Theorem eceqoveq 8862

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 5722	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
2	1	ad2antrr 726	. . . . . . 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 484	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) ∧ [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ )
6	4, 5	ereldm 8795	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) ∧ [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆)))
7	2, 6	mpbid 232	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) ∧ [⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ) → ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
8		opelxp2 5728	. . . . . 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 412	. . . 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 7627	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐵 + 𝐶) ∈ 𝑆)
13		eleq1 2829	. . . . . . 7 ⊢ ((𝐴 + 𝐷) = (𝐵 + 𝐶) → ((𝐴 + 𝐷) ∈ 𝑆 ↔ (𝐵 + 𝐶) ∈ 𝑆))
14	12, 13	imbitrrid 246	. . . . . 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 7619	. . . . . . 7 ⊢ ((𝐴 + 𝐷) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆))
18	17	simprd 495	. . . . . 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 714	. . . 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 480	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
23	21, 22	erth 8796	. . . . . 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 281	. . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
26	25	expr 456	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) → (𝐷 ∈ 𝑆 → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))))
27	10, 20, 26	pm5.21ndd 379	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
28	27	an32s 652	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ 𝐵 ∈ 𝑆) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
29		eqcom 2744	. . . 4 ⊢ (∅ = [⟨𝐶, 𝐷⟩] ∼ ↔ [⟨𝐶, 𝐷⟩] ∼ = ∅)
30		erdm 8755	. . . . . . . . . . . 12 ⊢ ( ∼ Er (𝑆 × 𝑆) → dom ∼ = (𝑆 × 𝑆))
31	3, 30	ax-mp 5	. . . . . . . . . . 11 ⊢ dom ∼ = (𝑆 × 𝑆)
32	31	eleq2i 2833	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ ∈ dom ∼ ↔ ⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆))
33		ecdmn0 8794	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ ∈ dom ∼ ↔ [⟨𝐶, 𝐷⟩] ∼ ≠ ∅)
34		opelxp 5721	. . . . . . . . . 10 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑆 × 𝑆) ↔ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆))
35	32, 33, 34	3bitr3i 301	. . . . . . . . 9 ⊢ ([⟨𝐶, 𝐷⟩] ∼ ≠ ∅ ↔ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆))
36	35	simplbi2 500	. . . . . . . 8 ⊢ (𝐶 ∈ 𝑆 → (𝐷 ∈ 𝑆 → [⟨𝐶, 𝐷⟩] ∼ ≠ ∅))
37	36	ad2antlr 727	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → (𝐷 ∈ 𝑆 → [⟨𝐶, 𝐷⟩] ∼ ≠ ∅))
38	37	necon2bd 2956	. . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ([⟨𝐶, 𝐷⟩] ∼ = ∅ → ¬ 𝐷 ∈ 𝑆))
39		simpr 484	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → 𝐷 ∈ 𝑆)
40	15	ndmov 7617	. . . . . . 7 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → (𝐴 + 𝐷) = ∅)
41	39, 40	nsyl5 159	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝑆 → (𝐴 + 𝐷) = ∅)
42	38, 41	syl6 35	. . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ([⟨𝐶, 𝐷⟩] ∼ = ∅ → (𝐴 + 𝐷) = ∅))
43		eleq1 2829	. . . . . . 7 ⊢ ((𝐴 + 𝐷) = ∅ → ((𝐴 + 𝐷) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
44	16, 43	mtbiri 327	. . . . . 6 ⊢ ((𝐴 + 𝐷) = ∅ → ¬ (𝐴 + 𝐷) ∈ 𝑆)
45	35	simprbi 496	. . . . . . . 8 ⊢ ([⟨𝐶, 𝐷⟩] ∼ ≠ ∅ → 𝐷 ∈ 𝑆)
46	11	caovcl 7627	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆) → (𝐴 + 𝐷) ∈ 𝑆)
47	46	ex 412	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑆 → (𝐷 ∈ 𝑆 → (𝐴 + 𝐷) ∈ 𝑆))
48	47	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → (𝐷 ∈ 𝑆 → (𝐴 + 𝐷) ∈ 𝑆))
49	45, 48	syl5 34	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ([⟨𝐶, 𝐷⟩] ∼ ≠ ∅ → (𝐴 + 𝐷) ∈ 𝑆))
50	49	necon1bd 2958	. . . . . 6 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → (¬ (𝐴 + 𝐷) ∈ 𝑆 → [⟨𝐶, 𝐷⟩] ∼ = ∅))
51	44, 50	syl5 34	. . . . 5 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ((𝐴 + 𝐷) = ∅ → [⟨𝐶, 𝐷⟩] ∼ = ∅))
52	42, 51	impbid 212	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ([⟨𝐶, 𝐷⟩] ∼ = ∅ ↔ (𝐴 + 𝐷) = ∅))
53	29, 52	bitrid 283	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → (∅ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = ∅))
54	31	eleq2i 2833	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom ∼ ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆))
55		ecdmn0 8794	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ dom ∼ ↔ [⟨𝐴, 𝐵⟩] ∼ ≠ ∅)
56		opelxp 5721	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑆 × 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
57	54, 55, 56	3bitr3i 301	. . . . . . 7 ⊢ ([⟨𝐴, 𝐵⟩] ∼ ≠ ∅ ↔ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
58	57	simprbi 496	. . . . . 6 ⊢ ([⟨𝐴, 𝐵⟩] ∼ ≠ ∅ → 𝐵 ∈ 𝑆)
59	58	necon1bi 2969	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝑆 → [⟨𝐴, 𝐵⟩] ∼ = ∅)
60	59	adantl 481	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → [⟨𝐴, 𝐵⟩] ∼ = ∅)
61	60	eqeq1d 2739	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ ∅ = [⟨𝐶, 𝐷⟩] ∼ ))
62		simpl 482	. . . . . 6 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → 𝐵 ∈ 𝑆)
63	15	ndmov 7617	. . . . . 6 ⊢ (¬ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐵 + 𝐶) = ∅)
64	62, 63	nsyl5 159	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝑆 → (𝐵 + 𝐶) = ∅)
65	64	adantl 481	. . . 4 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → (𝐵 + 𝐶) = ∅)
66	65	eqeq2d 2748	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ((𝐴 + 𝐷) = (𝐵 + 𝐶) ↔ (𝐴 + 𝐷) = ∅))
67	53, 61, 66	3bitr4d 311	. 2 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ ¬ 𝐵 ∈ 𝑆) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))
68	28, 67	pm2.61dan 813	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝐶, 𝐷⟩] ∼ ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∅c0 4333  ⟨cop 4632   class class class wbr 5143   × cxp 5683  dom cdm 5685  (class class class)co 7431   Er wer 8742  [cec 8743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569  df-ov 7434  df-er 8745  df-ec 8747

This theorem is referenced by: (None)