Theorem ecopovsymg 6612

Description: Assuming the operation 𝐹 is commutative, show that the relation ∼, specified by the first hypothesis, is symmetric. (Contributed by Jim Kingdon, 1-Sep-2019.)

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)))}
ecopoprg.com	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))

Assertion

Ref	Expression
ecopovsymg	⊢ (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 ecopovsymg

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 4685	. . . . 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 3179	. . . 4 ⊢ ∼ ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆))
4	3	brel 4663	. . 3 ⊢ (𝐴 ∼ 𝐵 → (𝐴 ∈ (𝑆 × 𝑆) ∧ 𝐵 ∈ (𝑆 × 𝑆)))
5		eqid 2170	. . . 4 ⊢ (𝑆 × 𝑆) = (𝑆 × 𝑆)
6		breq1 3992	. . . . 5 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ 𝐴 ∼ ⟨ℎ, 𝑡⟩))
7		breq2 3993	. . . . 5 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → (⟨ℎ, 𝑡⟩ ∼ ⟨𝑓, 𝑔⟩ ↔ ⟨ℎ, 𝑡⟩ ∼ 𝐴))
8	6, 7	bibi12d 234	. . . 4 ⊢ (⟨𝑓, 𝑔⟩ = 𝐴 → ((⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑓, 𝑔⟩) ↔ (𝐴 ∼ ⟨ℎ, 𝑡⟩ ↔ ⟨ℎ, 𝑡⟩ ∼ 𝐴)))
9		breq2 3993	. . . . 5 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → (𝐴 ∼ ⟨ℎ, 𝑡⟩ ↔ 𝐴 ∼ 𝐵))
10		breq1 3992	. . . . 5 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → (⟨ℎ, 𝑡⟩ ∼ 𝐴 ↔ 𝐵 ∼ 𝐴))
11	9, 10	bibi12d 234	. . . 4 ⊢ (⟨ℎ, 𝑡⟩ = 𝐵 → ((𝐴 ∼ ⟨ℎ, 𝑡⟩ ↔ ⟨ℎ, 𝑡⟩ ∼ 𝐴) ↔ (𝐴 ∼ 𝐵 ↔ 𝐵 ∼ 𝐴)))
12		ecopoprg.com	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
13	12	adantl 275	. . . . . . . 8 ⊢ ((((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
14		simpll 524	. . . . . . . 8 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → 𝑓 ∈ 𝑆)
15		simprr 527	. . . . . . . 8 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → 𝑡 ∈ 𝑆)
16	13, 14, 15	caovcomd 6009	. . . . . . 7 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → (𝑓 + 𝑡) = (𝑡 + 𝑓))
17		simplr 525	. . . . . . . 8 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → 𝑔 ∈ 𝑆)
18		simprl 526	. . . . . . . 8 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → ℎ ∈ 𝑆)
19	13, 17, 18	caovcomd 6009	. . . . . . 7 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → (𝑔 + ℎ) = (ℎ + 𝑔))
20	16, 19	eqeq12d 2185	. . . . . 6 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → ((𝑓 + 𝑡) = (𝑔 + ℎ) ↔ (𝑡 + 𝑓) = (ℎ + 𝑔)))
21		eqcom 2172	. . . . . 6 ⊢ ((𝑡 + 𝑓) = (ℎ + 𝑔) ↔ (ℎ + 𝑔) = (𝑡 + 𝑓))
22	20, 21	bitrdi 195	. . . . 5 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → ((𝑓 + 𝑡) = (𝑔 + ℎ) ↔ (ℎ + 𝑔) = (𝑡 + 𝑓)))
23	1	ecopoveq 6608	. . . . 5 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ (𝑓 + 𝑡) = (𝑔 + ℎ)))
24	1	ecopoveq 6608	. . . . . 6 ⊢ (((ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆) ∧ (𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆)) → (⟨ℎ, 𝑡⟩ ∼ ⟨𝑓, 𝑔⟩ ↔ (ℎ + 𝑔) = (𝑡 + 𝑓)))
25	24	ancoms 266	. . . . 5 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → (⟨ℎ, 𝑡⟩ ∼ ⟨𝑓, 𝑔⟩ ↔ (ℎ + 𝑔) = (𝑡 + 𝑓)))
26	22, 23, 25	3bitr4d 219	. . . 4 ⊢ (((𝑓 ∈ 𝑆 ∧ 𝑔 ∈ 𝑆) ∧ (ℎ ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) → (⟨𝑓, 𝑔⟩ ∼ ⟨ℎ, 𝑡⟩ ↔ ⟨ℎ, 𝑡⟩ ∼ ⟨𝑓, 𝑔⟩))
27	5, 8, 11, 26	2optocl 4688	. . 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 175	1 ⊢ (𝐴 ∼ 𝐵 → 𝐵 ∼ 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ⟨cop 3586   class class class wbr 3989  {copab 4049   × cxp 4609  (class class class)co 5853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  ecopoverg  6614