Theorem nadddilem4 36715

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

Hypotheses

Ref	Expression
nadddilem4.1	⊢ (𝜑 → 𝐴 ∈ On)
nadddilem4.2	⊢ (𝜑 → 𝐵 ∈ On)
nadddilem4.3	⊢ (𝜑 → 𝐶 ∈ On)
nadddilem4.4	⊢ (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))
nadddilem4.5	⊢ (𝜑 → ∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)))
nadddilem4.6	⊢ (𝜑 → ∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)))
nadddilem4.7	⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)))
nadddilem4.8	⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)))

Assertion

Ref	Expression
nadddilem4	⊢ (𝜑 → (𝐴 ·no (𝐵 +no 𝐶)) ⊆ ((𝐴 ·no 𝐵) +no (𝐴 ·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 nadddilem4

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

Step	Hyp	Ref	Expression
1		simprr 784	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) → 𝑦 ∈ (𝐵 +no 𝐶))
2		nadddilem4.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ On)
3		nadddilem4.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ On)
4	2, 3	naddcld 8662	. . . . . . . 8 ⊢ (𝜑 → (𝐵 +no 𝐶) ∈ On)
5	4	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) → (𝐵 +no 𝐶) ∈ On)
6	5, 1	onelond 36691	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) → 𝑦 ∈ On)
7	2	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) → 𝐵 ∈ On)
8	3	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) → 𝐶 ∈ On)
9		ltnadd 36710	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝑦 ∈ (𝐵 +no 𝐶) ↔ (∃𝑧 ∈ 𝐵 𝑦 ⊆ (𝑧 +no 𝐶) ∨ ∃𝑤 ∈ 𝐶 𝑦 ⊆ (𝐵 +no 𝑤))))
10	6, 7, 8, 9	syl3anc 1398	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) → (𝑦 ∈ (𝐵 +no 𝐶) ↔ (∃𝑧 ∈ 𝐵 𝑦 ⊆ (𝑧 +no 𝐶) ∨ ∃𝑤 ∈ 𝐶 𝑦 ⊆ (𝐵 +no 𝑤))))
11		nadddilem4.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ On)
12	11	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑧 +no 𝐶))) → 𝐴 ∈ On)
13	2	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑧 +no 𝐶))) → 𝐵 ∈ On)
14	3	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑧 +no 𝐶))) → 𝐶 ∈ On)
15		simplrl 788	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑧 +no 𝐶))) → 𝑥 ∈ 𝐴)
16		simplrr 789	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑧 +no 𝐶))) → 𝑦 ∈ (𝐵 +no 𝐶))
17		simprl 782	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑧 +no 𝐶))) → 𝑧 ∈ 𝐵)
18		simprr 784	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑧 +no 𝐶))) → 𝑦 ⊆ (𝑧 +no 𝐶))
19		nadddilem4.4	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))
20	19	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑧 +no 𝐶))) → ∀𝑑 ∈ 𝐴 (𝑑 ·no (𝐵 +no 𝐶)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)))
21		nadddilem4.5	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)))
22	21	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑧 +no 𝐶))) → ∀𝑒 ∈ 𝐵 (𝐴 ·no (𝑒 +no 𝐶)) = ((𝐴 ·no 𝑒) +no (𝐴 ·no 𝐶)))
23		nadddilem4.7	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)))
24	23	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑧 +no 𝐶))) → ∀𝑑 ∈ 𝐴 ∀𝑒 ∈ 𝐵 (𝑑 ·no (𝑒 +no 𝐶)) = ((𝑑 ·no 𝑒) +no (𝑑 ·no 𝐶)))
25	12, 13, 14, 15, 16, 17, 18, 20, 22, 24	nadddilem3 36714	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑧 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑧 +no 𝐶))) → ((𝑥 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑦)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑥 ·no 𝑦)))
26	25	rexlimdvaa 3167	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) → (∃𝑧 ∈ 𝐵 𝑦 ⊆ (𝑧 +no 𝐶) → ((𝑥 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑦)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑥 ·no 𝑦))))
27	11	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → 𝐴 ∈ On)
28	3	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → 𝐶 ∈ On)
29	2	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → 𝐵 ∈ On)
30		simplrl 788	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → 𝑥 ∈ 𝐴)
31		simplrr 789	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → 𝑦 ∈ (𝐵 +no 𝐶))
32	29, 28	naddcomd 36701	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → (𝐵 +no 𝐶) = (𝐶 +no 𝐵))
33	31, 32	eleqtrd 2865	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → 𝑦 ∈ (𝐶 +no 𝐵))
34		simprl 782	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → 𝑤 ∈ 𝐶)
35		simprr 784	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → 𝑦 ⊆ (𝐵 +no 𝑤))
36	28, 34	onelond 36691	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → 𝑤 ∈ On)
37	29, 36	naddcomd 36701	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → (𝐵 +no 𝑤) = (𝑤 +no 𝐵))
38	35, 37	sseqtrd 3973	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → 𝑦 ⊆ (𝑤 +no 𝐵))
39		oveq1 7417	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑝 → (𝑑 ·no (𝐵 +no 𝐶)) = (𝑝 ·no (𝐵 +no 𝐶)))
40		oveq1 7417	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑝 → (𝑑 ·no 𝐵) = (𝑝 ·no 𝐵))
41		oveq1 7417	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑝 → (𝑑 ·no 𝐶) = (𝑝 ·no 𝐶))
42	40, 41	oveq12d 7428	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑝 → ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝐶)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝐶)))
43	39, 42	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ (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))))
44	43	cbvralvw 3243	. . . . . . . . . . . 12 ⊢ (∀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)))
45	2, 3	naddcomd 36701	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 +no 𝐶) = (𝐶 +no 𝐵))
46	45	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝐵 +no 𝐶) = (𝐶 +no 𝐵))
47	46	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝 ·no (𝐵 +no 𝐶)) = (𝑝 ·no (𝐶 +no 𝐵)))
48	11	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝐴 ∈ On)
49		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐴)
50	48, 49	onelond 36691	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ On)
51	2	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝐵 ∈ On)
52	50, 51	nmulcld 36685	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝 ·no 𝐵) ∈ On)
53	3	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝐶 ∈ On)
54	50, 53	nmulcld 36685	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → (𝑝 ·no 𝐶) ∈ On)
55	52, 54	naddcomd 36701	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝐶)) = ((𝑝 ·no 𝐶) +no (𝑝 ·no 𝐵)))
56	47, 55	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ ((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))))
57	56	ralbidva 3186	. . . . . . . . . . . 12 ⊢ (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))))
58	44, 57	bitrid 286	. . . . . . . . . . 11 ⊢ (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))))
59	19, 58	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑝 ∈ 𝐴 (𝑝 ·no (𝐶 +no 𝐵)) = ((𝑝 ·no 𝐶) +no (𝑝 ·no 𝐵)))
60	59	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → ∀𝑝 ∈ 𝐴 (𝑝 ·no (𝐶 +no 𝐵)) = ((𝑝 ·no 𝐶) +no (𝑝 ·no 𝐵)))
61		nadddilem4.6	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)))
62		oveq2 7418	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑞 → (𝐵 +no 𝑓) = (𝐵 +no 𝑞))
63	62	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑞 → (𝐴 ·no (𝐵 +no 𝑓)) = (𝐴 ·no (𝐵 +no 𝑞)))
64		oveq2 7418	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑞 → (𝐴 ·no 𝑓) = (𝐴 ·no 𝑞))
65	64	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑞 → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑞)))
66	63, 65	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑞 → ((𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)) ↔ (𝐴 ·no (𝐵 +no 𝑞)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑞))))
67	66	cbvralvw 3243	. . . . . . . . . . . 12 ⊢ (∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)) ↔ ∀𝑞 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑞)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑞)))
68	2	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐶) → 𝐵 ∈ On)
69	3	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐶) → 𝐶 ∈ On)
70		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐶) → 𝑞 ∈ 𝐶)
71	69, 70	onelond 36691	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐶) → 𝑞 ∈ On)
72	68, 71	naddcomd 36701	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐶) → (𝐵 +no 𝑞) = (𝑞 +no 𝐵))
73	72	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐶) → (𝐴 ·no (𝐵 +no 𝑞)) = (𝐴 ·no (𝑞 +no 𝐵)))
74	11, 2	nmulcld 36685	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 ·no 𝐵) ∈ On)
75	74	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐶) → (𝐴 ·no 𝐵) ∈ On)
76	11	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐶) → 𝐴 ∈ On)
77	76, 71	nmulcld 36685	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐶) → (𝐴 ·no 𝑞) ∈ On)
78	75, 77	naddcomd 36701	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐶) → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑞)) = ((𝐴 ·no 𝑞) +no (𝐴 ·no 𝐵)))
79	73, 78	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐶) → ((𝐴 ·no (𝐵 +no 𝑞)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑞)) ↔ (𝐴 ·no (𝑞 +no 𝐵)) = ((𝐴 ·no 𝑞) +no (𝐴 ·no 𝐵))))
80	79	ralbidva 3186	. . . . . . . . . . . 12 ⊢ (𝜑 → (∀𝑞 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑞)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑞)) ↔ ∀𝑞 ∈ 𝐶 (𝐴 ·no (𝑞 +no 𝐵)) = ((𝐴 ·no 𝑞) +no (𝐴 ·no 𝐵))))
81	67, 80	bitrid 286	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑓 ∈ 𝐶 (𝐴 ·no (𝐵 +no 𝑓)) = ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝑓)) ↔ ∀𝑞 ∈ 𝐶 (𝐴 ·no (𝑞 +no 𝐵)) = ((𝐴 ·no 𝑞) +no (𝐴 ·no 𝐵))))
82	61, 81	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑞 ∈ 𝐶 (𝐴 ·no (𝑞 +no 𝐵)) = ((𝐴 ·no 𝑞) +no (𝐴 ·no 𝐵)))
83	82	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → ∀𝑞 ∈ 𝐶 (𝐴 ·no (𝑞 +no 𝐵)) = ((𝐴 ·no 𝑞) +no (𝐴 ·no 𝐵)))
84		nadddilem4.8	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)))
85		oveq1 7417	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑝 → (𝑑 ·no (𝐵 +no 𝑓)) = (𝑝 ·no (𝐵 +no 𝑓)))
86		oveq1 7417	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑝 → (𝑑 ·no 𝑓) = (𝑝 ·no 𝑓))
87	40, 86	oveq12d 7428	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑝 → ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝑓)))
88	85, 87	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑝 → ((𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)) ↔ (𝑝 ·no (𝐵 +no 𝑓)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝑓))))
89	62	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑞 → (𝑝 ·no (𝐵 +no 𝑓)) = (𝑝 ·no (𝐵 +no 𝑞)))
90		oveq2 7418	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑞 → (𝑝 ·no 𝑓) = (𝑝 ·no 𝑞))
91	90	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑞 → ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝑓)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝑞)))
92	89, 91	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑞 → ((𝑝 ·no (𝐵 +no 𝑓)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝑓)) ↔ (𝑝 ·no (𝐵 +no 𝑞)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝑞))))
93	88, 92	cbvral2vw 3247	. . . . . . . . . . . 12 ⊢ (∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)) ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐶 (𝑝 ·no (𝐵 +no 𝑞)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝑞)))
94	72	adantrl 728	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐶)) → (𝐵 +no 𝑞) = (𝑞 +no 𝐵))
95	94	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐶)) → (𝑝 ·no (𝐵 +no 𝑞)) = (𝑝 ·no (𝑞 +no 𝐵)))
96	52	adantrr 729	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐶)) → (𝑝 ·no 𝐵) ∈ On)
97	50	adantrr 729	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐶)) → 𝑝 ∈ On)
98	71	adantrl 728	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐶)) → 𝑞 ∈ On)
99	97, 98	nmulcld 36685	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐶)) → (𝑝 ·no 𝑞) ∈ On)
100	96, 99	naddcomd 36701	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐶)) → ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝑞)) = ((𝑝 ·no 𝑞) +no (𝑝 ·no 𝐵)))
101	95, 100	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ 𝑞 ∈ 𝐶)) → ((𝑝 ·no (𝐵 +no 𝑞)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝑞)) ↔ (𝑝 ·no (𝑞 +no 𝐵)) = ((𝑝 ·no 𝑞) +no (𝑝 ·no 𝐵))))
102	101	2ralbidva 3227	. . . . . . . . . . . 12 ⊢ (𝜑 → (∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐶 (𝑝 ·no (𝐵 +no 𝑞)) = ((𝑝 ·no 𝐵) +no (𝑝 ·no 𝑞)) ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐶 (𝑝 ·no (𝑞 +no 𝐵)) = ((𝑝 ·no 𝑞) +no (𝑝 ·no 𝐵))))
103	93, 102	bitrid 286	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑑 ∈ 𝐴 ∀𝑓 ∈ 𝐶 (𝑑 ·no (𝐵 +no 𝑓)) = ((𝑑 ·no 𝐵) +no (𝑑 ·no 𝑓)) ↔ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐶 (𝑝 ·no (𝑞 +no 𝐵)) = ((𝑝 ·no 𝑞) +no (𝑝 ·no 𝐵))))
104	84, 103	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐶 (𝑝 ·no (𝑞 +no 𝐵)) = ((𝑝 ·no 𝑞) +no (𝑝 ·no 𝐵)))
105	104	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐶 (𝑝 ·no (𝑞 +no 𝐵)) = ((𝑝 ·no 𝑞) +no (𝑝 ·no 𝐵)))
106	27, 28, 29, 30, 33, 34, 38, 60, 83, 105	nadddilem3 36714	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → ((𝑥 ·no (𝐶 +no 𝐵)) +no (𝐴 ·no 𝑦)) ∈ (((𝐴 ·no 𝐶) +no (𝐴 ·no 𝐵)) +no (𝑥 ·no 𝑦)))
107	32	oveq2d 7426	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → (𝑥 ·no (𝐵 +no 𝐶)) = (𝑥 ·no (𝐶 +no 𝐵)))
108	107	oveq1d 7425	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → ((𝑥 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑦)) = ((𝑥 ·no (𝐶 +no 𝐵)) +no (𝐴 ·no 𝑦)))
109	74	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → (𝐴 ·no 𝐵) ∈ On)
110	11, 3	nmulcld 36685	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ·no 𝐶) ∈ On)
111	110	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → (𝐴 ·no 𝐶) ∈ On)
112	109, 111	naddcomd 36701	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) = ((𝐴 ·no 𝐶) +no (𝐴 ·no 𝐵)))
113	112	oveq1d 7425	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑥 ·no 𝑦)) = (((𝐴 ·no 𝐶) +no (𝐴 ·no 𝐵)) +no (𝑥 ·no 𝑦)))
114	106, 108, 113	3eltr4d 2878	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) ∧ (𝑤 ∈ 𝐶 ∧ 𝑦 ⊆ (𝐵 +no 𝑤))) → ((𝑥 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑦)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑥 ·no 𝑦)))
115	114	rexlimdvaa 3167	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) → (∃𝑤 ∈ 𝐶 𝑦 ⊆ (𝐵 +no 𝑤) → ((𝑥 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑦)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑥 ·no 𝑦))))
116	26, 115	jaod 872	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) → ((∃𝑧 ∈ 𝐵 𝑦 ⊆ (𝑧 +no 𝐶) ∨ ∃𝑤 ∈ 𝐶 𝑦 ⊆ (𝐵 +no 𝑤)) → ((𝑥 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑦)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑥 ·no 𝑦))))
117	10, 116	sylbid 243	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) → (𝑦 ∈ (𝐵 +no 𝐶) → ((𝑥 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑦)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑥 ·no 𝑦))))
118	1, 117	mpd 16	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐵 +no 𝐶))) → ((𝑥 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑦)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑥 ·no 𝑦)))
119	118	ralrimivva 3208	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 +no 𝐶)((𝑥 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑦)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑥 ·no 𝑦)))
120	74, 110	naddcld 8662	. . 3 ⊢ (𝜑 → ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) ∈ On)
121		nmulle 36709	. . 3 ⊢ ((𝐴 ∈ On ∧ (𝐵 +no 𝐶) ∈ On ∧ ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) ∈ On) → ((𝐴 ·no (𝐵 +no 𝐶)) ⊆ ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 +no 𝐶)((𝑥 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑦)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑥 ·no 𝑦))))
122	11, 4, 120, 121	syl3anc 1398	. 2 ⊢ (𝜑 → ((𝐴 ·no (𝐵 +no 𝐶)) ⊆ ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐵 +no 𝐶)((𝑥 ·no (𝐵 +no 𝐶)) +no (𝐴 ·no 𝑦)) ∈ (((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)) +no (𝑥 ·no 𝑦))))
123	119, 122	mpbird 260	1 ⊢ (𝜑 → (𝐴 ·no (𝐵 +no 𝐶)) ⊆ ((𝐴 ·no 𝐵) +no (𝐴 ·no 𝐶)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ⊆ 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:  nadddi  36716