Theorem nadddilem3 36714

Description: Lemma for nadddi 36716. Prove a subcase of the forward implication. (Contributed by Scott Fenton, 3-Aug-2026.)

Hypotheses

Ref	Expression
nadddilem3.1	⊢ (𝜑 → 𝐴 ∈ On)
nadddilem3.2	⊢ (𝜑 → 𝐵 ∈ On)
nadddilem3.3	⊢ (𝜑 → 𝐶 ∈ On)
nadddilem3.4	⊢ (𝜑 → 𝑋 ∈ 𝐴)
nadddilem3.5	⊢ (𝜑 → 𝑌 ∈ (𝐵 +no 𝐶))
nadddilem3.6	⊢ (𝜑 → 𝑍 ∈ 𝐵)
nadddilem3.7	⊢ (𝜑 → 𝑌 ⊆ (𝑍 +no 𝐶))
nadddilem3.8	⊢ (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))
nadddilem3.9	⊢ (𝜑 → ∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)))
nadddilem3.10	⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)))

Assertion

Ref	Expression
nadddilem3	⊢ (𝜑 → ((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑋 ·no 𝑌)))

Distinct variable groups:   𝐴,𝑑,𝑒   𝐵,𝑑,𝑒   𝐶,𝑑,𝑒   𝑋,𝑑,𝑒   𝑌,𝑑,𝑒   𝑍,𝑑,𝑒

Allowed substitution hints:   𝜑(𝑒,𝑑)

Proof of Theorem nadddilem3

Step	Hyp	Ref	Expression
1		nadddilem3.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
2		nadddilem3.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ On)
3		nadddilem3.6	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
4	2, 3	onelond 36691	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ On)
5		nadddilem3.3	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On)
6	4, 5	naddcld 8662	. . . . . . 7 ⊢ (𝜑 → (𝑍 +no 𝐶) ∈ On)
7		nadddilem3.4	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
8	1, 7	onelond 36691	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ On)
9	2, 5	naddcld 8662	. . . . . . . 8 ⊢ (𝜑 → (𝐵 +no 𝐶) ∈ On)
10		nadddilem3.5	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝐵 +no 𝐶))
11	9, 10	onelond 36691	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ On)
12	1, 7	onelssd 36693	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝐴)
13		nadddilem3.7	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ (𝑍 +no 𝐶))
14		nmuladdss 36705	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ (𝑍 +no 𝐶) ∈ On) ∧ (𝑋 ∈ On ∧ 𝑌 ∈ On) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ (𝑍 +no 𝐶))) → ((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝑌)) ⊆ ((𝐴 ·no (𝑍 +no 𝐶)) +no (𝑋 ·no 𝑌)))
15	1, 6, 8, 11, 12, 13, 14	syl222anc 1413	. . . . . 6 ⊢ (𝜑 → ((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝑌)) ⊆ ((𝐴 ·no (𝑍 +no 𝐶)) +no (𝑋 ·no 𝑌)))
16	8, 6	nmulcld 36685	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ·no (𝑍 +no 𝐶)) ∈ On)
17	1, 11	nmulcld 36685	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ·no 𝑌) ∈ On)
18	16, 17	naddcld 8662	. . . . . . 7 ⊢ (𝜑 → ((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝑌)) ∈ On)
19	1, 6	nmulcld 36685	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ·no (𝑍 +no 𝐶)) ∈ On)
20	8, 11	nmulcld 36685	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ·no 𝑌) ∈ On)
21	19, 20	naddcld 8662	. . . . . . 7 ⊢ (𝜑 → ((𝐴 ·no (𝑍 +no 𝐶)) +no (𝑋 ·no 𝑌)) ∈ On)
22	1, 2	nmulcld 36685	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·no 𝐵) ∈ On)
23		naddss2 8673	. . . . . . 7 ⊢ ((((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝑌)) ∈ On ∧ ((𝐴 ·no (𝑍 +no 𝐶)) +no (𝑋 ·no 𝑌)) ∈ On ∧ (𝐴 ·no 𝐵) ∈ On) → (((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝑌)) ⊆ ((𝐴 ·no (𝑍 +no 𝐶)) +no (𝑋 ·no 𝑌)) ↔ ((𝐴 ·no 𝐵) +no ((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝑌))) ⊆ ((𝐴 ·no 𝐵) +no ((𝐴 ·no (𝑍 +no 𝐶)) +no (𝑋 ·no 𝑌)))))
24	18, 21, 22, 23	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → (((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝑌)) ⊆ ((𝐴 ·no (𝑍 +no 𝐶)) +no (𝑋 ·no 𝑌)) ↔ ((𝐴 ·no 𝐵) +no ((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝑌))) ⊆ ((𝐴 ·no 𝐵) +no ((𝐴 ·no (𝑍 +no 𝐶)) +no (𝑋 ·no 𝑌)))))
25	15, 24	mpbid 235	. . . . 5 ⊢ (𝜑 → ((𝐴 ·no 𝐵) +no ((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝑌))) ⊆ ((𝐴 ·no 𝐵) +no ((𝐴 ·no (𝑍 +no 𝐶)) +no (𝑋 ·no 𝑌))))
26		nmuladdel 36704	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑋 ∈ 𝐴 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ·no 𝐵) +no (𝐴 ·no 𝑍)) ∈ ((𝐴 ·no 𝐵) +no (𝑋 ·no 𝑍)))
27	1, 2, 7, 3, 26	syl22anc 851	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 ·no 𝐵) +no (𝐴 ·no 𝑍)) ∈ ((𝐴 ·no 𝐵) +no (𝑋 ·no 𝑍)))
28	8, 4	nmulcld 36685	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ·no 𝑍) ∈ On)
29	28, 22	naddcomd 36701	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 ·no 𝑍) +no (𝐴 ·no 𝐵)) = ((𝐴 ·no 𝐵) +no (𝑋 ·no 𝑍)))
30	27, 29	eleqtrrd 2866	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 ·no 𝐵) +no (𝐴 ·no 𝑍)) ∈ ((𝑋 ·no 𝑍) +no (𝐴 ·no 𝐵)))
31	8, 2	nmulcld 36685	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 ·no 𝐵) ∈ On)
32	1, 4	nmulcld 36685	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ·no 𝑍) ∈ On)
33	31, 32	naddcld 8662	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 ·no 𝐵) +no (𝐴 ·no 𝑍)) ∈ On)
34	28, 22	naddcld 8662	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 ·no 𝑍) +no (𝐴 ·no 𝐵)) ∈ On)
35	8, 5	nmulcld 36685	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ·no 𝐶) ∈ On)
36		naddel2 8671	. . . . . . . . . . 11 ⊢ ((((𝑋 ·no 𝐵) +no (𝐴 ·no 𝑍)) ∈ On ∧ ((𝑋 ·no 𝑍) +no (𝐴 ·no 𝐵)) ∈ On ∧ (𝑋 ·no 𝐶) ∈ On) → (((𝑋 ·no 𝐵) +no (𝐴 ·no 𝑍)) ∈ ((𝑋 ·no 𝑍) +no (𝐴 ·no 𝐵)) ↔ ((𝑋 ·no 𝐶) +no ((𝑋 ·no 𝐵) +no (𝐴 ·no 𝑍))) ∈ ((𝑋 ·no 𝐶) +no ((𝑋 ·no 𝑍) +no (𝐴 ·no 𝐵)))))
37	33, 34, 35, 36	syl3anc 1398	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑋 ·no 𝐵) +no (𝐴 ·no 𝑍)) ∈ ((𝑋 ·no 𝑍) +no (𝐴 ·no 𝐵)) ↔ ((𝑋 ·no 𝐶) +no ((𝑋 ·no 𝐵) +no (𝐴 ·no 𝑍))) ∈ ((𝑋 ·no 𝐶) +no ((𝑋 ·no 𝑍) +no (𝐴 ·no 𝐵)))))
38	30, 37	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 ·no 𝐶) +no ((𝑋 ·no 𝐵) +no (𝐴 ·no 𝑍))) ∈ ((𝑋 ·no 𝐶) +no ((𝑋 ·no 𝑍) +no (𝐴 ·no 𝐵))))
39	35, 28	naddcomd 36701	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 ·no 𝐶) +no (𝑋 ·no 𝑍)) = ((𝑋 ·no 𝑍) +no (𝑋 ·no 𝐶)))
40	39	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑋 ·no 𝐶) +no (𝑋 ·no 𝑍)) +no (𝐴 ·no 𝐵)) = (((𝑋 ·no 𝑍) +no (𝑋 ·no 𝐶)) +no (𝐴 ·no 𝐵)))
41	35, 28, 22	naddassd 36702	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑋 ·no 𝐶) +no (𝑋 ·no 𝑍)) +no (𝐴 ·no 𝐵)) = ((𝑋 ·no 𝐶) +no ((𝑋 ·no 𝑍) +no (𝐴 ·no 𝐵))))
42	28, 35, 22	naddassd 36702	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑋 ·no 𝑍) +no (𝑋 ·no 𝐶)) +no (𝐴 ·no 𝐵)) = ((𝑋 ·no 𝑍) +no ((𝑋 ·no 𝐶) +no (𝐴 ·no 𝐵))))
43	40, 41, 42	3eqtr3d 2806	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 ·no 𝐶) +no ((𝑋 ·no 𝑍) +no (𝐴 ·no 𝐵))) = ((𝑋 ·no 𝑍) +no ((𝑋 ·no 𝐶) +no (𝐴 ·no 𝐵))))
44	38, 43	eleqtrd 2865	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ·no 𝐶) +no ((𝑋 ·no 𝐵) +no (𝐴 ·no 𝑍))) ∈ ((𝑋 ·no 𝑍) +no ((𝑋 ·no 𝐶) +no (𝐴 ·no 𝐵))))
45		oveq1 7417	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑋 → (𝑑 ·no (𝐵 +no 𝐶)) = (𝑋 ·no (𝐵 +no 𝐶)))
46		oveq1 7417	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑋 → (𝑑 ·no 𝐵) = (𝑋 ·no 𝐵))
47		oveq1 7417	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑋 → (𝑑 ·no 𝐶) = (𝑋 ·no 𝐶))
48	46, 47	oveq12d 7428	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑋 → ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)) = ((𝑋 ·no 𝐵) +no (𝑋 ·no 𝐶)))
49	45, 48	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑋 → ((𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)) ↔ (𝑋 ·no (𝐵 +no 𝐶)) = ((𝑋 ·no 𝐵) +no (𝑋 ·no 𝐶))))
50		nadddilem3.8	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))
51	49, 50, 7	rspcdva 3582	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ·no (𝐵 +no 𝐶)) = ((𝑋 ·no 𝐵) +no (𝑋 ·no 𝐶)))
52	31, 35	naddcomd 36701	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑋 ·no 𝐵) +no (𝑋 ·no 𝐶)) = ((𝑋 ·no 𝐶) +no (𝑋 ·no 𝐵)))
53	51, 52	eqtrd 2798	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ·no (𝐵 +no 𝐶)) = ((𝑋 ·no 𝐶) +no (𝑋 ·no 𝐵)))
54	53	oveq1d 7425	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑍)) = (((𝑋 ·no 𝐶) +no (𝑋 ·no 𝐵)) +no (𝐴 ·no 𝑍)))
55	35, 31, 32	naddassd 36702	. . . . . . . . 9 ⊢ (𝜑 → (((𝑋 ·no 𝐶) +no (𝑋 ·no 𝐵)) +no (𝐴 ·no 𝑍)) = ((𝑋 ·no 𝐶) +no ((𝑋 ·no 𝐵) +no (𝐴 ·no 𝑍))))
56	54, 55	eqtrd 2798	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑍)) = ((𝑋 ·no 𝐶) +no ((𝑋 ·no 𝐵) +no (𝐴 ·no 𝑍))))
57	22, 16	naddcomd 36701	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ·no 𝐵) +no (𝑋 ·no (𝑍 +no 𝐶))) = ((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝐵)))
58		oveq1 7417	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑋 → (𝑑 ·no (𝑒 +no 𝐶)) = (𝑋 ·no (𝑒 +no 𝐶)))
59		oveq1 7417	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑋 → (𝑑 ·no 𝑒) = (𝑋 ·no 𝑒))
60	59, 47	oveq12d 7428	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑋 → ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)) = ((𝑋 ·no 𝑒) +no (𝑋 ·no 𝐶)))
61	58, 60	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑋 → ((𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)) ↔ (𝑋 ·no (𝑒 +no 𝐶)) = ((𝑋 ·no 𝑒) +no (𝑋 ·no 𝐶))))
62		oveq1 7417	. . . . . . . . . . . . . 14 ⊢ (𝑒 = 𝑍 → (𝑒 +no 𝐶) = (𝑍 +no 𝐶))
63	62	oveq2d 7426	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑍 → (𝑋 ·no (𝑒 +no 𝐶)) = (𝑋 ·no (𝑍 +no 𝐶)))
64		oveq2 7418	. . . . . . . . . . . . . 14 ⊢ (𝑒 = 𝑍 → (𝑋 ·no 𝑒) = (𝑋 ·no 𝑍))
65	64	oveq1d 7425	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑍 → ((𝑋 ·no 𝑒) +no (𝑋 ·no 𝐶)) = ((𝑋 ·no 𝑍) +no (𝑋 ·no 𝐶)))
66	63, 65	eqeq12d 2779	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑍 → ((𝑋 ·no (𝑒 +no 𝐶)) = ((𝑋 ·no 𝑒) +no (𝑋 ·no 𝐶)) ↔ (𝑋 ·no (𝑍 +no 𝐶)) = ((𝑋 ·no 𝑍) +no (𝑋 ·no 𝐶))))
67		nadddilem3.10	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)))
68	61, 66, 67, 7, 3	rspc2dv 3596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ·no (𝑍 +no 𝐶)) = ((𝑋 ·no 𝑍) +no (𝑋 ·no 𝐶)))
69	68	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝐵)) = (((𝑋 ·no 𝑍) +no (𝑋 ·no 𝐶)) +no (𝐴 ·no 𝐵)))
70	69, 42	eqtrd 2798	. . . . . . . . 9 ⊢ (𝜑 → ((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝐵)) = ((𝑋 ·no 𝑍) +no ((𝑋 ·no 𝐶) +no (𝐴 ·no 𝐵))))
71	57, 70	eqtrd 2798	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 ·no 𝐵) +no (𝑋 ·no (𝑍 +no 𝐶))) = ((𝑋 ·no 𝑍) +no ((𝑋 ·no 𝐶) +no (𝐴 ·no 𝐵))))
72	44, 56, 71	3eltr4d 2878	. . . . . . 7 ⊢ (𝜑 → ((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑍)) ∈ ((𝐴 ·no 𝐵) +no (𝑋 ·no (𝑍 +no 𝐶))))
73	8, 9	nmulcld 36685	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ·no (𝐵 +no 𝐶)) ∈ On)
74	73, 32	naddcld 8662	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑍)) ∈ On)
75	22, 16	naddcld 8662	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 ·no 𝐵) +no (𝑋 ·no (𝑍 +no 𝐶))) ∈ On)
76		naddel1 8670	. . . . . . . 8 ⊢ ((((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑍)) ∈ On ∧ ((𝐴 ·no 𝐵) +no (𝑋 ·no (𝑍 +no 𝐶))) ∈ On ∧ (𝐴 ·no 𝑌) ∈ On) → (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑍)) ∈ ((𝐴 ·no 𝐵) +no (𝑋 ·no (𝑍 +no 𝐶))) ↔ (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑍)) +no (𝐴 ·no 𝑌)) ∈ (((𝐴 ·no 𝐵) +no (𝑋 ·no (𝑍 +no 𝐶))) +no (𝐴 ·no 𝑌))))
77	74, 75, 17, 76	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑍)) ∈ ((𝐴 ·no 𝐵) +no (𝑋 ·no (𝑍 +no 𝐶))) ↔ (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑍)) +no (𝐴 ·no 𝑌)) ∈ (((𝐴 ·no 𝐵) +no (𝑋 ·no (𝑍 +no 𝐶))) +no (𝐴 ·no 𝑌))))
78	72, 77	mpbid 235	. . . . . 6 ⊢ (𝜑 → (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑍)) +no (𝐴 ·no 𝑌)) ∈ (((𝐴 ·no 𝐵) +no (𝑋 ·no (𝑍 +no 𝐶))) +no (𝐴 ·no 𝑌)))
79	73, 32, 17	nadd32d 36703	. . . . . 6 ⊢ (𝜑 → (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑍)) +no (𝐴 ·no 𝑌)) = (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) +no (𝐴 ·no 𝑍)))
80	22, 16, 17	naddassd 36702	. . . . . 6 ⊢ (𝜑 → (((𝐴 ·no 𝐵) +no (𝑋 ·no (𝑍 +no 𝐶))) +no (𝐴 ·no 𝑌)) = ((𝐴 ·no 𝐵) +no ((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝑌))))
81	78, 79, 80	3eltr3d 2877	. . . . 5 ⊢ (𝜑 → (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) +no (𝐴 ·no 𝑍)) ∈ ((𝐴 ·no 𝐵) +no ((𝑋 ·no (𝑍 +no 𝐶)) +no (𝐴 ·no 𝑌))))
82	25, 81	sseldd 3938	. . . 4 ⊢ (𝜑 → (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) +no (𝐴 ·no 𝑍)) ∈ ((𝐴 ·no 𝐵) +no ((𝐴 ·no (𝑍 +no 𝐶)) +no (𝑋 ·no 𝑌))))
83	22, 19, 20	naddassd 36702	. . . 4 ⊢ (𝜑 → (((𝐴 ·no 𝐵) +no (𝐴 ·no (𝑍 +no 𝐶))) +no (𝑋 ·no 𝑌)) = ((𝐴 ·no 𝐵) +no ((𝐴 ·no (𝑍 +no 𝐶)) +no (𝑋 ·no 𝑌))))
84	82, 83	eleqtrrd 2866	. . 3 ⊢ (𝜑 → (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) +no (𝐴 ·no 𝑍)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no (𝑍 +no 𝐶))) +no (𝑋 ·no 𝑌)))
85	62	oveq2d 7426	. . . . . . . . . 10 ⊢ (𝑒 = 𝑍 → (𝐴 ·no (𝑒 +no 𝐶)) = (𝐴 ·no (𝑍 +no 𝐶)))
86		oveq2 7418	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑍 → (𝐴 ·no 𝑒) = (𝐴 ·no 𝑍))
87	86	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝑒 = 𝑍 → ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)) = ((𝐴 ·no 𝑍) +no (𝐴 ·no 𝐶)))
88	85, 87	eqeq12d 2779	. . . . . . . . 9 ⊢ (𝑒 = 𝑍 → ((𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)) ↔ (𝐴 ·no (𝑍 +no 𝐶)) = ((𝐴 ·no 𝑍) +no (𝐴 ·no 𝐶))))
89		nadddilem3.9	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)))
90	88, 89, 3	rspcdva 3582	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ·no (𝑍 +no 𝐶)) = ((𝐴 ·no 𝑍) +no (𝐴 ·no 𝐶)))
91	1, 5	nmulcld 36685	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ·no 𝐶) ∈ On)
92	32, 91	naddcomd 36701	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 ·no 𝑍) +no (𝐴 ·no 𝐶)) = ((𝐴 ·no 𝐶) +no (𝐴 ·no 𝑍)))
93	90, 92	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → (𝐴 ·no (𝑍 +no 𝐶)) = ((𝐴 ·no 𝐶) +no (𝐴 ·no 𝑍)))
94	93	oveq2d 7426	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·no 𝐵) +no (𝐴 ·no (𝑍 +no 𝐶))) = ((𝐴 ·no 𝐵) +no ((𝐴 ·no 𝐶) +no (𝐴 ·no 𝑍))))
95	22, 91, 32	naddassd 36702	. . . . . 6 ⊢ (𝜑 → (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝐴 ·no 𝑍)) = ((𝐴 ·no 𝐵) +no ((𝐴 ·no 𝐶) +no (𝐴 ·no 𝑍))))
96	94, 95	eqtr4d 2801	. . . . 5 ⊢ (𝜑 → ((𝐴 ·no 𝐵) +no (𝐴 ·no (𝑍 +no 𝐶))) = (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝐴 ·no 𝑍)))
97	96	oveq1d 7425	. . . 4 ⊢ (𝜑 → (((𝐴 ·no 𝐵) +no (𝐴 ·no (𝑍 +no 𝐶))) +no (𝑋 ·no 𝑌)) = ((((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝐴 ·no 𝑍)) +no (𝑋 ·no 𝑌)))
98	22, 91	naddcld 8662	. . . . 5 ⊢ (𝜑 → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) ∈ On)
99	98, 32, 20	nadd32d 36703	. . . 4 ⊢ (𝜑 → ((((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝐴 ·no 𝑍)) +no (𝑋 ·no 𝑌)) = ((((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑋 ·no 𝑌)) +no (𝐴 ·no 𝑍)))
100	97, 99	eqtrd 2798	. . 3 ⊢ (𝜑 → (((𝐴 ·no 𝐵) +no (𝐴 ·no (𝑍 +no 𝐶))) +no (𝑋 ·no 𝑌)) = ((((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑋 ·no 𝑌)) +no (𝐴 ·no 𝑍)))
101	84, 100	eleqtrd 2865	. 2 ⊢ (𝜑 → (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) +no (𝐴 ·no 𝑍)) ∈ ((((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑋 ·no 𝑌)) +no (𝐴 ·no 𝑍)))
102	73, 17	naddcld 8662	. . 3 ⊢ (𝜑 → ((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) ∈ On)
103	98, 20	naddcld 8662	. . 3 ⊢ (𝜑 → (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑋 ·no 𝑌)) ∈ On)
104		naddel1 8670	. . 3 ⊢ ((((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) ∈ On ∧ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑋 ·no 𝑌)) ∈ On ∧ (𝐴 ·no 𝑍) ∈ On) → (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑋 ·no 𝑌)) ↔ (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) +no (𝐴 ·no 𝑍)) ∈ ((((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑋 ·no 𝑌)) +no (𝐴 ·no 𝑍))))
105	102, 103, 32, 104	syl3anc 1398	. 2 ⊢ (𝜑 → (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑋 ·no 𝑌)) ↔ (((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) +no (𝐴 ·no 𝑍)) ∈ ((((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑋 ·no 𝑌)) +no (𝐴 ·no 𝑍))))
106	101, 105	mpbird 260	1 ⊢ (𝜑 → ((𝑋 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑌)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑋 ·no 𝑌)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3905  Oncon0 6360  (class class class)co 7410   +no cnadd 8647   ·_no cnmul 36679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-ot 4598  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-frecs 8274  df-nadd 8648  df-nmul 36680

This theorem is referenced by:  nadddilem4  36715