Theorem nadddilem2 37185

Description: Lemma for nadddi 37188. Prove the reverse implication. (Contributed by Scott Fenton, 31-Jul-2026.)

Hypotheses

Ref	Expression
nadddilem2.1	⊢ (𝜑 → 𝐴 ∈ On)
nadddilem2.2	⊢ (𝜑 → 𝐵 ∈ On)
nadddilem2.3	⊢ (𝜑 → 𝐶 ∈ On)
nadddilem2.4	⊢ (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))
nadddilem2.5	⊢ (𝜑 → ∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)))
nadddilem2.6	⊢ (𝜑 → ∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)))
nadddilem2.7	⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)))
nadddilem2.8	⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)))

Assertion

Ref	Expression
nadddilem2	⊢ (𝜑 → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) ⊆ (𝐴 ·no (𝐵 +no 𝐶)))

Distinct variable groups:   𝐴,𝑑,𝑒,𝑓   𝐵,𝑑,𝑒,𝑓   𝐶,𝑑,𝑒,𝑓

Allowed substitution hints:   𝜑(𝑒, 𝑓, 𝑑)

Proof of Theorem nadddilem2

Dummy variables 𝑝 𝑞 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nadddilem2.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ On)
2		nadddilem2.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ On)
3		nadddilem2.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ On)
4		nadddilem2.4	. . . . . 6 ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))
5		oveq1 7458	. . . . . . . . 9 ⊢ (𝑑 = 𝑝 → (𝑑 ·no (𝐵 +no 𝐶)) = (𝑝 ·no (𝐵 +no 𝐶)))
6		oveq1 7458	. . . . . . . . . 10 ⊢ (𝑑 = 𝑝 → (𝑑 ·no 𝐵) = (𝑝 ·no 𝐵))
7		oveq1 7458	. . . . . . . . . 10 ⊢ (𝑑 = 𝑝 → (𝑑 ·no 𝐶) = (𝑝 ·no 𝐶))
8	6, 7	oveq12d 7469	. . . . . . . . 9 ⊢ (𝑑 = 𝑝 → ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝐶)))
9	5, 8	eqeq12d 2779	. . . . . . . 8 ⊢ (𝑑 = 𝑝 → ((𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)) ↔ (𝑝 ·no (𝐵 +no 𝐶)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝐶))))
10	9	cbvralvw 3244	. . . . . . 7 ⊢ (∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)) ↔ ∀𝑝 ∈ 𝐴 (𝑝 ·no (𝐵 +no 𝐶)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝐶)))
11	3, 2	naddcomd 37173	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 +no 𝐶) = (𝐶 +no 𝐵))
12	11	oveq2d 7467	. . . . . . . . . 10 ⊢ (𝜑 → (𝑝 ·no (𝐵 +no 𝐶)) = (𝑝 ·no (𝐶 +no 𝐵)))
13	12	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝 ·no (𝐵 +no 𝐶)) = (𝑝 ·no (𝐶 +no 𝐵)))
14	1	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝐴 ∈ On)
15		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐴)
16	14, 15	onelond 37163	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ On)
17	3	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝐵 ∈ On)
18	16, 17	nmulcld 37157	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝 ·no 𝐵) ∈ On)
19	2	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝐶 ∈ On)
20	16, 19	nmulcld 37157	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝 ·no 𝐶) ∈ On)
21	18, 20	naddcomd 37173	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝐶)) = ((𝑝 ·no 𝐶) +no (𝑝 ·no 𝐵)))
22	13, 21	eqeq12d 2779	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → ((𝑝 ·no (𝐵 +no 𝐶)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝐶)) ↔ (𝑝 ·no (𝐶 +no 𝐵)) = ((𝑝 ·no 𝐶) +no (𝑝 ·no 𝐵))))
23	22	ralbidva 3187	. . . . . . 7 ⊢ (𝜑 → (∀𝑝 ∈ 𝐴 (𝑝 ·no (𝐵 +no 𝐶)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝐶)) ↔ ∀𝑝 ∈ 𝐴 (𝑝 ·no (𝐶 +no 𝐵)) = ((𝑝 ·no 𝐶) +no (𝑝 ·no 𝐵))))
24	10, 23	bitrid 286	. . . . . 6 ⊢ (𝜑 → (∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)) ↔ ∀𝑝 ∈ 𝐴 (𝑝 ·no (𝐶 +no 𝐵)) = ((𝑝 ·no 𝐶) +no (𝑝 ·no 𝐵))))
25	4, 24	mpbid 235	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ 𝐴 (𝑝 ·no (𝐶 +no 𝐵)) = ((𝑝 ·no 𝐶) +no (𝑝 ·no 𝐵)))
26		nadddilem2.5	. . . . . 6 ⊢ (𝜑 → ∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)))
27		oveq1 7458	. . . . . . . . . 10 ⊢ (𝑒 = 𝑞 → (𝑒 +no 𝐶) = (𝑞 +no 𝐶))
28	27	oveq2d 7467	. . . . . . . . 9 ⊢ (𝑒 = 𝑞 → (𝐴 ·no (𝑒 +no 𝐶)) = (𝐴 ·no (𝑞 +no 𝐶)))
29		oveq2 7459	. . . . . . . . . 10 ⊢ (𝑒 = 𝑞 → (𝐴 ·no 𝑒) = (𝐴 ·no 𝑞))
30	29	oveq1d 7466	. . . . . . . . 9 ⊢ (𝑒 = 𝑞 → ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)) = ((𝐴 ·no 𝑞) +no (𝐴 ·no 𝐶)))
31	28, 30	eqeq12d 2779	. . . . . . . 8 ⊢ (𝑒 = 𝑞 → ((𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)) ↔ (𝐴 ·no (𝑞 +no 𝐶)) = ((𝐴 ·no 𝑞) +no (𝐴 ·no 𝐶))))
32	31	cbvralvw 3244	. . . . . . 7 ⊢ (∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)) ↔ ∀𝑞 ∈ 𝐵 (𝐴 ·no (𝑞 +no 𝐶)) = ((𝐴 ·no 𝑞) +no (𝐴 ·no 𝐶)))
33	3	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝐵 ∈ On)
34		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑞 ∈ 𝐵)
35	33, 34	onelond 37163	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑞 ∈ On)
36	2	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝐶 ∈ On)
37	35, 36	naddcomd 37173	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝑞 +no 𝐶) = (𝐶 +no 𝑞))
38	37	oveq2d 7467	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐴 ·no (𝑞 +no 𝐶)) = (𝐴 ·no (𝐶 +no 𝑞)))
39	1	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝐴 ∈ On)
40	39, 35	nmulcld 37157	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐴 ·no 𝑞) ∈ On)
41	39, 36	nmulcld 37157	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐴 ·no 𝐶) ∈ On)
42	40, 41	naddcomd 37173	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → ((𝐴 ·no 𝑞) +no (𝐴 ·no 𝐶)) = ((𝐴 ·no 𝐶) +no (𝐴 ·no 𝑞)))
43	38, 42	eqeq12d 2779	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → ((𝐴 ·no (𝑞 +no 𝐶)) = ((𝐴 ·no 𝑞) +no (𝐴 ·no 𝐶)) ↔ (𝐴 ·no (𝐶 +no 𝑞)) = ((𝐴 ·no 𝐶) +no (𝐴 ·no 𝑞))))
44	43	ralbidva 3187	. . . . . . 7 ⊢ (𝜑 → (∀𝑞 ∈ 𝐵 (𝐴 ·no (𝑞 +no 𝐶)) = ((𝐴 ·no 𝑞) +no (𝐴 ·no 𝐶)) ↔ ∀𝑞 ∈ 𝐵 (𝐴 ·no (𝐶 +no 𝑞)) = ((𝐴 ·no 𝐶) +no (𝐴 ·no 𝑞))))
45	32, 44	bitrid 286	. . . . . 6 ⊢ (𝜑 → (∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)) ↔ ∀𝑞 ∈ 𝐵 (𝐴 ·no (𝐶 +no 𝑞)) = ((𝐴 ·no 𝐶) +no (𝐴 ·no 𝑞))))
46	26, 45	mpbid 235	. . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ 𝐵 (𝐴 ·no (𝐶 +no 𝑞)) = ((𝐴 ·no 𝐶) +no (𝐴 ·no 𝑞)))
47		nadddilem2.7	. . . . . 6 ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)))
48		oveq1 7458	. . . . . . . . 9 ⊢ (𝑑 = 𝑝 → (𝑑 ·no (𝑒 +no 𝐶)) = (𝑝 ·no (𝑒 +no 𝐶)))
49		oveq1 7458	. . . . . . . . . 10 ⊢ (𝑑 = 𝑝 → (𝑑 ·no 𝑒) = (𝑝 ·no 𝑒))
50	49, 7	oveq12d 7469	. . . . . . . . 9 ⊢ (𝑑 = 𝑝 → ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)) = ((𝑝 ·no 𝑒) +no (𝑝 ·no 𝐶)))
51	48, 50	eqeq12d 2779	. . . . . . . 8 ⊢ (𝑑 = 𝑝 → ((𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)) ↔ (𝑝 ·no (𝑒 +no 𝐶)) = ((𝑝 ·no 𝑒) +no (𝑝 ·no 𝐶))))
52	27	oveq2d 7467	. . . . . . . . 9 ⊢ (𝑒 = 𝑞 → (𝑝 ·no (𝑒 +no 𝐶)) = (𝑝 ·no (𝑞 +no 𝐶)))
53		oveq2 7459	. . . . . . . . . 10 ⊢ (𝑒 = 𝑞 → (𝑝 ·no 𝑒) = (𝑝 ·no 𝑞))
54	53	oveq1d 7466	. . . . . . . . 9 ⊢ (𝑒 = 𝑞 → ((𝑝 ·no 𝑒) +no (𝑝 ·no 𝐶)) = ((𝑝 ·no 𝑞) +no (𝑝 ·no 𝐶)))
55	52, 54	eqeq12d 2779	. . . . . . . 8 ⊢ (𝑒 = 𝑞 → ((𝑝 ·no (𝑒 +no 𝐶)) = ((𝑝 ·no 𝑒) +no (𝑝 ·no 𝐶)) ↔ (𝑝 ·no (𝑞 +no 𝐶)) = ((𝑝 ·no 𝑞) +no (𝑝 ·no 𝐶))))
56	51, 55	cbvral2vw 3248	. . . . . . 7 ⊢ (∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)) ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐵 (𝑝 ·no (𝑞 +no 𝐶)) = ((𝑝 ·no 𝑞) +no (𝑝 ·no 𝐶)))
57	37	adantrl 729	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐵)) → (𝑞 +no 𝐶) = (𝐶 +no 𝑞))
58	57	oveq2d 7467	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐵)) → (𝑝 ·no (𝑞 +no 𝐶)) = (𝑝 ·no (𝐶 +no 𝑞)))
59	16	adantrr 730	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐵)) → 𝑝 ∈ On)
60	35	adantrl 729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐵)) → 𝑞 ∈ On)
61	59, 60	nmulcld 37157	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐵)) → (𝑝 ·no 𝑞) ∈ On)
62	2	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐵)) → 𝐶 ∈ On)
63	59, 62	nmulcld 37157	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐵)) → (𝑝 ·no 𝐶) ∈ On)
64	61, 63	naddcomd 37173	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐵)) → ((𝑝 ·no 𝑞) +no (𝑝 ·no 𝐶)) = ((𝑝 ·no 𝐶) +no (𝑝 ·no 𝑞)))
65	58, 64	eqeq12d 2779	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐵)) → ((𝑝 ·no (𝑞 +no 𝐶)) = ((𝑝 ·no 𝑞) +no (𝑝 ·no 𝐶)) ↔ (𝑝 ·no (𝐶 +no 𝑞)) = ((𝑝 ·no 𝐶) +no (𝑝 ·no 𝑞))))
66	65	2ralbidva 3228	. . . . . . 7 ⊢ (𝜑 → (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐵 (𝑝 ·no (𝑞 +no 𝐶)) = ((𝑝 ·no 𝑞) +no (𝑝 ·no 𝐶)) ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐵 (𝑝 ·no (𝐶 +no 𝑞)) = ((𝑝 ·no 𝐶) +no (𝑝 ·no 𝑞))))
67	56, 66	bitrid 286	. . . . . 6 ⊢ (𝜑 → (∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)) ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐵 (𝑝 ·no (𝐶 +no 𝑞)) = ((𝑝 ·no 𝐶) +no (𝑝 ·no 𝑞))))
68	47, 67	mpbid 235	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐵 (𝑝 ·no (𝐶 +no 𝑞)) = ((𝑝 ·no 𝐶) +no (𝑝 ·no 𝑞)))
69	1, 2, 3, 25, 46, 68	nadddilem1 37184	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ·no 𝐵)) → ((𝐴 ·no 𝐶) +no 𝑥) ∈ (𝐴 ·no (𝐶 +no 𝐵)))
70	1, 3	nmulcld 37157	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·no 𝐵) ∈ On)
71	70	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ·no 𝐵)) → (𝐴 ·no 𝐵) ∈ On)
72		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ·no 𝐵)) → 𝑥 ∈ (𝐴 ·no 𝐵))
73	71, 72	onelond 37163	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ·no 𝐵)) → 𝑥 ∈ On)
74	1, 2	nmulcld 37157	. . . . . 6 ⊢ (𝜑 → (𝐴 ·no 𝐶) ∈ On)
75	74	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ·no 𝐵)) → (𝐴 ·no 𝐶) ∈ On)
76	73, 75	naddcomd 37173	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ·no 𝐵)) → (𝑥 +no (𝐴 ·no 𝐶)) = ((𝐴 ·no 𝐶) +no 𝑥))
77	11	oveq2d 7467	. . . . 5 ⊢ (𝜑 → (𝐴 ·no (𝐵 +no 𝐶)) = (𝐴 ·no (𝐶 +no 𝐵)))
78	77	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ·no 𝐵)) → (𝐴 ·no (𝐵 +no 𝐶)) = (𝐴 ·no (𝐶 +no 𝐵)))
79	69, 76, 78	3eltr4d 2879	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ·no 𝐵)) → (𝑥 +no (𝐴 ·no 𝐶)) ∈ (𝐴 ·no (𝐵 +no 𝐶)))
80	79	ralrimiva 3158	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ·no 𝐵)(𝑥 +no (𝐴 ·no 𝐶)) ∈ (𝐴 ·no (𝐵 +no 𝐶)))
81		nadddilem2.6	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)))
82		nadddilem2.8	. . . 4 ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)))
83	1, 3, 2, 4, 81, 82	nadddilem1 37184	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ·no 𝐶)) → ((𝐴 ·no 𝐵) +no 𝑦) ∈ (𝐴 ·no (𝐵 +no 𝐶)))
84	83	ralrimiva 3158	. 2 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴 ·no 𝐶)((𝐴 ·no 𝐵) +no 𝑦) ∈ (𝐴 ·no (𝐵 +no 𝐶)))
85	3, 2	naddcld 8727	. . . 4 ⊢ (𝜑 → (𝐵 +no 𝐶) ∈ On)
86	1, 85	nmulcld 37157	. . 3 ⊢ (𝜑 → (𝐴 ·no (𝐵 +no 𝐶)) ∈ On)
87		naddle 37183	. . 3 ⊢ (((𝐴 ·no 𝐵) ∈ On ∧ (𝐴 ·no 𝐶) ∈ On ∧ (𝐴 ·no (𝐵 +no 𝐶)) ∈ On) → (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) ⊆ (𝐴 ·no (𝐵 +no 𝐶)) ↔ (∀𝑥 ∈ (𝐴 ·no 𝐵)(𝑥 +no (𝐴 ·no 𝐶)) ∈ (𝐴 ·no (𝐵 +no 𝐶)) ∧ ∀𝑦 ∈ (𝐴 ·no 𝐶)((𝐴 ·no 𝐵) +no 𝑦) ∈ (𝐴 ·no (𝐵 +no 𝐶)))))
88	70, 74, 86, 87	syl3anc 1399	. 2 ⊢ (𝜑 → (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) ⊆ (𝐴 ·no (𝐵 +no 𝐶)) ↔ (∀𝑥 ∈ (𝐴 ·no 𝐵)(𝑥 +no (𝐴 ·no 𝐶)) ∈ (𝐴 ·no (𝐵 +no 𝐶)) ∧ ∀𝑦 ∈ (𝐴 ·no 𝐶)((𝐴 ·no 𝐵) +no 𝑦) ∈ (𝐴 ·no (𝐵 +no 𝐶)))))
89	80, 84, 88	mpbir2and 726	1 ⊢ (𝜑 → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) ⊆ (𝐴 ·no (𝐵 +no 𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900  Oncon0 6388  (class class class)co 7451   +no cnadd 8712   ·_no cnmul 37151

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-frecs 8337  df-nadd 8713  df-nmul 37152

This theorem is used by:  nadddi  37188