Theorem addsdilem4 27538

Description: Lemma for addsdi 27539. 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 7401	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝐶)) = (𝑋 ·s (𝐵 +s 𝐶)))
2		oveq1 7401	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐵) = (𝑋 ·s 𝐵))
3		oveq1 7401	. . . . . . 7 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝐶) = (𝑋 ·s 𝐶))
4	2, 3	oveq12d 7412	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
5	1, 4	eqeq12d 2748	. . . . 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 481	. . . . 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 482	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ (( L ‘𝐴) ∪ ( R ‘𝐴)))
10	5, 7, 9	rspcdva 3611	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s (𝐵 +s 𝐶)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)))
11		oveq2 7402	. . . . . . 7 ⊢ (𝑧𝑂 = 𝑍 → (𝐵 +s 𝑧𝑂) = (𝐵 +s 𝑍))
12	11	oveq2d 7410	. . . . . 6 ⊢ (𝑧𝑂 = 𝑍 → (𝐴 ·s (𝐵 +s 𝑧𝑂)) = (𝐴 ·s (𝐵 +s 𝑍)))
13		oveq2 7402	. . . . . . 7 ⊢ (𝑧𝑂 = 𝑍 → (𝐴 ·s 𝑧𝑂) = (𝐴 ·s 𝑍))
14	13	oveq2d 7410	. . . . . 6 ⊢ (𝑧𝑂 = 𝑍 → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑧𝑂)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)))
15	12, 14	eqeq12d 2748	. . . . 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 481	. . . . 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 482	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑍 ∈ (( L ‘𝐶) ∪ ( R ‘𝐶)))
20	15, 17, 19	rspcdva 3611	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s (𝐵 +s 𝑍)) = ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)))
21	10, 20	oveq12d 7412	. . 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 7401	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s (𝐵 +s 𝑧𝑂)) = (𝑋 ·s (𝐵 +s 𝑧𝑂)))
23		oveq1 7401	. . . . . 6 ⊢ (𝑥𝑂 = 𝑋 → (𝑥𝑂 ·s 𝑧𝑂) = (𝑋 ·s 𝑧𝑂))
24	2, 23	oveq12d 7412	. . . . 5 ⊢ (𝑥𝑂 = 𝑋 → ((𝑥𝑂 ·s 𝐵) +s (𝑥𝑂 ·s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂)))
25	22, 24	eqeq12d 2748	. . . 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 7410	. . . . 5 ⊢ (𝑧𝑂 = 𝑍 → (𝑋 ·s (𝐵 +s 𝑧𝑂)) = (𝑋 ·s (𝐵 +s 𝑍)))
27		oveq2 7402	. . . . . 6 ⊢ (𝑧𝑂 = 𝑍 → (𝑋 ·s 𝑧𝑂) = (𝑋 ·s 𝑍))
28	27	oveq2d 7410	. . . . 5 ⊢ (𝑧𝑂 = 𝑍 → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑧𝑂)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍)))
29	26, 28	eqeq12d 2748	. . . 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 481	. . . 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 3623	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s (𝐵 +s 𝑍)) = ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝑍)))
33	21, 32	oveq12d 7412	. 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 27304	. . . . . . . . 9 ⊢ ( L ‘𝐴) ⊆ No
35		rightssno 27305	. . . . . . . . 9 ⊢ ( R ‘𝐴) ⊆ No
36	34, 35	unssi 4182	. . . . . . . 8 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) ⊆ No
37	36, 8	sselid 3977	. . . . . . 7 ⊢ (𝜓 → 𝑋 ∈ No )
38	37	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ No )
39		addsdilem4.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ No )
40	39	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ No )
41	38, 40	mulscld 27520	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝐵) ∈ No )
42		addsdilem4.3	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ No )
43	42	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ∈ No )
44	38, 43	mulscld 27520	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝐶) ∈ No )
45	41, 44	addscld 27393	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) ∈ No )
46		addsdilem4.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ No )
47	46, 39	mulscld 27520	. . . . . 6 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No )
48	47	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝐵) ∈ No )
49	46	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ No )
50		leftssno 27304	. . . . . . . . 9 ⊢ ( L ‘𝐶) ⊆ No
51		rightssno 27305	. . . . . . . . 9 ⊢ ( R ‘𝐶) ⊆ No
52	50, 51	unssi 4182	. . . . . . . 8 ⊢ (( L ‘𝐶) ∪ ( R ‘𝐶)) ⊆ No
53	52, 18	sselid 3977	. . . . . . 7 ⊢ (𝜓 → 𝑍 ∈ No )
54	53	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑍 ∈ No )
55	49, 54	mulscld 27520	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ·s 𝑍) ∈ No )
56	48, 55	addscld 27393	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍)) ∈ No )
57	45, 56	addscld 27393	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) +s ((𝐴 ·s 𝐵) +s (𝐴 ·s 𝑍))) ∈ No )
58	38, 54	mulscld 27520	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 ·s 𝑍) ∈ No )
59	57, 41, 58	subsubs4d 27489	. 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 27478	. . . . 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 27380	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) = ((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)))
62	61	oveq1d 7409	. . . . . . 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 27469	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐶) +s (𝑋 ·s 𝐵)) -s (𝑋 ·s 𝐵)) = (𝑋 ·s 𝐶))
65	62, 64	eqtrd 2772	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (((𝑋 ·s 𝐵) +s (𝑋 ·s 𝐶)) -s (𝑋 ·s 𝐵)) = (𝑋 ·s 𝐶))
66	65	oveq1d 7409	. . . . 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 27420	. . . . 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 2776	. . . 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 7409	. . 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 27393	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝑋 ·s 𝐶) +s (𝐴 ·s 𝑍)) ∈ No )
71	48, 70, 58	addsubsassd 27477	. . 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 2772	. 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 2778	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 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ∪ cun 3943  ‘cfv 6533  (class class class)co 7394   No csur 27072   L cleft 27269   R cright 27270   +_s cadds 27372   -_s csubs 27424   ·_s cmuls 27491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  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 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-1o 8450  df-2o 8451  df-nadd 8650  df-no 27075  df-slt 27076  df-bday 27077  df-sle 27177  df-sslt 27212  df-scut 27214  df-0s 27254  df-made 27271  df-old 27272  df-left 27274  df-right 27275  df-norec 27351  df-norec2 27362  df-adds 27373  df-negs 27425  df-subs 27426  df-muls 27492

This theorem is referenced by:  addsdi  27539