Theorem omeueq 35776

Description: The division algorithm for ordinal multiplication (omeu 8604) produces a unique quotient and remainder. (Contributed by BTernaryTau, 5-Oct-2026.)

Hypotheses

Ref	Expression
omeueq.1	⊢ (𝜑 → 𝐴 ∈ On)
omeueq.2	⊢ (𝜑 → 𝐴 ≠ ∅)
omeueq.3	⊢ (𝜑 → 𝐵 ∈ On)
omeueq.4	⊢ (𝜑 → 𝐶 ∈ 𝐴)
omeueq.5	⊢ (𝜑 → 𝐷 ∈ On)
omeueq.6	⊢ (𝜑 → 𝐸 ∈ 𝐴)
omeueq.7	⊢ (𝜑 → ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸))

Assertion

Ref	Expression
omeueq	⊢ (𝜑 → (𝐵 = 𝐷 ∧ 𝐶 = 𝐸))

Proof of Theorem omeueq

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

Step	Hyp	Ref	Expression
1		omeueq.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ On)
2		omeueq.3	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ On)
3		omcl 8555	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
4	1, 2, 3	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐴 ·o 𝐵) ∈ On)
5		omeueq.4	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
6		onelon 6393	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ On)
7	1, 5, 6	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ On)
8		oacl 8554	. . . . . 6 ⊢ (((𝐴 ·o 𝐵) ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ·o 𝐵) +o 𝐶) ∈ On)
9	4, 7, 8	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((𝐴 ·o 𝐵) +o 𝐶) ∈ On)
10		omeueq.2	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ ∅)
11		omeu 8604	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ((𝐴 ·o 𝐵) +o 𝐶) ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)))
12	1, 9, 10, 11	syl3anc 1398	. . . 4 ⊢ (𝜑 → ∃!𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)))
13		eumo 2606	. . . 4 ⊢ (∃!𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)) → ∃*𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)))
14	12, 13	syl 18	. . 3 ⊢ (𝜑 → ∃*𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)))
15		opeq1 4842	. . . . . 6 ⊢ (𝑥 = 𝐵 → ⟨𝑥, 𝑦⟩ = ⟨𝐵, 𝑦⟩)
16	15	eqeq2d 2774	. . . . 5 ⊢ (𝑥 = 𝐵 → (⟨𝐵, 𝐶⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐵, 𝐶⟩ = ⟨𝐵, 𝑦⟩))
17		oveq2 7435	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
18	17	oveq1d 7442	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝑦))
19	18	eqeq1d 2765	. . . . 5 ⊢ (𝑥 = 𝐵 → (((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶) ↔ ((𝐴 ·o 𝐵) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)))
20	16, 19	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝐵 → ((⟨𝐵, 𝐶⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)) ↔ (⟨𝐵, 𝐶⟩ = ⟨𝐵, 𝑦⟩ ∧ ((𝐴 ·o 𝐵) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶))))
21		opeq2 4843	. . . . . 6 ⊢ (𝑦 = 𝐶 → ⟨𝐵, 𝑦⟩ = ⟨𝐵, 𝐶⟩)
22	21	eqeq2d 2774	. . . . 5 ⊢ (𝑦 = 𝐶 → (⟨𝐵, 𝐶⟩ = ⟨𝐵, 𝑦⟩ ↔ ⟨𝐵, 𝐶⟩ = ⟨𝐵, 𝐶⟩))
23		oveq2 7435	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝐴 ·o 𝐵) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶))
24	23	eqeq1d 2765	. . . . 5 ⊢ (𝑦 = 𝐶 → (((𝐴 ·o 𝐵) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶) ↔ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐵) +o 𝐶)))
25	22, 24	anbi12d 644	. . . 4 ⊢ (𝑦 = 𝐶 → ((⟨𝐵, 𝐶⟩ = ⟨𝐵, 𝑦⟩ ∧ ((𝐴 ·o 𝐵) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)) ↔ (⟨𝐵, 𝐶⟩ = ⟨𝐵, 𝐶⟩ ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐵) +o 𝐶))))
26		eqid 2763	. . . . . 6 ⊢ ⟨𝐵, 𝐶⟩ = ⟨𝐵, 𝐶⟩
27		eqid 2763	. . . . . 6 ⊢ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐵) +o 𝐶)
28	26, 27	pm3.2i 476	. . . . 5 ⊢ (⟨𝐵, 𝐶⟩ = ⟨𝐵, 𝐶⟩ ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐵) +o 𝐶))
29	28	a1i 11	. . . 4 ⊢ (𝜑 → (⟨𝐵, 𝐶⟩ = ⟨𝐵, 𝐶⟩ ∧ ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐵) +o 𝐶)))
30	20, 25, 2, 5, 29	2rspcedvdw 3592	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (⟨𝐵, 𝐶⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)))
31		opeq1 4842	. . . . . 6 ⊢ (𝑥 = 𝐷 → ⟨𝑥, 𝑦⟩ = ⟨𝐷, 𝑦⟩)
32	31	eqeq2d 2774	. . . . 5 ⊢ (𝑥 = 𝐷 → (⟨𝐷, 𝐸⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐷, 𝐸⟩ = ⟨𝐷, 𝑦⟩))
33		oveq2 7435	. . . . . . 7 ⊢ (𝑥 = 𝐷 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐷))
34	33	oveq1d 7442	. . . . . 6 ⊢ (𝑥 = 𝐷 → ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐷) +o 𝑦))
35	34	eqeq1d 2765	. . . . 5 ⊢ (𝑥 = 𝐷 → (((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶) ↔ ((𝐴 ·o 𝐷) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)))
36	32, 35	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝐷 → ((⟨𝐷, 𝐸⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)) ↔ (⟨𝐷, 𝐸⟩ = ⟨𝐷, 𝑦⟩ ∧ ((𝐴 ·o 𝐷) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶))))
37		opeq2 4843	. . . . . 6 ⊢ (𝑦 = 𝐸 → ⟨𝐷, 𝑦⟩ = ⟨𝐷, 𝐸⟩)
38	37	eqeq2d 2774	. . . . 5 ⊢ (𝑦 = 𝐸 → (⟨𝐷, 𝐸⟩ = ⟨𝐷, 𝑦⟩ ↔ ⟨𝐷, 𝐸⟩ = ⟨𝐷, 𝐸⟩))
39		oveq2 7435	. . . . . 6 ⊢ (𝑦 = 𝐸 → ((𝐴 ·o 𝐷) +o 𝑦) = ((𝐴 ·o 𝐷) +o 𝐸))
40	39	eqeq1d 2765	. . . . 5 ⊢ (𝑦 = 𝐸 → (((𝐴 ·o 𝐷) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶) ↔ ((𝐴 ·o 𝐷) +o 𝐸) = ((𝐴 ·o 𝐵) +o 𝐶)))
41	38, 40	anbi12d 644	. . . 4 ⊢ (𝑦 = 𝐸 → ((⟨𝐷, 𝐸⟩ = ⟨𝐷, 𝑦⟩ ∧ ((𝐴 ·o 𝐷) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)) ↔ (⟨𝐷, 𝐸⟩ = ⟨𝐷, 𝐸⟩ ∧ ((𝐴 ·o 𝐷) +o 𝐸) = ((𝐴 ·o 𝐵) +o 𝐶))))
42		omeueq.5	. . . 4 ⊢ (𝜑 → 𝐷 ∈ On)
43		omeueq.6	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐴)
44		omeueq.7	. . . . . 6 ⊢ (𝜑 → ((𝐴 ·o 𝐵) +o 𝐶) = ((𝐴 ·o 𝐷) +o 𝐸))
45	44	eqcomd 2769	. . . . 5 ⊢ (𝜑 → ((𝐴 ·o 𝐷) +o 𝐸) = ((𝐴 ·o 𝐵) +o 𝐶))
46		eqid 2763	. . . . 5 ⊢ ⟨𝐷, 𝐸⟩ = ⟨𝐷, 𝐸⟩
47	45, 46	jctil 529	. . . 4 ⊢ (𝜑 → (⟨𝐷, 𝐸⟩ = ⟨𝐷, 𝐸⟩ ∧ ((𝐴 ·o 𝐷) +o 𝐸) = ((𝐴 ·o 𝐵) +o 𝐶)))
48	36, 41, 42, 43, 47	2rspcedvdw 3592	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (⟨𝐷, 𝐸⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)))
49		opex 5435	. . . . 5 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
50		opex 5435	. . . . 5 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
51	49, 50	pm3.2i 476	. . . 4 ⊢ (⟨𝐵, 𝐶⟩ ∈ V ∧ ⟨𝐷, 𝐸⟩ ∈ V)
52		eqeq1 2767	. . . . . . 7 ⊢ (𝑧 = ⟨𝐵, 𝐶⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐵, 𝐶⟩ = ⟨𝑥, 𝑦⟩))
53	52	anbi1d 643	. . . . . 6 ⊢ (𝑧 = ⟨𝐵, 𝐶⟩ → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)) ↔ (⟨𝐵, 𝐶⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶))))
54	53	2rexbidv 3230	. . . . 5 ⊢ (𝑧 = ⟨𝐵, 𝐶⟩ → (∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (⟨𝐵, 𝐶⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶))))
55		eqeq1 2767	. . . . . . 7 ⊢ (𝑧 = ⟨𝐷, 𝐸⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐷, 𝐸⟩ = ⟨𝑥, 𝑦⟩))
56	55	anbi1d 643	. . . . . 6 ⊢ (𝑧 = ⟨𝐷, 𝐸⟩ → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)) ↔ (⟨𝐷, 𝐸⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶))))
57	56	2rexbidv 3230	. . . . 5 ⊢ (𝑧 = ⟨𝐷, 𝐸⟩ → (∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)) ↔ ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (⟨𝐷, 𝐸⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶))))
58	54, 57	moi 3678	. . . 4 ⊢ (((⟨𝐵, 𝐶⟩ ∈ V ∧ ⟨𝐷, 𝐸⟩ ∈ V) ∧ ∃*𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)) ∧ (∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (⟨𝐵, 𝐶⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)) ∧ ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (⟨𝐷, 𝐸⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)))) → ⟨𝐵, 𝐶⟩ = ⟨𝐷, 𝐸⟩)
59	51, 58	mp3an1 1477	. . 3 ⊢ ((∃*𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)) ∧ (∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (⟨𝐵, 𝐶⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)) ∧ ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (⟨𝐷, 𝐸⟩ = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = ((𝐴 ·o 𝐵) +o 𝐶)))) → ⟨𝐵, 𝐶⟩ = ⟨𝐷, 𝐸⟩)
60	14, 30, 48, 59	syl12anc 850	. 2 ⊢ (𝜑 → ⟨𝐵, 𝐶⟩ = ⟨𝐷, 𝐸⟩)
61		opthg 5449	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ 𝐴) → (⟨𝐵, 𝐶⟩ = ⟨𝐷, 𝐸⟩ ↔ (𝐵 = 𝐷 ∧ 𝐶 = 𝐸)))
62	2, 5, 61	syl2anc 596	. 2 ⊢ (𝜑 → (⟨𝐵, 𝐶⟩ = ⟨𝐷, 𝐸⟩ ↔ (𝐵 = 𝐷 ∧ 𝐶 = 𝐸)))
63	60, 62	mpbid 235	1 ⊢ (𝜑 → (𝐵 = 𝐷 ∧ 𝐶 = 𝐸))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∃*wmo 2565  ∃!weu 2596   ≠ wne 2958  ∃wrex 3089  Vcvv 3453  ∅c0 4280  ⟨cop 4837  Oncon0 6368  (class class class)co 7427   +_o coa 8484   ·_o comu 8485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  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-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-oadd 8491  df-omul 8492

This theorem is used by:  jf1  35964