Theorem addsdilem4 27970

Description: Lemma for addsdi 27971. 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 7408	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝐶)) = (𝑋 ·s (𝐵 +s 𝐶)))
2		oveq1 7408	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐵) = (𝑋 ·s 𝐵))
3		oveq1 7408	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐶) = (𝑋 ·s 𝐶))
4	2, 3	oveq12d 7419	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
5	1, 4	eqeq12d 2740	. . . . 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 480	. . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
10	5, 7, 9	rspcdva 3605	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s (𝐵 +s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
11		oveq2 7409	. . . . . . 7 ⊢ (𝑧𝑂 = 𝑍 → (𝐵 +s 𝑧𝑂) = (𝐵 +s 𝑍))
12	11	oveq2d 7417	. . . . . 6 ⊢ (𝑧𝑂 = 𝑍 → (𝐴 ·s (𝐵 +s 𝑧𝑂)) = (𝐴 ·s (𝐵 +s 𝑍)))
13		oveq2 7409	. . . . . . 7 ⊢ (𝑧𝑂 = 𝑍 → (𝐴 ·s 𝑧𝑂) = (𝐴 ·s 𝑍))
14	13	oveq2d 7417	. . . . . 6 ⊢ (𝑧𝑂 = 𝑍 → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)))
15	12, 14	eqeq12d 2740	. . . . 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 480	. . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑍 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶)))
20	15, 17, 19	rspcdva 3605	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s (𝐵 +s 𝑍)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)))
21	10, 20	oveq12d 7419	. . 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 7408	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = (𝑋 ·s (𝐵 +s 𝑧𝑂)))
23		oveq1 7408	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝑧𝑂) = (𝑋 ·s 𝑧𝑂))
24	2, 23	oveq12d 7419	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂)))
25	22, 24	eqeq12d 2740	. . . 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 7417	. . . . 5 ⊢ (𝑧𝑂 = 𝑍 → (𝑋 ·s (𝐵 +s 𝑧𝑂)) = (𝑋 ·s (𝐵 +s 𝑍)))
27		oveq2 7409	. . . . . 6 ⊢ (𝑧𝑂 = 𝑍 → (𝑋 ·s 𝑧𝑂) = (𝑋 ·s 𝑍))
28	27	oveq2d 7417	. . . . 5 ⊢ (𝑧𝑂 = 𝑍 → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍)))
29	26, 28	eqeq12d 2740	. . . 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 480	. . . 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 3618	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s (𝐵 +s 𝑍)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍)))
33	21, 32	oveq12d 7419	. 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 27723	. . . . . . . . 9 ⊢ ( L ‘𝐴) ⊆ No
35		rightssno 27724	. . . . . . . . 9 ⊢ ( R ‘𝐴) ⊆ No
36	34, 35	unssi 4177	. . . . . . . 8 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No
37	36, 8	sselid 3972	. . . . . . 7 ⊢ (𝜓 → 𝑋 ∈ No )
38	37	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ No )
39		addsdilem4.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ No )
40	39	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ No )
41	38, 40	mulscld 27951	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝐵) ∈ No )
42		addsdilem4.3	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ No )
43	42	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ No )
44	38, 43	mulscld 27951	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝐶) ∈ No )
45	41, 44	addscld 27813	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) ∈ No )
46		addsdilem4.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ No )
47	46, 39	mulscld 27951	. . . . . 6 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No )
48	47	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝐵) ∈ No )
49	46	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ No )
50		leftssno 27723	. . . . . . . . 9 ⊢ ( L ‘𝐶) ⊆ No
51		rightssno 27724	. . . . . . . . 9 ⊢ ( R ‘𝐶) ⊆ No
52	50, 51	unssi 4177	. . . . . . . 8 ⊢ (( L ‘𝐶) ∪ ( R ‘𝐶)) ⊆ No
53	52, 18	sselid 3972	. . . . . . 7 ⊢ (𝜓 → 𝑍 ∈ No )
54	53	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑍 ∈ No )
55	49, 54	mulscld 27951	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝑍) ∈ No )
56	48, 55	addscld 27813	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)) ∈ No )
57	45, 56	addscld 27813	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) ∈ No )
58	38, 54	mulscld 27951	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝑍) ∈ No )
59	57, 41, 58	subsubs4d 27917	. 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 27906	. . . . 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 27800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) = ((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)))
62	61	oveq1d 7416	. . . . . . 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 27896	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝐵)) = (𝑋 ·s 𝐶))
65	62, 64	eqtrd 2764	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) = (𝑋 ·s 𝐶))
66	65	oveq1d 7416	. . . . 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 27841	. . . . 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 2768	. . . 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 7416	. . 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 27813	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) ∈ No )
71	48, 70, 58	addsubsassd 27905	. . 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 2764	. 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 2770	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))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3053   ∪ cun 3938  ‘cfv 6533  (class class class)co 7401   No csur 27489   L cleft 27688   R cright 27689   +_s cadds 27792   -_s csubs 27849   ·_s cmuls 27922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-ot 4629  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-1o 8461  df-2o 8462  df-nadd 8661  df-no 27492  df-slt 27493  df-bday 27494  df-sle 27594  df-sslt 27630  df-scut 27632  df-0s 27673  df-made 27690  df-old 27691  df-left 27693  df-right 27694  df-norec 27771  df-norec2 27782  df-adds 27793  df-negs 27850  df-subs 27851  df-muls 27923

This theorem is referenced by:  addsdi  27971