Theorem addsdilem3 28115

Description: Lemma for addsdi 28117. 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 7420	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝐶)) = (𝑋 ·s (𝐵 +s 𝐶)))
2		oveq1 7420	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐵) = (𝑋 ·s 𝐵))
3		oveq1 7420	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐶) = (𝑋 ·s 𝐶))
4	2, 3	oveq12d 7431	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
5	1, 4	eqeq12d 2750	. . . . 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 3606	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s (𝐵 +s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
11		oveq1 7420	. . . . . . 7 ⊢ (𝑦𝑂 = 𝑌 → (𝑦𝑂 +s 𝐶) = (𝑌 +s 𝐶))
12	11	oveq2d 7429	. . . . . 6 ⊢ (𝑦𝑂 = 𝑌 → (𝐴 ·s (𝑦𝑂 +s 𝐶)) = (𝐴 ·s (𝑌 +s 𝐶)))
13		oveq2 7421	. . . . . . 7 ⊢ (𝑦𝑂 = 𝑌 → (𝐴 ·s 𝑦𝑂) = (𝐴 ·s 𝑌))
14	13	oveq1d 7428	. . . . . 6 ⊢ (𝑦𝑂 = 𝑌 → ((𝐴 ·s 𝑦𝑂) +s (𝐴 ·s 𝐶)) = ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶)))
15	12, 14	eqeq12d 2750	. . . . 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 3606	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s (𝑌 +s 𝐶)) = ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶)))
21	10, 20	oveq12d 7431	. . 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 7420	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝑦𝑂 +s 𝐶)) = (𝑋 ·s (𝑦𝑂 +s 𝐶)))
23		oveq1 7420	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝑦𝑂) = (𝑋 ·s 𝑦𝑂))
24	23, 3	oveq12d 7431	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝑦𝑂) +s (𝑥𝑂 ·s 𝐶)) = ((𝑋 ·s 𝑦𝑂) +s (𝑋 ·s 𝐶)))
25	22, 24	eqeq12d 2750	. . . 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 7429	. . . . 5 ⊢ (𝑦𝑂 = 𝑌 → (𝑋 ·s (𝑦𝑂 +s 𝐶)) = (𝑋 ·s (𝑌 +s 𝐶)))
27		oveq2 7421	. . . . . 6 ⊢ (𝑦𝑂 = 𝑌 → (𝑋 ·s 𝑦𝑂) = (𝑋 ·s 𝑌))
28	27	oveq1d 7428	. . . . 5 ⊢ (𝑦𝑂 = 𝑌 → ((𝑋 ·s 𝑦𝑂) +s (𝑋 ·s 𝐶)) = ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶)))
29	26, 28	eqeq12d 2750	. . . 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 3620	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s (𝑌 +s 𝐶)) = ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶)))
33	21, 32	oveq12d 7431	. 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 27855	. . . . . . . . . . 11 ⊢ ( L ‘𝐴) ⊆ No
35		rightssno 27856	. . . . . . . . . . 11 ⊢ ( R ‘𝐴) ⊆ No
36	34, 35	unssi 4171	. . . . . . . . . 10 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No
37	36, 8	sselid 3961	. . . . . . . . 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 28097	. . . . . . 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 28097	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝐶) ∈ No )
45		pncans 28038	. . . . . . 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 7428	. . . . 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 27949	. . . . . 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 27855	. . . . . . . . . . 11 ⊢ ( L ‘𝐵) ⊆ No
52		rightssno 27856	. . . . . . . . . . 11 ⊢ ( R ‘𝐵) ⊆ No
53	51, 52	unssi 4171	. . . . . . . . . 10 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) ⊆ No
54	53, 18	sselid 3961	. . . . . . . . 9 ⊢ (𝜓 → 𝑌 ∈ No )
55	54	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑌 ∈ No )
56	50, 55	mulscld 28097	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝑌) ∈ No )
57	49, 42	mulscld 28097	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ·s 𝐶) ∈ No )
58	57	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝐶) ∈ No )
59	56, 58	addscld 27949	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶)) ∈ No )
60	48, 59, 44	addsubsd 28048	. . . . 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 27975	. . . . 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 2779	. . . 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 7428	. . 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 27949	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝑌) +s (𝐴 ·s 𝐶))) ∈ No )
65	37, 54	mulscld 28097	. . . . . 6 ⊢ (𝜓 → (𝑋 ·s 𝑌) ∈ No )
66	65	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝑌) ∈ No )
67	64, 44, 66	subsubs4d 28060	. . . 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 27936	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐶) +s (𝑋 ·s 𝑌)) = ((𝑋 ·s 𝑌) +s (𝑋 ·s 𝐶)))
69	68	oveq2d 7429	. . . 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 2769	. . 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 27949	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐵) +s (𝐴 ·s 𝑌)) ∈ No )
72	71, 58, 66	addsubsd 28048	. . 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 2777	. 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 2769	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 1539   ∈ wcel 2107  ∀wral 3050   ∪ cun 3929  ‘cfv 6541  (class class class)co 7413   No csur 27620   L cleft 27820   R cright 27821   +_s cadds 27928   -_s csubs 27988   ·_s cmuls 28068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-se 5618  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-1o 8488  df-2o 8489  df-nadd 8686  df-no 27623  df-slt 27624  df-bday 27625  df-sle 27726  df-sslt 27762  df-scut 27764  df-0s 27805  df-made 27822  df-old 27823  df-left 27825  df-right 27826  df-norec 27907  df-norec2 27918  df-adds 27929  df-negs 27989  df-subs 27990  df-muls 28069

This theorem is referenced by:  addsdi  28117