Theorem eceqoveq 8394

Description: Equality of equivalence relation in terms of an operation. (Contributed by NM, 15-Feb-1996.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)

Hypotheses

Ref	Expression
eceqoveq.5	⊢ ∼ Er (𝑆 × 𝑆)
eceqoveq.7	⊢ dom + = (𝑆 × 𝑆)
eceqoveq.8	⊢ ¬ ∅ ∈ 𝑆
eceqoveq.9	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
eceqoveq.10	⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩ ∼ ⟨𝐶, 𝐷⟩ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))

Assertion

Ref	Expression
eceqoveq	⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))

Distinct variable groups:   𝑥,𝑦, +   𝑥,𝑆,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦

Allowed substitution hints:   ∼ (𝑥,𝑦)

Proof of Theorem eceqoveq

Step	Hyp	Ref	Expression
1		opelxpi 5585	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
2	1	ad2antrr 724	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) ∧ [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
3		eceqoveq.5	. . . . . . . . 9 ⊢ ∼ Er (𝑆 × 𝑆)
4	3	a1i 11	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) ∧ [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → ∼ Er (𝑆 × 𝑆))
5		simpr 487	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) ∧ [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ )
6	4, 5	ereldm 8329	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) ∧ [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)))
7	2, 6	mpbid 234	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) ∧ [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
8		opelxp2 5590	. . . . . 6 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆) → 𝐷 ∈ 𝑆)
9	7, 8	syl 17	. . . . 5 ⊢ ((((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) ∧ [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → 𝐷 ∈ 𝑆)
10	9	ex 415	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ → 𝐷 ∈ 𝑆))
11		eceqoveq.9	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆)
12	11	caovcl 7334	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐵 + 𝐶) ∈ 𝑆)
13		eleq1 2898	. . . . . . 7 ⊢ ((𝐴 + 𝐷) = (𝐵 + 𝐶) → ((𝐴 + 𝐷) ∈ 𝑆 ↔ (𝐵 + 𝐶) ∈ 𝑆))
14	12, 13	syl5ibr 248	. . . . . 6 ⊢ ((𝐴 + 𝐷) = (𝐵 + 𝐶) → ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴 + 𝐷) ∈ 𝑆))
15		eceqoveq.7	. . . . . . . 8 ⊢ dom + = (𝑆 × 𝑆)
16		eceqoveq.8	. . . . . . . 8 ⊢ ¬ ∅ ∈ 𝑆
17	15, 16	ndmovrcl 7326	. . . . . . 7 ⊢ ((𝐴 + 𝐷) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆))
18	17	simprd 498	. . . . . 6 ⊢ ((𝐴 + 𝐷) ∈ 𝑆 → 𝐷 ∈ 𝑆)
19	14, 18	syl6com 37	. . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴 + 𝐷) = (𝐵 + 𝐶) → 𝐷 ∈ 𝑆))
20	19	adantll 712	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) → ((𝐴 + 𝐷) = (𝐵 + 𝐶) → 𝐷 ∈ 𝑆))
21	3	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ∼ Er (𝑆 × 𝑆))
22	1	adantr 483	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
23	21, 22	erth 8330	. . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩ ∼ ⟨𝐶, 𝐷⟩ ↔ [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ))
24		eceqoveq.10	. . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (⟨𝐴, 𝐵⟩ ∼ ⟨𝐶, 𝐷⟩ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
25	23, 24	bitr3d 283	. . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
26	25	expr 459	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) → (𝐷 ∈ 𝑆 → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))))
27	10, 20, 26	pm5.21ndd 383	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
28	27	an32s 650	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ 𝐵 ∈ 𝑆) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
29		eqcom 2826	. . . 4 ⊢ (∅ = [⟨𝐶, 𝐷⟩] ∼ ↔ [⟨𝐶, 𝐷⟩] ∼ = ∅)
30		erdm 8291	. . . . . . . . . . . 12 ⊢ ( ∼ Er (𝑆 × 𝑆) → dom ∼ = (𝑆 × 𝑆))
31	3, 30	ax-mp 5	. . . . . . . . . . 11 ⊢ dom ∼ = (𝑆 × 𝑆)
32	31	eleq2i 2902	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ ∈ dom ∼ ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
33		ecdmn0 8328	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ ∈ dom ∼ ↔ [⟨𝐶, 𝐷⟩] ∼ ≠ ∅)
34		opelxp 5584	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆) ↔ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆))
35	32, 33, 34	3bitr3i 303	. . . . . . . . 9 ⊢ ([⟨𝐶, 𝐷⟩] ∼ ≠ ∅ ↔ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆))
36	35	simplbi2 503	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑆 → (𝐷 ∈ 𝑆 → [⟨𝐶, 𝐷⟩] ∼ ≠ ∅))
37	36	ad2antlr 725	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → (𝐷 ∈ 𝑆 → [⟨𝐶, 𝐷⟩] ∼ ≠ ∅))
38	37	necon2bd 3030	. . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ([⟨𝐶, 𝐷⟩] ∼ = ∅ → ¬ 𝐷 ∈ 𝑆))
39		simpr 487	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → 𝐷 ∈ 𝑆)
40	39	con3i 157	. . . . . . 7 ⊢ (¬ 𝐷 ∈ 𝑆 → ¬ (𝐴 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆))
41	15	ndmov 7324	. . . . . . 7 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → (𝐴 + 𝐷) = ∅)
42	40, 41	syl 17	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝑆 → (𝐴 + 𝐷) = ∅)
43	38, 42	syl6 35	. . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ([⟨𝐶, 𝐷⟩] ∼ = ∅ → (𝐴 + 𝐷) = ∅))
44		eleq1 2898	. . . . . . 7 ⊢ ((𝐴 + 𝐷) = ∅ → ((𝐴 + 𝐷) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
45	16, 44	mtbiri 329	. . . . . 6 ⊢ ((𝐴 + 𝐷) = ∅ → ¬ (𝐴 + 𝐷) ∈ 𝑆)
46	35	simprbi 499	. . . . . . . 8 ⊢ ([⟨𝐶, 𝐷⟩] ∼ ≠ ∅ → 𝐷 ∈ 𝑆)
47	11	caovcl 7334	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → (𝐴 + 𝐷) ∈ 𝑆)
48	47	ex 415	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑆 → (𝐷 ∈ 𝑆 → (𝐴 + 𝐷) ∈ 𝑆))
49	48	ad2antrr 724	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → (𝐷 ∈ 𝑆 → (𝐴 + 𝐷) ∈ 𝑆))
50	46, 49	syl5 34	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ([⟨𝐶, 𝐷⟩] ∼ ≠ ∅ → (𝐴 + 𝐷) ∈ 𝑆))
51	50	necon1bd 3032	. . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → (¬ (𝐴 + 𝐷) ∈ 𝑆 → [⟨𝐶, 𝐷⟩] ∼ = ∅))
52	45, 51	syl5 34	. . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ((𝐴 + 𝐷) = ∅ → [⟨𝐶, 𝐷⟩] ∼ = ∅))
53	43, 52	impbid 214	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ([⟨𝐶, 𝐷⟩] ∼ = ∅ ↔ (𝐴 + 𝐷) = ∅))
54	29, 53	syl5bb 285	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → (∅ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = ∅))
55	31	eleq2i 2902	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom ∼ ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
56		ecdmn0 8328	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom ∼ ↔ [⟨𝐴, 𝐵⟩] ∼ ≠ ∅)
57		opelxp 5584	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
58	55, 56, 57	3bitr3i 303	. . . . . . 7 ⊢ ([⟨𝐴, 𝐵⟩] ∼ ≠ ∅ ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
59	58	simprbi 499	. . . . . 6 ⊢ ([⟨𝐴, 𝐵⟩] ∼ ≠ ∅ → 𝐵 ∈ 𝑆)
60	59	necon1bi 3042	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝑆 → [⟨𝐴, 𝐵⟩] ∼ = ∅)
61	60	adantl 484	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → [⟨𝐴, 𝐵⟩] ∼ = ∅)
62	61	eqeq1d 2821	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ ∅ = [⟨𝐶, 𝐷⟩] ∼ ))
63		simpl 485	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 𝐵 ∈ 𝑆)
64	63	con3i 157	. . . . . 6 ⊢ (¬ 𝐵 ∈ 𝑆 → ¬ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
65	15	ndmov 7324	. . . . . 6 ⊢ (¬ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐵 + 𝐶) = ∅)
66	64, 65	syl 17	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝑆 → (𝐵 + 𝐶) = ∅)
67	66	adantl 484	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → (𝐵 + 𝐶) = ∅)
68	67	eqeq2d 2830	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ((𝐴 + 𝐷) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐷) = ∅))
69	54, 62, 68	3bitr4d 313	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
70	28, 69	pm2.61dan 811	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1531   ∈ wcel 2108   ≠ wne 3014  ∅c0 4289  ⟨cop 4565   class class class wbr 5057   × cxp 5546  dom cdm 5548  (class class class)co 7148   Er wer 8278  [cec 8279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356  df-ov 7151  df-er 8281  df-ec 8283

This theorem is referenced by: (None)