Theorem addsdilem3 28179

Description: Lemma for addsdi 28181. 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 7438	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝐶)) = (𝑋 ·s (𝐵 +s 𝐶)))
2		oveq1 7438	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐵) = (𝑋 ·s 𝐵))
3		oveq1 7438	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐶) = (𝑋 ·s 𝐶))
4	2, 3	oveq12d 7449	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
5	1, 4	eqeq12d 2753	. . . . 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 3623	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s (𝐵 +s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
11		oveq1 7438	. . . . . . 7 ⊢ (𝑦𝑂 = 𝑌 → (𝑦𝑂 +s 𝐶) = (𝑌 +s 𝐶))
12	11	oveq2d 7447	. . . . . 6 ⊢ (𝑦𝑂 = 𝑌 → (𝐴 ·s (𝑦𝑂 +s 𝐶)) = (𝐴 ·s (𝑌 +s 𝐶)))
13		oveq2 7439	. . . . . . 7 ⊢ (𝑦𝑂 = 𝑌 → (𝐴 ·s 𝑦𝑂) = (𝐴 ·s 𝑌))
14	13	oveq1d 7446	. . . . . 6 ⊢ (𝑦𝑂 = 𝑌 → ((𝐴 ·s 𝑦𝑂) +s (𝐴 ·s 𝐶)) = ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶)))
15	12, 14	eqeq12d 2753	. . . . 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 3623	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s (𝑌 +s 𝐶)) = ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶)))
21	10, 20	oveq12d 7449	. . 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 7438	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝑦𝑂 +s 𝐶)) = (𝑋 ·s (𝑦𝑂 +s 𝐶)))
23		oveq1 7438	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝑦𝑂) = (𝑋 ·s 𝑦𝑂))
24	23, 3	oveq12d 7449	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝐶)) = ((𝑋 ·s 𝑦𝑂) +s (𝑋 ·s 𝐶)))
25	22, 24	eqeq12d 2753	. . . 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 7447	. . . . 5 ⊢ (𝑦𝑂 = 𝑌 → (𝑋 ·s (𝑦𝑂 +s 𝐶)) = (𝑋 ·s (𝑌 +s 𝐶)))
27		oveq2 7439	. . . . . 6 ⊢ (𝑦𝑂 = 𝑌 → (𝑋 ·s 𝑦𝑂) = (𝑋 ·s 𝑌))
28	27	oveq1d 7446	. . . . 5 ⊢ (𝑦𝑂 = 𝑌 → ((𝑋 ·s 𝑦𝑂) +s (𝑋 ·s 𝐶)) = ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶)))
29	26, 28	eqeq12d 2753	. . . 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 3637	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s (𝑌 +s 𝐶)) = ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶)))
33	21, 32	oveq12d 7449	. 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 27919	. . . . . . . . . . 11 ⊢ ( L ‘𝐴) ⊆ No
35		rightssno 27920	. . . . . . . . . . 11 ⊢ ( R ‘𝐴) ⊆ No
36	34, 35	unssi 4191	. . . . . . . . . 10 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No
37	36, 8	sselid 3981	. . . . . . . . 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 28161	. . . . . . 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 28161	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝐶) ∈ No )
45		pncans 28102	. . . . . . 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 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐶)) = (𝑋 ·s 𝐵))
47	46	oveq1d 7446	. . . . 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 28013	. . . . . 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 27919	. . . . . . . . . . 11 ⊢ ( L ‘𝐵) ⊆ No
52		rightssno 27920	. . . . . . . . . . 11 ⊢ ( R ‘𝐵) ⊆ No
53	51, 52	unssi 4191	. . . . . . . . . 10 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) ⊆ No
54	53, 18	sselid 3981	. . . . . . . . 9 ⊢ (𝜓 → 𝑌 ∈ No )
55	54	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ No )
56	50, 55	mulscld 28161	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝑌) ∈ No )
57	49, 42	mulscld 28161	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No )
58	57	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝐶) ∈ No )
59	56, 58	addscld 28013	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶)) ∈ No )
60	48, 59, 44	addsubsd 28112	. . . . 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 28039	. . . . 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 2787	. . . 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 7446	. . 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 28013	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) ∈ No )
65	37, 54	mulscld 28161	. . . . . 6 ⊢ (𝜓 → (𝑋 ·s 𝑌) ∈ No )
66	65	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝑌) ∈ No )
67	64, 44, 66	subsubs4d 28124	. . . 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 28000	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐶) +s (𝑋 ·s 𝑌)) = ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶)))
69	68	oveq2d 7447	. . . 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 2777	. . 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 28013	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) ∈ No )
72	71, 58, 66	addsubsd 28112	. . 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 2785	. 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 2777	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 1540   ∈ wcel 2108  ∀wral 3061   ∪ cun 3949  ‘cfv 6561  (class class class)co 7431   No csur 27684   L cleft 27884   R cright 27885   +_s cadds 27992   -_s csubs 28052   ·_s cmuls 28132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-muls 28133

This theorem is referenced by:  addsdi  28181