Theorem addsdilem3 28197

Description: Lemma for addsdi 28199. Show one of the equalities involved in the final expression. (Contributed by Scott Fenton, 9-Mar-2025.)

Hypotheses

Ref	Expression
addsdilem3.1	⊢ (𝜑 → 𝐴 ∈ No )
addsdilem3.2	⊢ (𝜑 → 𝐵 ∈ No )
addsdilem3.3	⊢ (𝜑 → 𝐶 ∈ No )
addsdilem3.4	⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))(𝑥𝑂 ·s (𝐵 +s 𝐶)) = ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)))
addsdilem3.5	⊢ (𝜑 → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))(𝐴 ·s (𝑦𝑂 +s 𝐶)) = ((𝐴 ·s 𝑦𝑂) +s (𝐴 ·s 𝐶)))
addsdilem3.6	⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))(𝑥𝑂 ·s (𝑦𝑂 +s 𝐶)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝐶)))
addsdilem3.7	⊢ (𝜓 → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
addsdilem3.8	⊢ (𝜓 → 𝑌 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))

Assertion

Ref	Expression
addsdilem3	⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑌 +s 𝐶))) -s (𝑋 ·s (𝑌 +s 𝐶))) = ((((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) -s (𝑋 ·s 𝑌)) +s (𝐴 ·s 𝐶)))

Distinct variable groups:   𝐴,𝑥_𝑂,𝑦_𝑂   𝐵,𝑥_𝑂,𝑦_𝑂   𝐶,𝑥_𝑂,𝑦_𝑂   𝑋,𝑥_𝑂,𝑦_𝑂   𝑌,𝑦_𝑂

Allowed substitution hints:   𝜑(𝑥_𝑂,𝑦_𝑂)   𝜓(𝑥_𝑂,𝑦_𝑂)   𝑌(𝑥_𝑂)

Proof of Theorem addsdilem3

Step	Hyp	Ref	Expression
1		oveq1 7455	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝐶)) = (𝑋 ·s (𝐵 +s 𝐶)))
2		oveq1 7455	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐵) = (𝑋 ·s 𝐵))
3		oveq1 7455	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐶) = (𝑋 ·s 𝐶))
4	2, 3	oveq12d 7466	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
5	1, 4	eqeq12d 2756	. . . . 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		addsdilem3.4	. . . . . 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		addsdilem3.7	. . . . . 6 ⊢ (𝜓 → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
9	8	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
10	5, 7, 9	rspcdva 3636	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s (𝐵 +s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
11		oveq1 7455	. . . . . . 7 ⊢ (𝑦𝑂 = 𝑌 → (𝑦𝑂 +s 𝐶) = (𝑌 +s 𝐶))
12	11	oveq2d 7464	. . . . . 6 ⊢ (𝑦𝑂 = 𝑌 → (𝐴 ·s (𝑦𝑂 +s 𝐶)) = (𝐴 ·s (𝑌 +s 𝐶)))
13		oveq2 7456	. . . . . . 7 ⊢ (𝑦𝑂 = 𝑌 → (𝐴 ·s 𝑦𝑂) = (𝐴 ·s 𝑌))
14	13	oveq1d 7463	. . . . . 6 ⊢ (𝑦𝑂 = 𝑌 → ((𝐴 ·s 𝑦𝑂) +s (𝐴 ·s 𝐶)) = ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶)))
15	12, 14	eqeq12d 2756	. . . . 5 ⊢ (𝑦𝑂 = 𝑌 → ((𝐴 ·s (𝑦𝑂 +s 𝐶)) = ((𝐴 ·s 𝑦𝑂) +s (𝐴 ·s 𝐶)) ↔ (𝐴 ·s (𝑌 +s 𝐶)) = ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))))
16		addsdilem3.5	. . . . . 6 ⊢ (𝜑 → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))(𝐴 ·s (𝑦𝑂 +s 𝐶)) = ((𝐴 ·s 𝑦𝑂) +s (𝐴 ·s 𝐶)))
17	16	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))(𝐴 ·s (𝑦𝑂 +s 𝐶)) = ((𝐴 ·s 𝑦𝑂) +s (𝐴 ·s 𝐶)))
18		addsdilem3.8	. . . . . 6 ⊢ (𝜓 → 𝑌 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
19	18	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
20	15, 17, 19	rspcdva 3636	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s (𝑌 +s 𝐶)) = ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶)))
21	10, 20	oveq12d 7466	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑌 +s 𝐶))) = (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))))
22		oveq1 7455	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝑦𝑂 +s 𝐶)) = (𝑋 ·s (𝑦𝑂 +s 𝐶)))
23		oveq1 7455	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝑦𝑂) = (𝑋 ·s 𝑦𝑂))
24	23, 3	oveq12d 7466	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝐶)) = ((𝑋 ·s 𝑦𝑂) +s (𝑋 ·s 𝐶)))
25	22, 24	eqeq12d 2756	. . . 4 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s (𝑦𝑂 +s 𝐶)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝐶)) ↔ (𝑋 ·s (𝑦𝑂 +s 𝐶)) = ((𝑋 ·s 𝑦𝑂) +s (𝑋 ·s 𝐶))))
26	11	oveq2d 7464	. . . . 5 ⊢ (𝑦𝑂 = 𝑌 → (𝑋 ·s (𝑦𝑂 +s 𝐶)) = (𝑋 ·s (𝑌 +s 𝐶)))
27		oveq2 7456	. . . . . 6 ⊢ (𝑦𝑂 = 𝑌 → (𝑋 ·s 𝑦𝑂) = (𝑋 ·s 𝑌))
28	27	oveq1d 7463	. . . . 5 ⊢ (𝑦𝑂 = 𝑌 → ((𝑋 ·s 𝑦𝑂) +s (𝑋 ·s 𝐶)) = ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶)))
29	26, 28	eqeq12d 2756	. . . 4 ⊢ (𝑦𝑂 = 𝑌 → ((𝑋 ·s (𝑦𝑂 +s 𝐶)) = ((𝑋 ·s 𝑦𝑂) +s (𝑋 ·s 𝐶)) ↔ (𝑋 ·s (𝑌 +s 𝐶)) = ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶))))
30		addsdilem3.6	. . . . 5 ⊢ (𝜑 → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))(𝑥𝑂 ·s (𝑦𝑂 +s 𝐶)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝐶)))
31	30	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥𝑂 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴))∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))(𝑥𝑂 ·s (𝑦𝑂 +s 𝐶)) = ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝐶)))
32	25, 29, 31, 9, 19	rspc2dv 3650	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s (𝑌 +s 𝐶)) = ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶)))
33	21, 32	oveq12d 7466	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑌 +s 𝐶))) -s (𝑋 ·s (𝑌 +s 𝐶))) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) -s ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶))))
34		leftssno 27937	. . . . . . . . . . 11 ⊢ ( L ‘𝐴) ⊆ No
35		rightssno 27938	. . . . . . . . . . 11 ⊢ ( R ‘𝐴) ⊆ No
36	34, 35	unssi 4214	. . . . . . . . . 10 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No
37	36, 8	sselid 4006	. . . . . . . . 9 ⊢ (𝜓 → 𝑋 ∈ No )
38	37	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ No )
39		addsdilem3.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ No )
40	39	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ No )
41	38, 40	mulscld 28179	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝐵) ∈ No )
42		addsdilem3.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ No )
43	42	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ No )
44	38, 43	mulscld 28179	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝐶) ∈ No )
45		pncans 28120	. . . . . . 7 ⊢ (((𝑋 ·s 𝐵) ∈ No ∧ (𝑋 ·s 𝐶) ∈ No ) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐶)) = (𝑋 ·s 𝐵))
46	41, 44, 45	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐶)) = (𝑋 ·s 𝐵))
47	46	oveq1d 7463	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) = ((𝑋 ·s 𝐵) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))))
48	41, 44	addscld 28031	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) ∈ No )
49		addsdilem3.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ No )
50	49	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ No )
51		leftssno 27937	. . . . . . . . . . 11 ⊢ ( L ‘𝐵) ⊆ No
52		rightssno 27938	. . . . . . . . . . 11 ⊢ ( R ‘𝐵) ⊆ No
53	51, 52	unssi 4214	. . . . . . . . . 10 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) ⊆ No
54	53, 18	sselid 4006	. . . . . . . . 9 ⊢ (𝜓 → 𝑌 ∈ No )
55	54	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ No )
56	50, 55	mulscld 28179	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝑌) ∈ No )
57	49, 42	mulscld 28179	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No )
58	57	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝐶) ∈ No )
59	56, 58	addscld 28031	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶)) ∈ No )
60	48, 59, 44	addsubsd 28130	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) -s (𝑋 ·s 𝐶)) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))))
61	41, 56, 58	addsassd 28057	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) +s (𝐴 ·s 𝐶)) = ((𝑋 ·s 𝐵) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))))
62	47, 60, 61	3eqtr4d 2790	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) -s (𝑋 ·s 𝐶)) = (((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) +s (𝐴 ·s 𝐶)))
63	62	oveq1d 7463	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) -s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝑌)) = ((((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) +s (𝐴 ·s 𝐶)) -s (𝑋 ·s 𝑌)))
64	48, 59	addscld 28031	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) ∈ No )
65	37, 54	mulscld 28179	. . . . . 6 ⊢ (𝜓 → (𝑋 ·s 𝑌) ∈ No )
66	65	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝑌) ∈ No )
67	64, 44, 66	subsubs4d 28142	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) -s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝑌)) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) -s ((𝑋 ·s 𝐶) +s (𝑋 ·s 𝑌))))
68	44, 66	addscomd 28018	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐶) +s (𝑋 ·s 𝑌)) = ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶)))
69	68	oveq2d 7464	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) -s ((𝑋 ·s 𝐶) +s (𝑋 ·s 𝑌))) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) -s ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶))))
70	67, 69	eqtrd 2780	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) -s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝑌)) = ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) -s ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶))))
71	41, 56	addscld 28031	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) ∈ No )
72	71, 58, 66	addsubsd 28130	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) +s (𝐴 ·s 𝐶)) -s (𝑋 ·s 𝑌)) = ((((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) -s (𝑋 ·s 𝑌)) +s (𝐴 ·s 𝐶)))
73	63, 70, 72	3eqtr3d 2788	. 2 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) -s ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶))) = ((((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) -s (𝑋 ·s 𝑌)) +s (𝐴 ·s 𝐶)))
74	33, 73	eqtrd 2780	1 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s (𝐵 +s 𝐶)) +s (𝐴 ·s (𝑌 +s 𝐶))) -s (𝑋 ·s (𝑌 +s 𝐶))) = ((((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) -s (𝑋 ·s 𝑌)) +s (𝐴 ·s 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∪ cun 3974  ‘cfv 6573  (class class class)co 7448   No csur 27702   L cleft 27902   R cright 27903   +_s cadds 28010   -_s csubs 28070   ·_s cmuls 28150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-muls 28151

This theorem is referenced by:  addsdi  28199