Theorem ecopovsym 6776

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 4792	. . . . 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 3256	. . . 4 ⊢ ∼ ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
4	3	brel 4770	. . 3 ⊢ (𝐴 ∼ 𝐵 → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)))
5		eqid 2229	. . . 4 ⊢ (𝑆 × 𝑆) = (𝑆 × 𝑆)
6		breq1 4085	. . . . 5 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ 𝐴 ∼ ⟨ℎ, 𝑡⟩))
7		breq2 4086	. . . . 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 4086	. . . . 5 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → (𝐴 ∼ ⟨ℎ, 𝑡⟩ ↔ 𝐴 ∼ 𝐵))
10		breq1 4085	. . . . 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 6775	. . . . . 6 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + ℎ)))
13		vex 2802	. . . . . . . . 9 ⊢ 𝑓 ∈ V
14		vex 2802	. . . . . . . . 9 ⊢ 𝑡 ∈ V
15		ecopopr.com	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥)
16	13, 14, 15	caovcom 6162	. . . . . . . 8 ⊢ (𝑓 + 𝑡) = (𝑡 + 𝑓)
17		vex 2802	. . . . . . . . 9 ⊢ 𝑔 ∈ V
18		vex 2802	. . . . . . . . 9 ⊢ ℎ ∈ V
19	17, 18, 15	caovcom 6162	. . . . . . . 8 ⊢ (𝑔 + ℎ) = (ℎ + 𝑔)
20	16, 19	eqeq12i 2243	. . . . . . 7 ⊢ ((𝑓 + 𝑡) = (𝑔 + ℎ) ↔ (𝑡 + 𝑓) = (ℎ + 𝑔))
21		eqcom 2231	. . . . . . 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 6775	. . . . . 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 4795	. . 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)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ⟨cop 3669   class class class wbr 4082  {copab 4143   × cxp 4716  (class class class)co 6000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-xp 4724  df-iota 5277  df-fv 5325  df-ov 6003

This theorem is referenced by:  ecopover  6778