Theorem nadddilem1 37184

Description: Lemma for nadddi 37188. 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 37157	. . . . 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 37163	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝐴 ·no 𝐶)) → 𝑌 ∈ On)
8		ltnmul 37180	. . . 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 1399	. . 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 37157	. . . . . . . 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 37163	. . . . . . . . 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 37163	. . . . . . . . 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 37157	. . . . . . . 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 37174	. . . . . . 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 8727	. . . . . . . . 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 8727	. . . . . . . 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 8727	. . . . . . . . . 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 37157	. . . . . . . . 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 8727	. . . . . . . 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 37157	. . . . . . . . . . 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 37157	. . . . . . . . . . 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 8727	. . . . . . . . . 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 8738	. . . . . . . . . 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 1399	. . . . . . . . 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 7459	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑤 → (𝐵 +no 𝑓) = (𝐵 +no 𝑤))
35	34	oveq2d 7467	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑤 → (𝐴 ·no (𝐵 +no 𝑓)) = (𝐴 ·no (𝐵 +no 𝑤)))
36		oveq2 7459	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑤 → (𝐴 ·no 𝑓) = (𝐴 ·no 𝑤))
37	36	oveq2d 7467	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑤 → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑤)))
38	35, 37	eqeq12d 2779	. . . . . . . . . . . 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 3581	. . . . . . . . . . 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 7466	. . . . . . . . . 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 37175	. . . . . . . . . 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 37174	. . . . . . . . . 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 2803	. . . . . . . . 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 8736	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ On ∧ 𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝑤 ∈ 𝐶 ↔ (𝐵 +no 𝑤) ∈ (𝐵 +no 𝐶)))
47	19, 17, 12, 46	syl3anc 1399	. . . . . . . . . . . . 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 37176	. . . . . . . . . . . 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 8727	. . . . . . . . . . . . . . 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 37157	. . . . . . . . . . . . . 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 37157	. . . . . . . . . . . . . 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 37174	. . . . . . . . . . . . 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 7458	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑧 → (𝑑 ·no (𝐵 +no 𝐶)) = (𝑧 ·no (𝐵 +no 𝐶)))
56		oveq1 7458	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑧 → (𝑑 ·no 𝐵) = (𝑧 ·no 𝐵))
57		oveq1 7458	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑧 → (𝑑 ·no 𝐶) = (𝑧 ·no 𝐶))
58	56, 57	oveq12d 7469	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑧 → ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)) = ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝐶)))
59	55, 58	eqeq12d 2779	. . . . . . . . . . . . . . . 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 3581	. . . . . . . . . . . . . . 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 37173	. . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . 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 7467	. . . . . . . . . . . . 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 2801	. . . . . . . . . . . 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 37157	. . . . . . . . . . . . 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 37173	. . . . . . . . . . . 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 2798	. . . . . . . . . . 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 37174	. . . . . . . . . . . 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 37173	. . . . . . . . . . . . . 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 7458	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑧 → (𝑑 ·no (𝐵 +no 𝑓)) = (𝑧 ·no (𝐵 +no 𝑓)))
73		oveq1 7458	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 = 𝑧 → (𝑑 ·no 𝑓) = (𝑧 ·no 𝑓))
74	56, 73	oveq12d 7469	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑧 → ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)) = ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝑓)))
75	72, 74	eqeq12d 2779	. . . . . . . . . . . . . . 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 7467	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑤 → (𝑧 ·no (𝐵 +no 𝑓)) = (𝑧 ·no (𝐵 +no 𝑤)))
77		oveq2 7459	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑤 → (𝑧 ·no 𝑓) = (𝑧 ·no 𝑤))
78	77	oveq2d 7467	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑤 → ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝑓)) = ((𝑧 ·no 𝐵) +no (𝑧 ·no 𝑤)))
79	76, 78	eqeq12d 2779	. . . . . . . . . . . . . . 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 3594	. . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . 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 7467	. . . . . . . . . . . 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 2798	. . . . . . . . . . 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 2879	. . . . . . . . . 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 8727	. . . . . . . . . . 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 8735	. . . . . . . . . . 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 1399	. . . . . . . . . 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 2864	. . . . . . . 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 37164	. . . . . . 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 2864	. . . . . 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 8727	. . . . . . 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 8735	. . . . . . 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 1399	. . . . . 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 3223	. . 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 2147  ∀wral 3080  ∃wrex 3090   ⊆ 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:  nadddilem2  37185