Theorem addsdilem3 28521

Description: Lemma for addsdi 28523. 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 7419	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝐶)) = (𝑋 ·s (𝐵 +s 𝐶)))
2		oveq1 7419	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐵) = (𝑋 ·s 𝐵))
3		oveq1 7419	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐶) = (𝑋 ·s 𝐶))
4	2, 3	oveq12d 7430	. . . . . 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		addsdilem3.4	. . . . . 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		addsdilem3.7	. . . . . 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		oveq1 7419	. . . . . . 7 ⊢ (𝑦𝑂 = 𝑌 → (𝑦𝑂 +s 𝐶) = (𝑌 +s 𝐶))
12	11	oveq2d 7428	. . . . . 6 ⊢ (𝑦𝑂 = 𝑌 → (𝐴 ·s (𝑦𝑂 +s 𝐶)) = (𝐴 ·s (𝑌 +s 𝐶)))
13		oveq2 7420	. . . . . . 7 ⊢ (𝑦𝑂 = 𝑌 → (𝐴 ·s 𝑦𝑂) = (𝐴 ·s 𝑌))
14	13	oveq1d 7427	. . . . . 6 ⊢ (𝑦𝑂 = 𝑌 → ((𝐴 ·s 𝑦𝑂) +s (𝐴 ·s 𝐶)) = ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶)))
15	12, 14	eqeq12d 2777	. . . . 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 486	. . . . 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 487	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
20	15, 17, 19	rspcdva 3578	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s (𝑌 +s 𝐶)) = ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶)))
21	10, 20	oveq12d 7430	. . 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 7419	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝑦𝑂 +s 𝐶)) = (𝑋 ·s (𝑦𝑂 +s 𝐶)))
23		oveq1 7419	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝑦𝑂) = (𝑋 ·s 𝑦𝑂))
24	23, 3	oveq12d 7430	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝐶)) = ((𝑋 ·s 𝑦𝑂) +s (𝑋 ·s 𝐶)))
25	22, 24	eqeq12d 2777	. . . 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 7428	. . . . 5 ⊢ (𝑦𝑂 = 𝑌 → (𝑋 ·s (𝑦𝑂 +s 𝐶)) = (𝑋 ·s (𝑌 +s 𝐶)))
27		oveq2 7420	. . . . . 6 ⊢ (𝑦𝑂 = 𝑌 → (𝑋 ·s 𝑦𝑂) = (𝑋 ·s 𝑌))
28	27	oveq1d 7427	. . . . 5 ⊢ (𝑦𝑂 = 𝑌 → ((𝑋 ·s 𝑦𝑂) +s (𝑋 ·s 𝐶)) = ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶)))
29	26, 28	eqeq12d 2777	. . . 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 486	. . . 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 3591	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s (𝑌 +s 𝐶)) = ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶)))
33	21, 32	oveq12d 7430	. 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 28241	. . . . . . . . . . 11 ⊢ ( L ‘𝐴) ⊆ No
35		rightssno 28242	. . . . . . . . . . 11 ⊢ ( R ‘𝐴) ⊆ No
36	34, 35	unssi 4137	. . . . . . . . . 10 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No
37	36, 8	sselid 3929	. . . . . . . . 9 ⊢ (𝜓 → 𝑋 ∈ No )
38	37	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ No )
39		addsdilem3.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ No )
40	39	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ No )
41	38, 40	mulscld 28503	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝐵) ∈ No )
42		addsdilem3.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ No )
43	42	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ No )
44	38, 43	mulscld 28503	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝐶) ∈ No )
45		pncans 28440	. . . . . . 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 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐶)) = (𝑋 ·s 𝐵))
47	46	oveq1d 7427	. . . . 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 28348	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) ∈ No )
49		addsdilem3.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ No )
50	49	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ No )
51		leftssno 28241	. . . . . . . . . . 11 ⊢ ( L ‘𝐵) ⊆ No
52		rightssno 28242	. . . . . . . . . . 11 ⊢ ( R ‘𝐵) ⊆ No
53	51, 52	unssi 4137	. . . . . . . . . 10 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) ⊆ No
54	53, 18	sselid 3929	. . . . . . . . 9 ⊢ (𝜓 → 𝑌 ∈ No )
55	54	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ No )
56	50, 55	mulscld 28503	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝑌) ∈ No )
57	49, 42	mulscld 28503	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No )
58	57	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝐶) ∈ No )
59	56, 58	addscld 28348	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶)) ∈ No )
60	48, 59, 44	addsubsd 28450	. . . . 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 28374	. . . . 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 2806	. . . 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 7427	. . 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 28348	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) ∈ No )
65	37, 54	mulscld 28503	. . . . . 6 ⊢ (𝜓 → (𝑋 ·s 𝑌) ∈ No )
66	65	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝑌) ∈ No )
67	64, 44, 66	subsubs4d 28462	. . . 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 28335	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐶) +s (𝑋 ·s 𝑌)) = ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶)))
69	68	oveq2d 7428	. . . 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 2796	. . 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 28348	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) ∈ No )
72	71, 58, 66	addsubsd 28450	. . 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 2804	. 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 2796	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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ∪ cun 3897  ‘cfv 6531  (class class class)co 7412   No csur 27979   L cleft 28193   R cright 28194   +_s cadds 28327   -_s csubs 28388   ·_s cmuls 28474

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 7740

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 6297  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8460  df-2o 8461  df-nadd 8659  df-no 27982  df-lts 27983  df-bday 27984  df-les 28084  df-slts 28126  df-cuts 28128  df-0s 28175  df-made 28195  df-old 28196  df-left 28198  df-right 28199  df-norec 28306  df-norec2 28317  df-adds 28328  df-negs 28389  df-subs 28390  df-muls 28475

This theorem is used by:  addsdi  28523