^st ‘𝐵) ∧ 𝑧 ∈ (1^st ‘𝐶)) → (𝑦 +_Q 𝑧) ∈ (1^st ‘(𝐵 +_P 𝐶))))
52	51	imp 122	. . . . . . . 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 1171	. . . . . . 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 6930	. . . . . . . . 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 6837	. . . . . . . . 9 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 ·_Q ℎ) ∈ Q)
56	54, 55	genpprecll 6975	. . . . . . . 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 122	. . . . . . 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 1171	. . . . . 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 2160	. . . . 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 6936	. . . . . 6 ⊢ ((𝐴 ·_P (𝐵 +_P 𝐶)) ∈ P → ⟨(1^st ‘(𝐴 ·_P (𝐵 +_P 𝐶))), (2^nd ‘(𝐴 ·_P (𝐵 +_P 𝐶)))⟩ ∈ P)
61		prcdnql 6945	. . . . . 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 277	. . . . 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 403	. . . 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 148	. . 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 5748	. . . . 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 6837	. . . . . . 7 ⊢ ((𝑥 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 ·_Q 𝑧) ∈ Q)
67	8, 33, 66	syl2anc 403	. . . . . 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 6861	. . . . . 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 1170	. . . . 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 186	. . . 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 6848	. . . . . . 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 1170	. . . . . 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 504	. . . . . . 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 6975	. . . . . . . 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 122	. . . . . . 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 1171	. . . . . 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 2160	. . . . 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 6945	. . . . . 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 277	. . . . 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 403	. . . 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 148	. . 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 670	. 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 6859	. . . . 5 ⊢ <_Q Or Q
84		nqtri3or 6857	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 <_Q 𝑓 ∨ 𝑥 = 𝑓 ∨ 𝑓 <_Q 𝑥))
85	83, 84	sotritrieq 4115	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥)))
86	8, 12, 85	syl2anc 403	. . 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 5597	. . . . . . 7 ⊢ (𝑥 = 𝑓 → (𝑥 ·_Q 𝑧) = (𝑓 ·_Q 𝑧))
88	87	oveq2d 5606	. . . . . 6 ⊢ (𝑥 = 𝑓 → ((𝑥 ·_Q 𝑦) +_Q (𝑥 ·_Q 𝑧)) = ((𝑥 ·_Q 𝑦) +_Q (𝑓 ·_Q 𝑧)))
89	72, 88	sylan9eq 2135	. . . . 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 270	. . . . 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 2160	. . . 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 113	. . 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 168	. 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 6858	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID 𝑥 <_Q 𝑓)
95		ltdcnq 6858	. . . . . 6 ⊢ ((𝑓 ∈ Q ∧ 𝑥 ∈ Q) → DECID 𝑓 <_Q 𝑥)
96	95	ancoms 264	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID 𝑓 <_Q 𝑥)
97		dcor 877	. . . . 5 ⊢ (DECID 𝑥 <_Q 𝑓 → (DECID 𝑓 <_Q 𝑥 → DECID (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥)))
98	94, 96, 97	sylc 61	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑓 ∈ Q) → DECID (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥))
99	8, 12, 98	syl2anc 403	. . 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 777	. . 3 ⊢ (DECID (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥) ↔ ((𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥) ∨ ¬ (𝑥 <_Q 𝑓 ∨ 𝑓 <_Q 𝑥)))
101	99, 100	sylib 120	. 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 671	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 102 ↔ wb 103 ∨ wo 662 DECID wdc 776 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 ⟨cop 3425 class class class wbr 3811 ‘cfv 4968 (class class class)co 5590 1^st c1st 5843 2^nd c2nd 5844 Qcnq 6741 +_Q cplq 6743 ·_Q cmq 6744 <_Q cltq 6746 Pcnp 6752 +_P cpp 6754 ·_P cmp 6755

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365

This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-eprel 4079 df-id 4083 df-po 4086 df-iso 4087 df-iord 4156 df-on 4158 df-suc 4161 df-iom 4368 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-rn 4411 df-res 4412 df-ima 4413 df-iota 4933 df-fun 4970 df-fn 4971 df-f 4972 df-f1 4973 df-fo 4974 df-f1o 4975 df-fv 4976 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-1st 5845 df-2nd 5846 df-recs 6001 df-irdg 6066 df-1o 6112 df-2o 6113 df-oadd 6116 df-omul 6117 df-er 6221 df-ec 6223 df-qs 6227 df-ni 6765 df-pli 6766 df-mi 6767 df-lti 6768 df-plpq 6805 df-mpq 6806 df-enq 6808 df-nqqs 6809 df-plqqs 6810 df-mqqs 6811 df-1nqqs 6812 df-rq 6813 df-ltnqqs 6814 df-enq0 6885 df-nq0 6886 df-0nq0 6887 df-plq0 6888 df-mq0 6889 df-inp 6927 df-iplp 6929 df-imp 6930

This theorem is referenced by: distrlem5prl 7047

W3C validator