Theorem addsdilem4 28533

Description: Lemma for addsdi 28534. Show one of the equalities involved in the final expression. (Contributed by Scott Fenton, 9-Mar-2025.)

Hypotheses

Ref	Expression
addsdilem4.1	⊢ (𝜑 → 𝐴 ∈ No )
addsdilem4.2	⊢ (𝜑 → 𝐵 ∈ No )
addsdilem4.3	⊢ (𝜑 → 𝐶 ∈ No )
addsdilem4.4	⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))
addsdilem4.5	⊢ (𝜑 → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)))
addsdilem4.6	⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)))
addsdilem4.7	⊢ (𝜓 → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
addsdilem4.8	⊢ (𝜓 → 𝑍 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶)))

Assertion

Ref	Expression
addsdilem4	⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) -s (𝑋 ·s (𝐵 +s 𝑍))) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))

Distinct variable groups:   𝐴,𝑥_𝑂,𝑧_𝑂   𝐵,𝑥_𝑂,𝑧_𝑂   𝐶,𝑥_𝑂,𝑧_𝑂   𝑋,𝑥_𝑂,𝑧_𝑂   𝑍,𝑧_𝑂

Allowed substitution hints:   𝜑(𝑥_𝑂, 𝑧_𝑂)   𝜓(𝑥_𝑂, 𝑧_𝑂)   𝑍(𝑥_𝑂)

Proof of Theorem addsdilem4

Step	Hyp	Ref	Expression
1		oveq1 7425	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝐶)) = (𝑋 ·s (𝐵 +s 𝐶)))
2		oveq1 7425	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐵) = (𝑋 ·s 𝐵))
3		oveq1 7425	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐶) = (𝑋 ·s 𝐶))
4	2, 3	oveq12d 7436	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
5	1, 4	eqeq12d 2777	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)) ↔ (𝑋 ·s (𝐵 +s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶))))
6		addsdilem4.4	. . . . . 6 ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))
7	6	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))
8		addsdilem4.7	. . . . . 6 ⊢ (𝜓 → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
9	8	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
10	5, 7, 9	rspcdva 3578	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s (𝐵 +s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
11		oveq2 7426	. . . . . . 7 ⊢ (𝑧𝑂 = 𝑍 → (𝐵 +s 𝑧𝑂) = (𝐵 +s 𝑍))
12	11	oveq2d 7434	. . . . . 6 ⊢ (𝑧𝑂 = 𝑍 → (𝐴 ·s (𝐵 +s 𝑧𝑂)) = (𝐴 ·s (𝐵 +s 𝑍)))
13		oveq2 7426	. . . . . . 7 ⊢ (𝑧𝑂 = 𝑍 → (𝐴 ·s 𝑧𝑂) = (𝐴 ·s 𝑍))
14	13	oveq2d 7434	. . . . . 6 ⊢ (𝑧𝑂 = 𝑍 → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)))
15	12, 14	eqeq12d 2777	. . . . 5 ⊢ (𝑧𝑂 = 𝑍 → ((𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)) ↔ (𝐴 ·s (𝐵 +s 𝑍)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))))
16		addsdilem4.5	. . . . . 6 ⊢ (𝜑 → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)))
17	16	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝐴 ·s (𝐵 +s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)))
18		addsdilem4.8	. . . . . 6 ⊢ (𝜓 → 𝑍 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶)))
19	18	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑍 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶)))
20	15, 17, 19	rspcdva 3578	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s (𝐵 +s 𝑍)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)))
21	10, 20	oveq12d 7436	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) = (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))))
22		oveq1 7425	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = (𝑋 ·s (𝐵 +s 𝑧𝑂)))
23		oveq1 7425	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝑧𝑂) = (𝑋 ·s 𝑧𝑂))
24	2, 23	oveq12d 7436	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂)))
25	22, 24	eqeq12d 2777	. . . 4 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)) ↔ (𝑋 ·s (𝐵 +s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂))))
26	11	oveq2d 7434	. . . . 5 ⊢ (𝑧𝑂 = 𝑍 → (𝑋 ·s (𝐵 +s 𝑧𝑂)) = (𝑋 ·s (𝐵 +s 𝑍)))
27		oveq2 7426	. . . . . 6 ⊢ (𝑧𝑂 = 𝑍 → (𝑋 ·s 𝑧𝑂) = (𝑋 ·s 𝑍))
28	27	oveq2d 7434	. . . . 5 ⊢ (𝑧𝑂 = 𝑍 → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍)))
29	26, 28	eqeq12d 2777	. . . 4 ⊢ (𝑧𝑂 = 𝑍 → ((𝑋 ·s (𝐵 +s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂)) ↔ (𝑋 ·s (𝐵 +s 𝑍)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍))))
30		addsdilem4.6	. . . . 5 ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)))
31	30	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑧𝑂 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶))(𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)))
32	25, 29, 31, 9, 19	rspc2dv 3591	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s (𝐵 +s 𝑍)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍)))
33	21, 32	oveq12d 7436	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) -s (𝑋 ·s (𝐵 +s 𝑍))) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍))))
34		leftssno 28252	. . . . . . . . 9 ⊢ ( L ‘𝐴) ⊆ No
35		rightssno 28253	. . . . . . . . 9 ⊢ ( R ‘𝐴) ⊆ No
36	34, 35	unssi 4137	. . . . . . . 8 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No
37	36, 8	sselid 3929	. . . . . . 7 ⊢ (𝜓 → 𝑋 ∈ No )
38	37	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ No )
39		addsdilem4.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ No )
40	39	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ No )
41	38, 40	mulscld 28514	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝐵) ∈ No )
42		addsdilem4.3	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ No )
43	42	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ No )
44	38, 43	mulscld 28514	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝐶) ∈ No )
45	41, 44	addscld 28359	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) ∈ No )
46		addsdilem4.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ No )
47	46, 39	mulscld 28514	. . . . . 6 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No )
48	47	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝐵) ∈ No )
49	46	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ No )
50		leftssno 28252	. . . . . . . . 9 ⊢ ( L ‘𝐶) ⊆ No
51		rightssno 28253	. . . . . . . . 9 ⊢ ( R ‘𝐶) ⊆ No
52	50, 51	unssi 4137	. . . . . . . 8 ⊢ (( L ‘𝐶) ∪ ( R ‘𝐶)) ⊆ No
53	52, 18	sselid 3929	. . . . . . 7 ⊢ (𝜓 → 𝑍 ∈ No )
54	53	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑍 ∈ No )
55	49, 54	mulscld 28514	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝑍) ∈ No )
56	48, 55	addscld 28359	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)) ∈ No )
57	45, 56	addscld 28359	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) ∈ No )
58	38, 54	mulscld 28514	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝑍) ∈ No )
59	57, 41, 58	subsubs4d 28473	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝑍)) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍))))
60	45, 56, 41	addsubsd 28461	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))))
61	41, 44	addscomd 28346	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) = ((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)))
62	61	oveq1d 7433	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) = (((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝐵)))
63		pncans 28451	. . . . . . . 8 ⊢ (((𝑋 ·s 𝐶) ∈ No ∧ (𝑋 ·s 𝐵) ∈ No ) → (((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝐵)) = (𝑋 ·s 𝐶))
64	44, 41, 63	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝐵)) = (𝑋 ·s 𝐶))
65	62, 64	eqtrd 2796	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) = (𝑋 ·s 𝐶))
66	65	oveq1d 7433	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) = ((𝑋 ·s 𝐶) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))))
67	44, 48, 55	adds12d 28387	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐶) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) = ((𝐴 ·s 𝐵) +s ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍))))
68	60, 66, 67	3eqtrd 2800	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) = ((𝐴 ·s 𝐵) +s ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍))))
69	68	oveq1d 7433	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝑍)) = (((𝐴 ·s 𝐵) +s ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝑍)))
70	44, 55	addscld 28359	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) ∈ No )
71	48, 70, 58	addsubsassd 28460	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 ·s 𝐵) +s ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝑍)) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))
72	69, 71	eqtrd 2796	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) -s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝑍)) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))
73	33, 59, 72	3eqtr2d 2802	1 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝐵 +s 𝑍))) -s (𝑋 ·s (𝐵 +s 𝑍))) = ((𝐴 ·s 𝐵) +s (((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) -s (𝑋 ·s 𝑍))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ∪ cun 3897  ‘cfv 6537  (class class class)co 7418   No csur 27990   L cleft 28204   R cright 28205   +_s cadds 28338   -_s csubs 28399   ·_s cmuls 28485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-1o 8469  df-2o 8470  df-nadd 8668  df-no 27993  df-lts 27994  df-bday 27995  df-les 28095  df-slts 28137  df-cuts 28139  df-0s 28186  df-made 28206  df-old 28207  df-left 28209  df-right 28210  df-norec 28317  df-norec2 28328  df-adds 28339  df-negs 28400  df-subs 28401  df-muls 28486

This theorem is used by:  addsdi  28534