Theorem nadddilem1 36969

Description: Lemma for nadddi 36973. Prove a subcase of the reverse implication. (Contributed by Scott Fenton, 31-Jul-2026.)

Hypotheses

Ref	Expression
nadddilem1.1	⊢ (𝜑 → 𝐴 ∈ On)
nadddilem1.2	⊢ (𝜑 → 𝐵 ∈ On)
nadddilem1.3	⊢ (𝜑 → 𝐶 ∈ On)
nadddilem1.4	⊢ (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))
nadddilem1.5	⊢ (𝜑 → ∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)))
nadddilem1.6	⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)))

Assertion

Ref	Expression
nadddilem1	⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) → ((𝐴 ·no 𝐵) +no 𝑌) ∈ (𝐴 ·no (𝐵 +no 𝐶)))

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

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

Proof of Theorem nadddilem1

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nadddilem1.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
2	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) → 𝐴 ∈ On)
3		nadddilem1.3	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ On)
4	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) → 𝐶 ∈ On)
5	2, 4	nmulcld 36942	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) → (𝐴 ·no 𝐶) ∈ On)
6		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) → 𝑌 ∈ (𝐴 ·no 𝐶))
7	5, 6	onelond 36948	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) → 𝑌 ∈ On)
8		ltnmul 36965	. . . 4 ⊢ ((𝑌 ∈ On ∧ 𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝑌 ∈ (𝐴 ·no 𝐶) ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤))))
9	7, 2, 4, 8	syl3anc 1398	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) → (𝑌 ∈ (𝐴 ·no 𝐶) ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤))))
10	1	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → 𝐴 ∈ On)
11		nadddilem1.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ On)
12	11	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → 𝐵 ∈ On)
13	10, 12	nmulcld 36942	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝐴 ·no 𝐵) ∈ On)
14	7	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → 𝑌 ∈ On)
15		simprll 791	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → 𝑧 ∈ 𝐴)
16	10, 15	onelond 36948	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → 𝑧 ∈ On)
17	3	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → 𝐶 ∈ On)
18		simprlr 792	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → 𝑤 ∈ 𝐶)
19	17, 18	onelond 36948	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → 𝑤 ∈ On)
20	16, 19	nmulcld 36942	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝑧 ·no 𝑤) ∈ On)
21	13, 14, 20	naddassd 36959	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (((𝐴 ·no 𝐵) +no 𝑌) +no (𝑧 ·no 𝑤)) = ((𝐴 ·no 𝐵) +no (𝑌 +no (𝑧 ·no 𝑤))))
22	14, 20	naddcld 8689	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝑌 +no (𝑧 ·no 𝑤)) ∈ On)
23	13, 22	naddcld 8689	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no 𝐵) +no (𝑌 +no (𝑧 ·no 𝑤))) ∈ On)
24	12, 17	naddcld 8689	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝐵 +no 𝐶) ∈ On)
25	10, 24	nmulcld 36942	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝐴 ·no (𝐵 +no 𝐶)) ∈ On)
26	25, 20	naddcld 8689	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤)) ∈ On)
27		simprr 785	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))
28	16, 17	nmulcld 36942	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝑧 ·no 𝐶) ∈ On)
29	10, 19	nmulcld 36942	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝐴 ·no 𝑤) ∈ On)
30	28, 29	naddcld 8689	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)) ∈ On)
31		naddss2 8700	. . . . . . . . . 10 ⊢ (((𝑌 +no (𝑧 ·no 𝑤)) ∈ On ∧ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)) ∈ On ∧ (𝐴 ·no 𝐵) ∈ On) → ((𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)) ↔ ((𝐴 ·no 𝐵) +no (𝑌 +no (𝑧 ·no 𝑤))) ⊆ ((𝐴 ·no 𝐵) +no ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))))
32	22, 30, 13, 31	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)) ↔ ((𝐴 ·no 𝐵) +no (𝑌 +no (𝑧 ·no 𝑤))) ⊆ ((𝐴 ·no 𝐵) +no ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))))
33	27, 32	mpbid 235	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no 𝐵) +no (𝑌 +no (𝑧 ·no 𝑤))) ⊆ ((𝐴 ·no 𝐵) +no ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤))))
34		oveq2 7428	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑤 → (𝐵 +no 𝑓) = (𝐵 +no 𝑤))
35	34	oveq2d 7436	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑤 → (𝐴 ·no (𝐵 +no 𝑓)) = (𝐴 ·no (𝐵 +no 𝑤)))
36		oveq2 7428	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑤 → (𝐴 ·no 𝑓) = (𝐴 ·no 𝑤))
37	36	oveq2d 7436	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑤 → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑤)))
38	35, 37	eqeq12d 2777	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑤 → ((𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)) ↔ (𝐴 ·no (𝐵 +no 𝑤)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑤))))
39		nadddilem1.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)))
40	39	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)))
41	38, 40, 18	rspcdva 3578	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝐴 ·no (𝐵 +no 𝑤)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑤)))
42	41	oveq1d 7435	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no 𝐶)) = (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑤)) +no (𝑧 ·no 𝐶)))
43	13, 29, 28	nadd32d 36960	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑤)) +no (𝑧 ·no 𝐶)) = (((𝐴 ·no 𝐵) +no (𝑧 ·no 𝐶)) +no (𝐴 ·no 𝑤)))
44	13, 28, 29	naddassd 36959	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (((𝐴 ·no 𝐵) +no (𝑧 ·no 𝐶)) +no (𝐴 ·no 𝑤)) = ((𝐴 ·no 𝐵) +no ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤))))
45	42, 43, 44	3eqtrrd 2801	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no 𝐵) +no ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤))) = ((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no 𝐶)))
46		naddel2 8698	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ On ∧ 𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝑤 ∈ 𝐶 ↔ (𝐵 +no 𝑤) ∈ (𝐵 +no 𝐶)))
47	19, 17, 12, 46	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝑤 ∈ 𝐶 ↔ (𝐵 +no 𝑤) ∈ (𝐵 +no 𝐶)))
48	18, 47	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝐵 +no 𝑤) ∈ (𝐵 +no 𝐶))
49		nmuladdel 36961	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ (𝐵 +no 𝐶) ∈ On) ∧ (𝑧 ∈ 𝐴 ∧ (𝐵 +no 𝑤) ∈ (𝐵 +no 𝐶))) → ((𝑧 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no (𝐵 +no 𝑤))) ∈ ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no (𝐵 +no 𝑤))))
50	10, 24, 15, 48, 49	syl22anc 852	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝑧 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no (𝐵 +no 𝑤))) ∈ ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no (𝐵 +no 𝑤))))
51	12, 19	naddcld 8689	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝐵 +no 𝑤) ∈ On)
52	10, 51	nmulcld 36942	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝐴 ·no (𝐵 +no 𝑤)) ∈ On)
53	16, 12	nmulcld 36942	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝑧 ·no 𝐵) ∈ On)
54	52, 28, 53	naddassd 36959	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no 𝐶)) +no (𝑧 ·no 𝐵)) = ((𝐴 ·no (𝐵 +no 𝑤)) +no ((𝑧 ·no 𝐶) +no (𝑧 ·no 𝐵))))
55		oveq1 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑧 → (𝑑 ·no (𝐵 +no 𝐶)) = (𝑧 ·no (𝐵 +no 𝐶)))
56		oveq1 7427	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑧 → (𝑑 ·no 𝐵) = (𝑧 ·no 𝐵))
57		oveq1 7427	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑧 → (𝑑 ·no 𝐶) = (𝑧 ·no 𝐶))
58	56, 57	oveq12d 7438	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑧 → ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)) = ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝐶)))
59	55, 58	eqeq12d 2777	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑧 → ((𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)) ↔ (𝑧 ·no (𝐵 +no 𝐶)) = ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝐶))))
60		nadddilem1.4	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))
61	60	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))
62	59, 61, 15	rspcdva 3578	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝑧 ·no (𝐵 +no 𝐶)) = ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝐶)))
63	53, 28	naddcomd 36958	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝐶)) = ((𝑧 ·no 𝐶) +no (𝑧 ·no 𝐵)))
64	62, 63	eqtrd 2796	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝑧 ·no (𝐵 +no 𝐶)) = ((𝑧 ·no 𝐶) +no (𝑧 ·no 𝐵)))
65	64	oveq2d 7436	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no (𝐵 +no 𝐶))) = ((𝐴 ·no (𝐵 +no 𝑤)) +no ((𝑧 ·no 𝐶) +no (𝑧 ·no 𝐵))))
66	54, 65	eqtr4d 2799	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no 𝐶)) +no (𝑧 ·no 𝐵)) = ((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no (𝐵 +no 𝐶))))
67	16, 24	nmulcld 36942	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝑧 ·no (𝐵 +no 𝐶)) ∈ On)
68	52, 67	naddcomd 36958	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no (𝐵 +no 𝐶))) = ((𝑧 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no (𝐵 +no 𝑤))))
69	66, 68	eqtrd 2796	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no 𝐶)) +no (𝑧 ·no 𝐵)) = ((𝑧 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no (𝐵 +no 𝑤))))
70	25, 20, 53	naddassd 36959	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤)) +no (𝑧 ·no 𝐵)) = ((𝐴 ·no (𝐵 +no 𝐶)) +no ((𝑧 ·no 𝑤) +no (𝑧 ·no 𝐵))))
71	20, 53	naddcomd 36958	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝑧 ·no 𝑤) +no (𝑧 ·no 𝐵)) = ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝑤)))
72		oveq1 7427	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑧 → (𝑑 ·no (𝐵 +no 𝑓)) = (𝑧 ·no (𝐵 +no 𝑓)))
73		oveq1 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑧 → (𝑑 ·no 𝑓) = (𝑧 ·no 𝑓))
74	56, 73	oveq12d 7438	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑧 → ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)) = ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝑓)))
75	72, 74	eqeq12d 2777	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑧 → ((𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)) ↔ (𝑧 ·no (𝐵 +no 𝑓)) = ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝑓))))
76	34	oveq2d 7436	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑤 → (𝑧 ·no (𝐵 +no 𝑓)) = (𝑧 ·no (𝐵 +no 𝑤)))
77		oveq2 7428	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑤 → (𝑧 ·no 𝑓) = (𝑧 ·no 𝑤))
78	77	oveq2d 7436	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑤 → ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝑓)) = ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝑤)))
79	76, 78	eqeq12d 2777	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑤 → ((𝑧 ·no (𝐵 +no 𝑓)) = ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝑓)) ↔ (𝑧 ·no (𝐵 +no 𝑤)) = ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝑤))))
80		nadddilem1.6	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)))
81	80	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)))
82	75, 79, 81, 15, 18	rspc2dv 3591	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (𝑧 ·no (𝐵 +no 𝑤)) = ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝑤)))
83	71, 82	eqtr4d 2799	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝑧 ·no 𝑤) +no (𝑧 ·no 𝐵)) = (𝑧 ·no (𝐵 +no 𝑤)))
84	83	oveq2d 7436	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no (𝐵 +no 𝐶)) +no ((𝑧 ·no 𝑤) +no (𝑧 ·no 𝐵))) = ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no (𝐵 +no 𝑤))))
85	70, 84	eqtrd 2796	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤)) +no (𝑧 ·no 𝐵)) = ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no (𝐵 +no 𝑤))))
86	50, 69, 85	3eltr4d 2876	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no 𝐶)) +no (𝑧 ·no 𝐵)) ∈ (((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤)) +no (𝑧 ·no 𝐵)))
87	52, 28	naddcld 8689	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no 𝐶)) ∈ On)
88		naddel1 8697	. . . . . . . . . . 11 ⊢ ((((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no 𝐶)) ∈ On ∧ ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤)) ∈ On ∧ (𝑧 ·no 𝐵) ∈ On) → (((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no 𝐶)) ∈ ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤)) ↔ (((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no 𝐶)) +no (𝑧 ·no 𝐵)) ∈ (((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤)) +no (𝑧 ·no 𝐵))))
89	87, 26, 53, 88	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no 𝐶)) ∈ ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤)) ↔ (((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no 𝐶)) +no (𝑧 ·no 𝐵)) ∈ (((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤)) +no (𝑧 ·no 𝐵))))
90	86, 89	mpbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no (𝐵 +no 𝑤)) +no (𝑧 ·no 𝐶)) ∈ ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤)))
91	45, 90	eqeltrd 2861	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no 𝐵) +no ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤))) ∈ ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤)))
92	23, 26, 33, 91	ontr2d 36949	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no 𝐵) +no (𝑌 +no (𝑧 ·no 𝑤))) ∈ ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤)))
93	21, 92	eqeltrd 2861	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (((𝐴 ·no 𝐵) +no 𝑌) +no (𝑧 ·no 𝑤)) ∈ ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤)))
94	13, 14	naddcld 8689	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no 𝐵) +no 𝑌) ∈ On)
95		naddel1 8697	. . . . . . 7 ⊢ ((((𝐴 ·no 𝐵) +no 𝑌) ∈ On ∧ (𝐴 ·no (𝐵 +no 𝐶)) ∈ On ∧ (𝑧 ·no 𝑤) ∈ On) → (((𝐴 ·no 𝐵) +no 𝑌) ∈ (𝐴 ·no (𝐵 +no 𝐶)) ↔ (((𝐴 ·no 𝐵) +no 𝑌) +no (𝑧 ·no 𝑤)) ∈ ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤))))
96	94, 25, 20, 95	syl3anc 1398	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → (((𝐴 ·no 𝐵) +no 𝑌) ∈ (𝐴 ·no (𝐵 +no 𝐶)) ↔ (((𝐴 ·no 𝐵) +no 𝑌) +no (𝑧 ·no 𝑤)) ∈ ((𝐴 ·no (𝐵 +no 𝐶)) +no (𝑧 ·no 𝑤))))
97	93, 96	mpbird 260	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶) ∧ (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)))) → ((𝐴 ·no 𝐵) +no 𝑌) ∈ (𝐴 ·no (𝐵 +no 𝐶)))
98	97	expr 462	. . . 4 ⊢ (((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐶)) → ((𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)) → ((𝐴 ·no 𝐵) +no 𝑌) ∈ (𝐴 ·no (𝐵 +no 𝐶))))
99	98	rexlimdvva 3220	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) → (∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 (𝑌 +no (𝑧 ·no 𝑤)) ⊆ ((𝑧 ·no 𝐶) +no (𝐴 ·no 𝑤)) → ((𝐴 ·no 𝐵) +no 𝑌) ∈ (𝐴 ·no (𝐵 +no 𝐶))))
100	9, 99	sylbid 243	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) → (𝑌 ∈ (𝐴 ·no 𝐶) → ((𝐴 ·no 𝐵) +no 𝑌) ∈ (𝐴 ·no (𝐵 +no 𝐶))))
101	100	syldbl2 855	1 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) → ((𝐴 ·no 𝐵) +no 𝑌) ∈ (𝐴 ·no (𝐵 +no 𝐶)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  Oncon0 6362  (class class class)co 7420   +no cnadd 8674   ·_no cnmul 36936

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8299  df-nadd 8675  df-nmul 36937

This theorem is used by:  nadddilem2  36970