Theorem ecopovsym 6905

Description: Assuming the operation 𝐹 is commutative, show that the relation ∼, specified by the first hypothesis, is symmetric. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypotheses

Ref	Expression
ecopopr.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}
ecopopr.com	⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥)

Assertion

Ref	Expression
ecopovsym	⊢ (𝐴 ∼ 𝐵 → 𝐵 ∼ 𝐴)

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢, +   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣,𝑢

Allowed substitution hints:   𝐴(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢)   𝐵(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢)   ∼ (𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢)

Proof of Theorem ecopovsym

Dummy variables 𝑓 𝑔 ℎ 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecopopr.1	. . . . 5 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))}
2		opabssxp 4849	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = ⟨𝑧, 𝑤⟩ ∧ 𝑦 = ⟨𝑣, 𝑢⟩) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
3	1, 2	eqsstri 3280	. . . 4 ⊢ ∼ ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
4	3	brel 4827	. . 3 ⊢ (𝐴 ∼ 𝐵 → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)))
5		eqid 2238	. . . 4 ⊢ (𝑆 × 𝑆) = (𝑆 × 𝑆)
6		breq1 4133	. . . . 5 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ 𝐴 ∼ ⟨ℎ, 𝑡⟩))
7		breq2 4134	. . . . 5 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨ℎ, 𝑡⟩ ∼ ⟨𝑓, 𝑔⟩ ↔ ⟨ℎ, 𝑡⟩ ∼ 𝐴))
8	6, 7	bibi12d 235	. . . 4 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → ((⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑓, 𝑔⟩) ↔ (𝐴 ∼ ⟨ℎ, 𝑡⟩ ↔ ⟨ℎ, 𝑡⟩ ∼ 𝐴)))
9		breq2 4134	. . . . 5 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → (𝐴 ∼ ⟨ℎ, 𝑡⟩ ↔ 𝐴 ∼ 𝐵))
10		breq1 4133	. . . . 5 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → (⟨ℎ, 𝑡⟩ ∼ 𝐴 ↔ 𝐵 ∼ 𝐴))
11	9, 10	bibi12d 235	. . . 4 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → ((𝐴 ∼ ⟨ℎ, 𝑡⟩ ↔ ⟨ℎ, 𝑡⟩ ∼ 𝐴) ↔ (𝐴 ∼ 𝐵 ↔ 𝐵 ∼ 𝐴)))
12	1	ecopoveq 6904	. . . . . 6 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + ℎ)))
13		vex 2824	. . . . . . . . 9 ⊢ 𝑓 ∈ V
14		vex 2824	. . . . . . . . 9 ⊢ 𝑡 ∈ V
15		ecopopr.com	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥)
16	13, 14, 15	caovcom 6247	. . . . . . . 8 ⊢ (𝑓 + 𝑡) = (𝑡 + 𝑓)
17		vex 2824	. . . . . . . . 9 ⊢ 𝑔 ∈ V
18		vex 2824	. . . . . . . . 9 ⊢ ℎ ∈ V
19	17, 18, 15	caovcom 6247	. . . . . . . 8 ⊢ (𝑔 + ℎ) = (ℎ + 𝑔)
20	16, 19	eqeq12i 2252	. . . . . . 7 ⊢ ((𝑓 + 𝑡) = (𝑔 + ℎ) ↔ (𝑡 + 𝑓) = (ℎ + 𝑔))
21		eqcom 2240	. . . . . . 7 ⊢ ((𝑡 + 𝑓) = (ℎ + 𝑔) ↔ (ℎ + 𝑔) = (𝑡 + 𝑓))
22	20, 21	bitri 184	. . . . . 6 ⊢ ((𝑓 + 𝑡) = (𝑔 + ℎ) ↔ (ℎ + 𝑔) = (𝑡 + 𝑓))
23	12, 22	bitrdi 196	. . . . 5 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ (ℎ + 𝑔) = (𝑡 + 𝑓)))
24	1	ecopoveq 6904	. . . . . 6 ⊢ (((ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆)) → (⟨ℎ, 𝑡⟩ ∼ ⟨𝑓, 𝑔⟩ ↔ (ℎ + 𝑔) = (𝑡 + 𝑓)))
25	24	ancoms 268	. . . . 5 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → (⟨ℎ, 𝑡⟩ ∼ ⟨𝑓, 𝑔⟩ ↔ (ℎ + 𝑔) = (𝑡 + 𝑓)))
26	23, 25	bitr4d 191	. . . 4 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑓, 𝑔⟩))
27	5, 8, 11, 26	2optocl 4852	. . 3 ⊢ ((𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)) → (𝐴 ∼ 𝐵 ↔ 𝐵 ∼ 𝐴))
28	4, 27	syl 14	. 2 ⊢ (𝐴 ∼ 𝐵 → (𝐴 ∼ 𝐵 ↔ 𝐵 ∼ 𝐴))
29	28	ibi 176	1 ⊢ (𝐴 ∼ 𝐵 → 𝐵 ∼ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ⟨cop 3712   class class class wbr 4130  {copab 4191   × cxp 4772  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-iota 5337  df-fv 5385  df-ov 6088

This theorem is used by:  ecopover  6907